diff --git "a/9dA0T4oBgHgl3EQfO_9E/content/tmp_files/load_file.txt" "b/9dA0T4oBgHgl3EQfO_9E/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/9dA0T4oBgHgl3EQfO_9E/content/tmp_files/load_file.txt" @@ -0,0 +1,1601 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf,len=1600 +page_content='On the Approximation Accuracy of Gaussian Variational Inference Anya Katsevich akatsevi@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='edu Philippe Rigollet rigollet@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='edu January 6, 2023 Abstract The main quantities of interest in Bayesian inference are arguably the first two moments of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the past decades, variational inference (VI) has emerged as a tractable approach to approximate these summary statistics, and a viable alternative to the more established paradigm of Markov Chain Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, little is known about the approximation accuracy of VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In this work, we bound the mean and covariance approximation error of Gaussian VI in terms of dimension and sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Our results indicate that Gaussian VI outperforms significantly the classical Gaussian approximation obtained from the ubiquitous Laplace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Our error analysis relies on a Hermite series expansion of the log posterior whose first terms are precisely cancelled out by the first order optimality conditions associated to the Gaussian VI optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 1 Introduction A central challenge in Bayesian inference is to sample from, or compute summary statistics of, a posterior distribution π on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The classical approach to sampling is Markov Chain Monte Carlo (MCMC), in which a Markov chain designed to converge to π is simulated for sufficiently long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, MCMC can be expensive, and it is notoriously difficult to identify clear-cut stopping criteria for the algorithm [CC96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Besides, if one is only interested in summary statistics of π such as the mean and covariance, then generating samples from π may not be the most efficient way to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' An alternative, often computationally cheaper, approach is variational inference (VI) [BKM17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The idea of VI is to find, among all measures in a certain parameterized family P, the closest measure to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' While various measures of proximity have been proposed since the introduction of VI [DD21, DDP21], we employ here KL divergence, which is, by far, the most common choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Typically, statistics of interest, chiefly its first two moments, for measures in the family P are either readily available or else easily computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In this work, we consider the family of normal distributions, which are directly parameterized by their mean and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We define ˆπ = N( ˆm, ˆS) ∈ argmin p∈PGauss KL( p ∥ π), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and take ˆm, ˆS as our estimates of the true mean mπ and covariance Sπ of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' PGauss denotes the family of non-degenerate Gaussian distributions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' A key difference between MCMC and VI is that unbiased MCMC algorithms yield arbitrarily accurate samples from π if they are run for long enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' On the other hand, the output of a perfect VI algorithm is ˆπ, which is itself only an approximation to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, a fundamental question in VI is to understand the quality of the approximation ˆπ ≈ π, particularly in terms of the statistics of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In this work, we bound the mean and covariance estimation errors ∥ ˆm − mπ∥ and ∥ ˆS − Sπ∥ for the Gaussian VI estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Of course, we cannot expect an arbitrary, potentially multimodal π to be well-approximated by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the setting of Bayesian inference, however, the Bernstein-von Mises theorem guarantees that under certain regularity conditions, a posterior distribution converges to a Gaussian density in the limit of large sample size [VdV00, Chapter 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To understand why this is the case, consider a generic posterior 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='02168v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='ST] 5 Jan 2023 π = πn with density of the form πn(θ | x1:n) ∝ ν(θ) n � i=1 pθ(xi) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) Here, ν is the prior, pθ is the model density, and x1:n = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='d observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Provided ν and pθ are everywhere positive, we can write πn as πn(θ) ∝ e−nvn(θ), vn(θ) := − 1 n n � i=1 log pθ(xi) − 1 n log ν(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If n is large and vn has a strict global minimum at θ = m∗, then πn will place most of its mass in a neighborhood of m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, πn is effectively unimodal, and hence a Gaussian approximation is reasonable in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This reasoning drives a second, so-called Laplace approximation to πn, which is a Gaussian centered at m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence, the mode m∗ can also serve as an approximation to the true mean mπn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, as we discuss below, Gaussian VI yields a more accurate estimate of mπn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Our main result quantifies the mean and covariance estimation errors of Gaussian VI for a target measure πn ∝ e−nvn, in terms of sample size n and dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In line with the above reasoning, the key assumption is that vn has a unique global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It is useful at this point to think of vn as being a quantity of order 1, and for the purpose of readability, we write simply vn = v in the rest of this introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It is easy to see that πn ∝ e−nv has variance of order 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To account for this vanishing variance, we rescale the approximation errors appropriately in the statement of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let πn ∝ exp(−nv) have mean and covariance mπn, Sπn respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assume that d3 ≤ n and that v ∈ C4(Rd) has a unique strict minimum at m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If ∥∇3v∥ and ∥∇4v∥ grow at most polynomially away from m∗, and if v grows at least logarithmically away from m∗, then the mean and covariance ˆmn, ˆSn of the variational Gaussian approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) to π satisfy √n∥ ˆmn − mπn∥ ≲ �d3 n �3/2 n∥ ˆSn − Sπn∥ ≲ d3 n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) Here, ≲ means the inequalities hold up to an absolute (d, n-independent) constant, as well as a factor depending on second and third order derivatives of v in a neighborhood of the mode m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This v-dependent factor is made explicit in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The theorem shows that both the mean and covariance VI estimates, and especially the mean estimate ˆmn, are remarkably accurate approximations to the true mean and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' As such, it is a compelling endorsement of Gaussian VI for estimating the posterior mean and covariance in the finite sample regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Although the condition n ≥ d3 is restrictive when d is very large, we believe that it is unavoidable without further assumptions and note that it also appears in existing bounds for the Laplace method [Spo22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' As mentioned above, the Laplace method is a competing Gaussian approximation to πn that is widespread in practice for its computational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We use it as a benchmark to put the above error bounds into context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The Laplace approximation to πn ∝ e−nv is given by πn ≈ N(m∗, (n∇2v(m∗))−1), where m∗ is the global minimizer of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This approximation simply replaces v by its second order Taylor expansion around m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The recent works [Spo22] and [KGB22] derive error bounds for the Laplace approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Spokoiny [Spo22] shows that √n∥m∗ − mπn∥ ≲ (d3/n)1/2 assuming v is strongly convex, and [KGB22] similarly shows that √n∥m∗ − mπn∥ ≲ 1/√n with implicit dependence on d, under weaker assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For the covariance approximation, an explicit error bound is stated only in [KGB22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' the authors show that n∥(n∇2v(m∗))−1 − Sπn∥ ≲ 1/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Meanwhile, Spokoiny states lemmas in the appendix from which one can derive a d-dependent covariance error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In a companion paper [Kat23], we extend the techniques developed in the present work to obtain the following tighter n dependence of the Laplace covariance error: n∥(n∇2v(m∗))−1 − Sπn∥ ≲ 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) 2 This n dependence can also be obtained using the approach in [Spo22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let us summarize the n-dependence of these bounds, incorporating the 1/√n and 1/n scaling of the mean and covariance errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The Gaussian VI mean approximation error is n−1/2 × n−3/2, which is a factor of n−1 more accurate than the Laplace mean error of n−1/2 × n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The covariance approximation error is the same for both methods (using the tighter covariance bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4)): n−1 × n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' VI’s improved mean approximation accuracy is confirmed in our simulations of a simple Bayesian logistic regression example in d = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' see Figure 1, and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Figure 1: Gaussian VI yields a more accurate mean estimate than does Laplace, while the two covariance estimates are on the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Here, πn is the likelihood of logistic regression given n observations in dimension d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For the left-hand plot, the slopes of the best-fit lines are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='04 for the Laplace approximation and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='02 for Gaussian VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For covariance: the slopes of the best-fit lines are -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='09 for Laplace, -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12 for VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We note that the Laplace approximation error bounds in the companion work [Kat23] are also tighter in their dimension dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First-order optimality conditions and Hermite series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The improvement of Gaussian VI over the Laplace method to estimate the mean a posteriori rests on a remarkable interaction between first-order optimality conditions and a Hermite series expansion of the potential v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hereafter, we replace θ by x and let V = nv, π ∝ e−V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The focal point of this work are the first order optimality equations for the minimization (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1): ∇m,SKL( N(m, S) ∥ π) �� (m,S)=( ˆm, ˆS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This is also equivalent to setting the Bures-Wasserstein gradient of KL( p ∥ π) to zero at p = N( ˆm, ˆS) as in [LCB+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Explicitly, we obtain that (m, S) = ( ˆm, ˆS) is a solution to E [∇V (m + S1/2Z)] = 0, E [∇2V (m + S1/2Z)] = S−1, (EV ) where Z ∼ N(0, Id) and S1/2 is the positive definite symmetric square root of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' see [LCB+22] for this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In some sense, the fact that N( ˆm, ˆS) minimizes the KL divergence to π does not explain why ˆm is such an accurate estimate of mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Rather, the true reason has to do with properties of solutions to the fixed point equations (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To see why, consider the function ¯V (x) = V ( ˆm + ˆS1/2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If π ∝ e−V is close to the density of N( ˆm, ˆS), then ¯π ∝ e− ¯V should be close to the density of N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, we should have that ¯V (x) ≈ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + ∥x∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This is ensured by the first order optimality equations (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, note that (EV ) can be written in terms of ¯V as E [∇ ¯V (Z)] = 0, E [∇2 ¯V (Z)] = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) 3 Mean approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' error m - mπ ( 10-2 10-3 m = m* (Laplace) m = mn (Gaussian VI) 10- 10-6 102 103 nCovariance approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' error IlS - Shm S = (n-v(m×))-1 (Laplace) S = Sn (Gaussian VI) 10-3 10- 10-5 102 103 nAs we explain in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4, the equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) set the first and second order coefficients in the Hermite series expansion of ¯V to 0 and Id, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' As a result, ¯V (x) − ∥x∥2/2 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + r3(x), where r3 is a Hermite series containing only third and higher order Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The accuracy of the Gaussian VI mean and covariance estimates stems from the fact that the Hermite remainder r3 is of order r3 ∼ 1/√n, and the fact that r3 is orthogonal to linear and quadratic functions with respect to the Gaussian measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 for a high-level summary of this Hermite series based error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The literature on VI can be roughly divided into statistical and algorithmic works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Works on the statistical side have focused on the contraction of variational posteriors around a ground truth parameter in the large n (sample size) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (We use “variational posterior” as an abbreviation for variational approximation to the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=') For example, [WB19] prove an analogue of the Bernstein-von Mises theorem for the variational posterior, [ZG20] study the contraction rate of the variational posterior around the ground truth in a nonparametric setting, and [AR20] study the contraction rate of variational approximations to tempered posteriors, in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' A key difference between these works and ours is that here, we determine how well the statistics of the variational posterior match those of the posterior itself, rather than those of a limiting (n → ∞) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We are only aware of one other work studying the problem of quantifying posterior approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In [HY19], the authors consider a Bayesian model with “local” latent variables (one per data point) and global latent variables, and they study the mean field variational approximation, given by the product measure closest to the true posterior in terms of KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' They show that the the mean ˆm of their approximation satisfies √n∥ ˆm − mπ∥ ≲ 1/n1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since the algorithmic side of VI is not our focus here, we simply refer the reader to the work [LCB+22] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This work complements our analysis in that it provides rigorous convergence guarantees for an algorithm that solves the optimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In Section 2, we first redefine ( ˆm, ˆS) as a certain “canonical” solution to the first order optimality conditions (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then state our assumptions and main result on the Gaussian VI mean and covariance approximation errors, and present a numerical result confirming the n scaling of our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In Section 3, we give an overview of the proof, and in Section 4 we flesh out the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Section 5 outlines the proof of the existence and uniqueness of the aforementioned “canonical” solution ( ˆm, ˆS) to (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the Appendix, we derive a multivariate Her- mite series remainder formula and then prove a number of supplementary results omitted from the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For two k-tensors T, Q ∈ (Rd)⊗k, we define ⟨T, Q⟩ = d � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',ik=1 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='ikQi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='ik, and let ∥T∥F = ⟨T, T⟩1/2 be the Frobenius norm of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We will more often make use of the operator norm of T, denoted simply by ∥ · ∥: ∥T∥ = sup ∥u1∥≤1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',∥uk∥≤1 ⟨u1 ⊗ · · · ⊗ uk, T⟩, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) where the supremum is over vectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , uk ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For positive scalars a, b, we write a ≲ b to denote that a ≤ Cb for an absolute constant C (the only exception to this notation is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) above, in which ≲ also incorporated a v dependent factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We let mπ = E π[X], Sπ = Covπ(X) = E π[(X − mπ)(X − mπ)T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Finally, for a function V with a unique global minimizer m∗, we let HV denote ∇2V (m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 2 Statement of Main Result Throughout the rest of the paper, we write π ∝ e−nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that v may depend on n in a mild fashion as is often the case for Bayesian posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We also define V = nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 4 In light of the centrality of the fixed point equations (EV ), we begin the section by redefining ( ˆm, ˆS) as solutions to (EV ) rather than minimizers of the KL divergence objective (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' These definitions diverge only in the case that V is not strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, if V is strongly convex then KL( · ∥ π) is strongly geodesically convex in the submanifold of normal distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=', [LCB+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, in this case, there is a unique minimizer ˆπ of the KL divergence, corresponding to a unique solution ( ˆm, ˆS) ∈ Rd × Sd ++ to (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In general, however, if (m, S) solve (EV ) this does not guarantee that m is a good estimator of mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To see this, consider the equations in the following form, recalling that v = V/n: E [∇v(m + S1/2Z)] = 0, S E [∇2v(m + S1/2Z)] = 1 nId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) Let x ̸= m∗ be a critical point of v, that is, ∇v(x) = 0, and consider the pair (m, S) = (x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For this (m, S) we have E [∇v(m + S1/2Z)] = ∇v(x) = 0, S E [∇2v(m + S1/2Z)] = 0 ≈ 1 nId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus (x, 0) is an approximate solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1), and by continuity, we expect that there is an exact solution nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, to each critical point x of v is associated a solution (m, S) ≈ (x, 0) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The solution (m, S) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) which we are interested in, then, is the one near (m∗, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 1 below formalizes this intuition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' we show there is a unique solution (m, S) to (EV ) in the set RV = � (m, S) ∈ Rd×Sd ++ : S ⪯ 2H−1, ∥ √ H √ S∥2 + ∥ √ H(m − m∗)∥2 ≤ 8 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) where H = ∇2V (m∗) = n∇2v(m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that due to the scaling of H with n, the set RV is a small neighborhood of (m∗, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We call this unique solution (m, S) in RV the “canonical” solution of (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We expect the Gaussian distribution corresponding to this canonical solution to be the minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1), although we have not proved this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Regardless of whether it is true, we will redefine ( ˆm, ˆS) to denote the canonical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, whether or not N( ˆm, ˆS) actually minimizes the KL divergence or is only a local minimizer is immaterial for the purpose of estimating mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the rest of this section, we state our assumptions on v, a lemma guaranteeing a canonical solution ˆm, ˆS to (EV ), and our main results bounding the mean and covariance errors of the Laplace and Gaussian VI approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 Assumptions Our main theorem rests on rather mild assumptions on the regularity of the potential v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The function v is at least C3 and has a unique global minimizer x = m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let α2 be a lower bound on λmin(∇2v(m∗)) and β2 be an upper bound on λmax(∇2v(m∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' There exists r > 0 such that N := nr ≥ d3 and √r α2√α2 sup ∥y∥≤1 ���∇3v � m∗ + � r/α2 y ���� ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) Note that the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) is monotonically increasing with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, changing variables to z = � r/α2 y, we see that the supremum is taken over the domain {∥z∥ ≤ � r/α2}, which grows with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, the left-hand side equals zero when r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, this assumption states that we can increase r from 0 up to a large multiple of d3/n, while keeping the left-hand side below 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Define β3 = 1 2α2√α2/√r, so that r = 1 4α3 2/β2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By Assumption V1, we have sup ∥y∥≤1 ∥∇3v(m∗ + � r/α2 y)∥ ≤ β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 5 Hence, we can also think of β3 as an upper bound on ∥∇3v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For future reference, we also define C2,3 := 1/r = 4β2 3 α3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) Assumption V2 (Polynomial growth of ∥∇kv∥, k = 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For some 0 < q ≲ 1 we have √r α2√α2 ���∇3v � m∗ + � r/α2 y ���� ≤ 1 + ∥y∥q, ∀y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) Here, r is from Assumption V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If v is C4, we additionally assume that r α2 2 ���∇4v � m∗ + � r/α2 y ���� ≲ 1 + ∥y∥q, ∀y ∈ Rd with the same q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that Assumption V1 guarantees that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) is satisfied inside the unit ball {∥y∥ ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) simply states that we can extend the constant bound 1/2 to a polynomial bound outside the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Also, note that if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) is satisfied for some q only up to a constant factor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ≲) in the region {∥y∥ ≥ 1} then we can always increase q to ensure the inequality is satisfied exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption V3 (Growth of v away from the minimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let q be as in Assumption V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then v (m∗ + x) ≥ d + 12q + 36 n log( � nβ2∥x∥), ∀∥x∥ ≥ � r/β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) See Section 3 below for further explanation of the intuition behind and consequences of the above as- sumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 Main result We are now ready to state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First, we characterize the Gaussian VI parameters ( ˆmπ, ˆSπ): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let Assumptions V0, V1, V2 be satisfied and assume √nr/d ≥ 40 √ 2( √ 3 + � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ), where r, q are from Assumptions V1, V2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Define H = ∇2V (m∗) = n∇2v(m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then there exists a unique (m, S) = ( ˆmπ, ˆSπ) in the set RV = � (m, S) ∈ Rd × Sd ++ : S ⪯ 2H−1, ∥ √ H √ S∥2 + ∥ √ H(m − m∗)∥2 ≤ 8 � which solves (EV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Moreover, ˆSπ satisfies 2/3 nβ2 Id ⪯ ˆSπ ⪯ 2 nα2 Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) We now state our bounds on the mean and covariance errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For simplicity, we restrict ourselves to the case v ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' See Theorem 1-W for results in the case v ∈ C3 \\ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Theorem 1 (Accuracy of Gaussian VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let Assumption V3 and the assumptions from Lemma 1 be satisfied, and ˆmπ, ˆSπ be as in this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Recall the definition of C2,3 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If v ∈ C4, then ∥ ˆmπ − mπ∥ ≲ 1 √nα2 �C2,3d3 n �3/2 ∥ ˆSπ − Sπ∥ ≲ 1 nα2 C2,3d3 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, we prove that Lemma 1 and Theorem 1 are a consequence of analogous statements for a certain affine invariant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' See that subsection, and Section 3 more generally, for proof overviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 An example: Logistic Regression As noted in the introduction, our results show that Gaussian VI yields very accurate mean and covariance approximations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' in fact, the mean estimate is a full factor of 1/n more accurate than the mean estimate given by the Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Neither our bounds nor those on the Laplace error in [Spo22] and [KGB22] are proven to be tight, but we will now confirm numerically that the bounds give the correct asymptotic scalings with n for a logistic regression example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We also show how to check the assumptions for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In logistic regression, we observe n covariates xi ∈ Rd and corresponding labels yi ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The labels are generated randomly from the covariates and a parameter z ∈ Rd via p(yi | xi, z) = s(xT i z)yi(1 − s(xT i z))1−yi, where s(a) = (1 + e−a)−1 is the sigmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, yi ∼ Bern(s(xT i z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We take the ground truth z to be z = e1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , 0), and we generate the xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' from N(0, λ2Id), so in particular the covariates themselves do not depend on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We take a flat prior, so that the posterior distribution of z is simply the likelihood, π(z) = πn(z | x1:n) ∝ e−nv(z), where v(z) = − 1 n n � i=1 log p(yi | xi, z) = − 1 n n � i=1 � yi log s(xT i z) + (1 − yi) log(1 − s(xT i z)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) Numerical Simulation For the numerical simulation displayed in Figure 1, we take d = 2 and n = 100, 200, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For each n, we draw ten sets of covariates xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , n from N(0, λ2Id) with λ = √ 5, yielding ten posterior distributions πn(· | x1:n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then compute the Laplace and VI mean and covariance approximation errors for each n and each of the ten posteriors at a given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The solid lines in Figure 1 depict the average approximation errors over the ten distributions at each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The shaded regions depict the spread of the middle six out of ten approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' See Appendix D for details about the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the left panel of Figure 1 depicting the mean error, the slopes of the best fit lines are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='04 and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='02 for Laplace and Gaussian VI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For the covariance error in the righthand panel, the slopes of the best fit lines are −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='09 and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12 for Laplace and Gaussian VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This confirms that our bounds, the mean bound of [KGB22] and the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) (also implied by results in [Spo22]) are tight in their n dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Verification of Assumptions It is well known that the likelihood (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) is convex, and has a finite global minimizer z = m∗ (the MLE) provided the data xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , n are not linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption V0 is satisfied in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For simplicity, we verify the remaining assumptions in the case that n is large enough that we can approximate v by the population log likelihood v∞, whose global minimizer is m∗ = e1, the ground truth vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using this approximation, we show in Appendix D that α2 ≳ λ2s′(λ), β2 ≤ λ2 4 , and ∥∇3v∞(z)∥ ≤ β3 := 2λ3, ∀z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) To verify Assumption V1, we need to find r such that sup ∥z−m∗∥≤√ r/α2 ∥∇3v(z)∥ ≤ α3/2 2 2√r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using the uniform bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) on ∥∇3v∥, it suffices to take r = α3 2 4β2 3 , in which case C2,3 = 1 r = 4β2 3 α3 2 ≲ 1 s′(λ)3 ≲ (1 + cosh(λ))3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) 7 using that s′(λ) = s(λ)(1 − s(λ)) = 1 2(1 + cosh(λ))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus Assumption V1 is satisfied as long as n ≥ d3/r, which is true provided n is larger than a constant multiple of (1 + cosh(λ))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next, we can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) and a similar bound on ∥∇4v∥ (which is also bounded uniformly over Rd) to show that Assumption V2 is satisfied with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It remains to check Assumption V3, which we do in Appendix D using the convexity of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, convexity immediately implies at least linear growth away from any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We conclude that the conditions of Theorem 1 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 3 Proof Overview: Affine Invariant Rescaling and Hermite Ex- pansion In this section, we overview the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We start in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 by explaining the affine invariance inherent to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This motivates us to rescale V = nv to obtain a new affine invariant function W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2, we state Assumptions W0-W3 on W, which include the definition of a scale-free parameter N intrinsic to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then state our main results for W: Lemma 1-W and Theorem 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, we deduce Lemma 1 and Theorem 1 for V from the lemma and theorem for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We outline the proof of Theorem 1-W in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The proof of Lemma 1-W is of a different flavor, and is postponed to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 Affine Invariance To prove Theorem 1, we will bound the quantities ∥ ˆS−1/2 π ( ˆmπ − mπ)∥, ∥ ˆS−1/2 π Sπ ˆS−1/2 π − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) As shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, combining the bounds on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) with bounds on ∥ ˆSπ∥ will give the desired estimates in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The reason for considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) rather than directly bounding the quantities in the theorems is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the following lemma, we show that the quantities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) are affine invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We discuss the implications of this fact at the end of the subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First, define Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f be a C2 function with unique global minimizer m∗f, and let Hf = ∇2f(m∗f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then Rf = � (m, S) ∈ Rd×Sd ++ : S ⪯ 2H−1 f , ∥ √ Hf √ S∥2 + ∥ √ Hf(m − m∗f)∥2 ≤ 8 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let V1, V2 ∈ C2(Rd), where V2(x) = V1(Ax + b) for some b ∈ Rd and invertible A ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let πi ∝ e−Vi, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then the pair ( ˆm1, ˆS1) is a unique solution to (EV1) in the set RV1 if and only if the pair ( ˆm2, ˆS2) given by ˆm2 = A−1( ˆm1 − b), ˆS2 = A−1 ˆS1A−T (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) is a unique solution to (EV2) in the set RV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, ∥ ˆS−1/2 2 ( ˆm2 − mπ2)∥ = ∥ ˆS−1/2 1 ( ˆm1 − mπ1)∥, ∥ ˆS−1/2 2 Sπ2 ˆS−1/2 2 − Id∥ = ∥ ˆS−1/2 1 Sπ1 ˆS−1/2 1 − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) See Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 of Appendix C for the proof of the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3), the fact that mπ2 = A−1(mπ1 − b), Sπ2 = A−1Sπ1A−T , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) and the following lemma Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let C, D ∈ Sd ++ be symmetric positive definite matrices and x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then ∥C−1/2x∥ = √ xT C−1x and ∥C−1/2DC−1/2 − Id∥ = sup u̸=0 uT Du uT Cu − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This is a simple linear algebra result, so we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 8 Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 shows that our bounds on the quantities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) should themselves be affine invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' the same bounds should hold if we replace V = nv by any function in the set {V (A·+b) : A ∈ Rd×d invertible, b ∈ Rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This motivates us to identify an affine-invariant large parameter N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It is clear that n itself cannot be the correct parameter N because n is not well-defined: nv = (n/c)(cv) for any c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Another natural candidate for N, which removes this degree of freedom, is N = λmin(∇2V (m∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, λmin(∇2V (m∗)) is not affine-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, replacing V (x) by V (cx), for example, changes λmin by a factor of c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To obtain an affine invariant bound, we will define N in Assumption W1 below as a parameter intrinsic to the function W = W[V ] given by W(x) = V (H−1/2 V x + m∗V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) It is straightforward to show that for any other V2(x) = V (Ax + b), we have W[V2](x) = V2(H−1/2 V2 x + m∗V2) = V (H−1/2 V x + m∗V ) = W[V ](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, any function V2 in the set {V (A · +b) : A ∈ Rd×d invertible, b ∈ Rd} maps to the same, affine invariant W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This function is the “correct” object of study, and any bounds we obtain must follow from properties intrinsic to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 Assumptions and Results for W In this section, we state assumptions on W, one of which identifies an appropriate affine invariant parameter N intrinsic to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This parameter is such that as N increases, the measure ρ ∝ e−W is more closely approximated by a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then state results on the existence and uniqueness of solutions ˆmρ, ˆSρ to the first order optimality equations (EW ), and obtain bounds in terms of d and N on the quality of the VI approximation to the mean and covariance of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let W be at least C3, with unique global minimizer x = 0, and ∇2W(0) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Moreover, assume without loss of generality that W(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next, we identify N as a parameter quantifying the size of ∥∇3W∥ in a certain neighborhood of zero: Assumption W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' There exists N ≥ d3 such that √ N sup ∥x∥≤1 ∥∇3W( √ Nx)∥ ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) This definition ensures that N scales proportionally to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, suppose W1 is the affine invariant function corresponding to the equivalence class containing n1v, and let W1 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) with N = N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then the affine invariant W2 corresponding to the equivalence class containing n2v is given by W2(x) = n2 n1 W1( √n1 √n2 x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' From here it is straightforward to see that W2 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) with N = N2, where N2/N1 = n2/n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To further understand the intuition behind this assumption, consider the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let W satisfy Assumptions W0 and W1 and let C ≤ � N/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then ����W(x) − ∥x∥2 2 ���� ≤ C3 12 d √ d √ N , ∀∥x∥ ≤ C √ d, W(x) ≥ ∥x∥2 4 , ∀∥x∥ ≤ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) The lemma shows that N quantifies how close W is to a quadratic, and therefore how close ρ ∝ e−W is to being Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Taylor expanding W(x) to second order for ∥x∥ ≤ C √ d and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7), we have |W(x) − ∥x∥2/2| ≤ 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' sup ∥x∥≤C √ d ∥x∥3∥∇3W(x)∥ ≤ C3 12 d √ d √ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) 9 The second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) follows from the fact that ∇2W(x) ⪰ 1 2Id for all ∥x∥ ≤ √ N, as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Taylor expanding ∇2W(x) to zeroth order, we get that ∥∇2W(x) − ∇2W(0)∥ ≤ sup ∥x∥≤ √ N ∥x∥∥∇3W(x)∥ ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since ∇2W(0) = Id it follows that ∇2W(x) ⪰ 1 2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption W2 (Polynomial growth of ∥∇kW∥, k = 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' There exists 0 < q ≲ 1 such that √ N ���∇3W �√ Nx ���� ≤ 1 + ∥x∥q ∀x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) If W is C4, then the following bound also holds with the same q: N ���∇4W �√ Nx ���� ≲ 1 + ∥x∥q, ∀x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) The N 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) is also chosen to respect the proportional scaling of N with n: if the affine invariant W1 corresponding to n1v satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) with N = N1, then the affine invariant W2 corresponding to n2v satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) with the same q and N = N2, where N2/N1 = n2/n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that Assumption W1 guarantees that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) is satisfied inside the unit ball;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) simply states that we can extend the constant bound 1/2 to a polynomial bound outside of this ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Also, note that if the inequality is satisfied for some q only up to a constant factor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ≲) in the region {∥x∥ ≥ 1}, then we can always increase q to ensure the inequality is satisfied exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption W2 implies that expectations of the form E [∥∇kW(Y )∥p], k = 3, 4, decay with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, we have Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let p ≥ 0 and Y ∈ Rd be a random variable such that E [∥Y ∥pq] < ∞, where q is from Assumption W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let k = 3 or 4, corresponding to the cases W ∈ C3 or W ∈ C4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then E [∥∇kW(Y )∥p] ≲ N −p( k 2 −1) � 1 + E � ∥Y/ √ d∥pq�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By Assumption W2, E [∥∇kW(Y )∥p] ≲ N −p( k 2 −1)E �� 1 + ∥Y/ √ N∥q�p� ≤ N −p( k 2 −1)E �� 1 + ∥Y/ √ d∥q�p� ≲ N −p( k 2 −1) � 1 + E � ∥Y/ √ d∥pq�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12) In the second line we used that d ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If E [∥Y/ √ d∥pq] is d-independent, as for Gaussian random variables, then the above bound reduces to E [∥∇kW(Y )∥p] ≲ N −p( k 2 −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumption W3 (Separation from Zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Growth at Infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have W(x) ≥ (d + 12q + 36) log ∥x∥, ∀∥x∥ ≥ √ N, where q is from Assumption W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For consistency with the previous assumptions, let us also reformulate this one in terms of W( √ Nx): W( √ Nx) ≥ (d + 12q + 36) log √ N + (d + 12q + 36) log ∥x∥, ∀∥x∥ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13) Recall that inside the unit ball, W( √ Nx) is no less than N∥x∥2/4, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, the value of W( √ Nx) increases up to at least N/4 as x approaches unit norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We can interpret (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13) as saying that outside the unit ball, we must maintain constant separation of order d log N from zero, and W(x) must grow at least logarithmically in ∥x∥ as ∥x∥ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 10 We now state the existence and uniqueness of solutions to (EW ) in the region RW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Take Assumptions W0, W1, and W2 to be true, and assume √ N/d ≥ 40 √ 2( √ 3 + � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ), where q is from Assumption W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then there exists a unique (m, S) = ( ˆmρ, ˆSρ) ∈ RW , RW = {(m, S) ∈ Rd × Sd ++ : S ⪯ 2Id, ∥S∥ + ∥m∥2 ≤ 8}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='14) solving (EW ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The matrix ˆSρ furthermore satisfies 2 3Id ⪯ ˆSρ ⪯ 2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15) See Section 5 for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that RW as defined here is the same as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1, since m∗W = 0 and HW = ∇2W(0) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We will make frequent use of the following inequality, which summarizes the bounds on ˆmρ, ˆSρ guaranteed by the lemma: ∥ ˆmρ∥ ≤ 2 √ 2, 2 3Id ⪯ ˆSρ ⪯ 2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='16) Theorem 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Take Assumptions W0, W1, W2, and W3 to be true, and let ( ˆmρ, ˆSρ) be as in the above lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then ∥ ˆS−1/2 ρ ( ˆmρ − mρ)∥ ≲ � � � d3 N if W ∈ C3, � d3 N �3/2 , if W ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , ∥ ˆS−1/2 ρ Sρ ˆS−1/2 ρ − Id∥ ≲ d3 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 From V to W and back In the following sections, we prove Lemma 1-W and Theorem 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 in the appendix, we show that Assumptions V0-V3 imply Assumptions W0-W3 with N = nr and the same q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' From these results, Lemma 1 and Theorem 1 easily follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let ρ ∝ e−W , where W is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2, the assumptions on V imply the assumptions on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence, we can apply Lemma 1-W to conclude there is a unique ( ˆmρ, ˆSρ) ∈ RW solving (EW ), with 2 3Id ⪯ ˆSρ ⪯ 2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since W is an affine transformation of V , it follows by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 that there exists a unique ( ˆmπ, ˆSπ) ∈ RV solving (EV ), with ˆSπ = H−1/2 V ˆSρH−1/2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) for π can be deduced from the corresponding inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15) for ˆSρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First note that ∥ ˆmπ − mπ∥ ≤ ∥ ˆS1/2 π ∥∥ ˆS−1/2 π ( ˆmπ − mπ)∥ ≲ 1 √nα2 ∥ ˆS−1/2 π ( ˆmπ − mπ)∥, and ∥ ˆSπ − Sπ∥ = ∥ ˆS1/2 π ( ˆS−1/2 π Sπ ˆS−1/2 π − Id) ˆS1/2 π ∥ ≲ 1 nα2 ∥ ˆS−1/2 π Sπ ˆS−1/2 π − Id∥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='17) using the bound on ˆSπ from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next note that by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 (affine invariance) we have ∥ ˆS−1/2 π ( ˆmπ − mπ)∥ = ∥ ˆS−1/2 ρ ( ˆmρ − mρ)∥, ∥ ˆS−1/2 π Sπ ˆS−1/2 π − Id∥ = ∥ ˆS−1/2 ρ Sρ ˆS−1/2 ρ − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='18) Apply Theorem 1-W to conclude, recalling that N = nr and C2,3 = 1/r so that d3/N = d3/nr = C2,3d3/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 Overview of Theorem 1-W proof For brevity let m = ˆmρ, S = ˆSρ, and σ = S1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We continue to denote the mean and covariance of ρ by mρ and Sρ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let ¯W(x) = W(m + σx) and note that the optimality equations (EW ) can be written as E [∇ ¯W(Z)] = 0, E [∇2 ¯W(Z)] = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) The proof of Theorem 1-W is based on several key observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 1) The optimality conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) imply that the Hermite series expansion of ¯W is given by ¯W(x) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + 1 2∥x∥2 + r3(x), where r3(x) = � k≥3 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨ck( ¯W), Hk(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='20) 2) The assumptions on W imply that r3 ∼ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 3) We can represent the quantities of interest from Theorem 1-W as expectations with respect to ¯X ∼ ¯ρ ∝ e− ¯ W : ∥σ−1(mρ − m)∥ = sup ∥u∥=1 E [f1,u( ¯X)], ∥σ−1Sρσ−1 − Id∥ ≤ sup ∥u∥=1 E [f2,u( ¯X)] + ∥σ−1(mρ − m)∥2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21) where f1,u(x) = uT x, f2,u(x) = (uT x)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 4) We have E [f( ¯X)] = E [f(Z)e−r3(Z)] E [e−r3(Z)] = E [f(Z)(1 − r3(Z) + r3(Z)2/2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' )] E [e−r3(Z)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='22) 5) We have E [f(Z)] = 0 and E [f(Z)r3(Z)] = 0 for f = f1,u, f2,u, because the remainder r3 is orthogonal to linear and quadratic f with respect to the Gaussian measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, the leading order term in E [f( ¯X)] is 1 2E [f(Z)r3(Z)2] ∼ N −1 for both f = f1,u and f = f2,u, and hence by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21) the quantities of interest are no larger than N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This is the essence of the proof when W ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now that we have given this overview, let us go into a few more details about the above points, and consider the case W ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 1) We can write W(x) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + 1 2∥x∥2 + r3(x), where r3 is the third order Hermite series remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The Hermite series expansion of ¯W is defined as ¯W(x) = ∞ � k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨ck( ¯W), Hk(x)⟩, ck( ¯W) := E [ ¯W(Z)Hk(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) Here, the ck and Hk(x) are tensors in (Rd)⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Specifically, Hk(x) is the tensor of all order k Hermite polynomials, enumerated as H(α) k , α ∈ [d]k with some entries repeating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For k = 0, 1, 2, the Hermite tensors are given by H0(x) = 1, H1(x) = x, H2(x) = xxT − Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 for further details on Hermite series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Distinct Hermite polynomials are orthogonal to each other with respect to the Gaussian weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In particular, if f is an order k polynomial and ℓ > k then E [f(Z)H(α) ℓ (Z)] = 0, ∀α ∈ [d]ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 12 In general, the Hk are given by Hk(x)e−∥x∥2/2 = (−1)k∇ke−∥x∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='24) This representation of the Hermite polynomials leads to the following, “Gaussian integration by parts” identity for a k-times differentiable function f: E [f(Z)Hk(Z)] = E [∇kf(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='25) This is a generalization of Stein’s identity, E [Zif(Z)] = E [∂xif(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since ¯W is at least three times differentiable, we can use Gaussian integration by parts to write c1, c2 as c1( ¯W) := E [ ¯W(Z)H1(Z)] = E [∇ ¯W(Z)] = 0, c2( ¯W) := E [ ¯W(Z)H2(Z)] = E [∇2 ¯W(Z)] = Id, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='26) where the last equality in each line comes from the optimality conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore the Hermite series expansion of ¯W takes the form ¯W(x) = E [ ¯W(Z)] + ⟨0, H1(x)⟩ + 1 2⟨Id, H2(x)⟩ + r3(x) = E [ ¯W(Z)] + 0 + 1 2(∥x∥2 − d) + r3(x) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + 1 2∥x∥2 + r3(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='27) where r3 is the third order remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 2) The assumptions imply r3 ∼ 1/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, since W is C3 and k ≥ 3, we can apply “partial” Gaussian integration by parts to express ck as ck = E [Hk(Z) ¯W(Z)] = E [Hk−3(Z) ⊗ ∇3 ¯W(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' But by Assumptions W1 and W2 we have that ∥∇3W∥ ∼ 1/ √ N, and hence ∥∇3 ¯W∥ ≤ ∥σ∥3∥∇W∥ ∼ 1/ √ N, since σ ⪯ √ 2Id by Lemma 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore each ck ∼ 1/ √ N for k ≥ 3, so r3 ∼ 1/ √ N as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now suppose W ∈ C4, and write r3 as r3(x) = 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨c3, H3(x)⟩ + r4(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We know ⟨c3, H3(x)⟩ ∼ 1/ √ N, and by an analogous argument as for r3, we can show that each of the coefficients ck, k ≥ 4 has order 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence r4 ∼ 1/N, so that r3 = O(N −1/2) + O(N −1) and r2 3 = O(N −1) + O(N −3/2) + O(N −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We can then show that the order N −1 term in r2 3 is orthogonal to f1,u with respect to the Gaussian weight, and hence E [f1,u(Z)r3(Z)2] is order N −3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This is why the mean error is smaller when W ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We will prove 3) in the next section, and 4) follows directly from the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 5) follows from the definition of r3 as a sum of third and higher order Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This discussion explains how the N −1 and N −3/2 scalings arise in Theorem 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Obtaining the correct scaling with dimension d requires a bit more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The scaling with d of the overall error bound depends, among other things, on the scaling with d of expectations of the form E [rk(Z)p], k = 3, 4 (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 below for further discussion of the bound’s d-dependence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We show that E [rk(Z)p] ∼ E [ � ∥Z∥k�p] ∼ dpk/2 using the following explicit formula for rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This result is known in one dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15 in [Leb72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, we could not find the multidimensional version in the literature, so we have proved it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assume ¯W ∈ Ck for k = 3 or k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let ¯W(x) = �∞ j=0 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨cj( ¯W), Hj(x)⟩ be the Hermite series expansion of ¯W, and define rk(x) = ¯W(x) − k−1 � j=0 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨cj( ¯W), Hj(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='28) Then rk(x) = � 1 0 (1 − t)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' E �� ∇k ¯W ((1 − t)Z + tx) , Hk(x) − Z ⊗ Hk−1(x) �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='29) 13 Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='29) is analogous to the integral form of the remainder of a Taylor series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We state and prove this proposition in greater generality in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Carefully applying Cauchy-Schwarz to the inner product in this formula (using the operator norm rather than the Frobenius norm, which would incur additional dimension dependence), allows us to bound E [|rk(Z)|p] by a product of expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' One ex- pectation involves ∥∇k ¯W∥p ∼ N p(1−k/2), and the other expectation, stemming from the Hk and Z ⊗ Hk−1 on the right-hand side of the inner product, involves a (pk)th degree polynomial in ∥Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This explains the dpk/2 scaling of E [|rk(Z)|p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 4 Proof of Theorem 1-W Let ¯X ∼ ¯ρ ∝ e− ¯ W , where ¯W(x) = W(m + σx), and σ = ˆS1/2 ρ , m = ˆmρ are from Lemma 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Also, let r3(x) = � k≥3 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨ck( ¯W), Hk(x)⟩ be the remainder of the Hermite expansion of ¯W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 (Preliminary Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If v ∈ C3, then we have ∥σ−1(m − mρ)∥ ≲ � E r3(Z)4 + � E r3(Z)6 + � E r3( ¯X)6 sup ∥u∥=1 � E (uT ¯X)2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and ∥σ−1Sρσ−1 − Id∥ ≲∥σ−1(m − mρ)∥2 + � E r3(Z)4 + � E r3(Z)6 + � E r3( ¯X)6 sup ∥u∥=1 � E ((uT ¯X)2 − 1)2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) If v ∈ C4, then ∥σ−1(m − mρ)∥ ≲ � E r3(Z)6 + � E r3( ¯X)6 sup ∥u∥=1 � E (uT ¯X)2 + sup ∥u∥=1 ��� u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)] ��� + � E r4(Z)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' From the discussion in the previous section, we know c3, r3 ∼ N −1/2 and c4, r4 ∼ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, we can easily read off the N-dependence of the overall error bound from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The d-dependence of the terms of the form � E [rk(Z)p] can be computed from our explicit formula for rk, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, simple Laplace-type integral bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 show that E [f( ¯X)] ≲ E [f(Z)], so the d-dependence of the ¯X expectations is the same as that of the Z expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Finally, the d-dependence of ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3 ⊗ H4]⟩ can be estimated using the structure of the Hermite tensors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' in particular, we show at most O(d4) of the d8 entries of E [Z ⊗ H3 ⊗ H4] are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First, we prove point 3) from the above proof overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Recall that f1,u(x) = uT x and f2,u(x) = (uT x)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that we can write ¯X = σ−1(X − m), where X ∼ ρ ∝ e−W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, E ¯X = σ−1(mρ − m) and hence ∥σ−1(mρ − m)∥ = ∥E ¯X∥ = sup ∥u∥=1 E [uT ¯X] = sup ∥u∥=1 E [f1,u( ¯X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next, note that Cov( ¯X) = σ−1Sρσ−1, and hence ∥σ−1Sρσ−1 − Id∥ = ∥Cov( ¯X) − Id∥ ≤ ∥E [ ¯X ¯XT − Id]∥ + ∥E ¯XE ¯XT ∥ ≤ sup ∥u∥=1 E [uT ( ¯X ¯XT − Id)u] + ∥E ¯X∥2 = sup ∥u∥=1 E [(uT ¯X)2 − 1] + ∥E ¯X∥2 = sup ∥u∥=1 E [f2,u( ¯X)] + ∥σ−1(mρ − m)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) 14 Now, recalling that ¯W(x) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + ∥x∥2/2 + r3(x), note that E [f( ¯X)] = E [f(Z)e−r3(Z)] E [e−r3(Z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) Write e−r3(Z) = 1 − r3(Z) + 1 2r3(Z)2 − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='r3(Z)3eξ(Z), where ξ(Z) lies on the interval between 0 and −r3(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The key insight is that for f = f1,u and f = f2,u (at most second order polynomials), f is orthogonal to both 1 and r3, since r3 is a series of Hermite polynomials of order greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, E � f(Z)e−r3(Z)� = E � f(Z) � 1 − r3(Z) + 1 2r3(Z)2 − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='r3(Z)3eξ(Z) �� = E � f(Z) �1 2r3(Z)2 − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='r3(Z)3eξ(Z) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5), we get E[f(Z)] = 1 2 E � f(Z)r3(Z)2� E � e−r3(Z)� − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' E [f(Z)r3(Z)3eξ(Z)] E � e−r3(Z)� =: I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) Using Jensen’s inequality and that E [r3(Z)] = 0, we have E [e−r3(Z)] ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence, |I1| ≲ ��E � f(Z)r3(Z)2��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) To bound I2, note that eξ ≤ 1 + e−r3, since ξ ≤ 0 if r3 ≥ 0 and ξ ≤ −r3 if r3 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence, |I2| ≲ E � |f(Z)| |r3(Z)|3� E � e−r3(Z)� + E � |f(Z)| |r3(Z)|3 e−r3(Z)� E � e−r3(Z)� ≤ E � |f(Z)| |r3(Z)|3� + E � |f(Z)| |r3(Z)|3 e−r3(Z)� E � e−r3(Z)� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) again using that E [e−r3(Z)] ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, using the conversion between Z and ¯X expectations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5), observe that E � |f(Z)| |r3(Z)|3 e−r3(Z)� E � e−r3(Z)� = E ���f( ¯X) �� ��r3( ¯X) ��3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Incorporating this into the above bound on |I2| we get |I2| ≤ E � |f(Z)| |r3(Z)|3� + E ���f( ¯X) �� ��r3( ¯X) ��3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) Applying Cauchy-Schwarz to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) we get |I2| ≤ � E [r3(Z)6] � E f(Z)2 + � E r3( ¯X)6 � E f( ¯X)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Adding this inequality to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8), we get ��E [f( ¯X)] �� ≲ ��E � f(Z)r3(Z)2��� + � E [r3(Z)6] � E f(Z)2 + � E � r3( ¯X)6�� E � f( ¯X)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) Taking f(x) = uT x and f(x) = (uT x)2−1 and applying Cauchy-Schwarz to the first term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If v ∈ C4 and f(x) = uT x, we can refine the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11), specifically the first term E [(uT Z)r3(Z)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Write r3(x) = 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ⟨c3, H3(x)⟩ + r4(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 15 Then r3(x)2 = 1 (3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' )2 � c⊗2 3 , H3(x)⊗2� + 2 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='r4(x) ⟨c3, H3(x)⟩ + r4(x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12) To get the first summand on the right we use the fact that ⟨T, S⟩2 = ⟨T ⊗2, S⊗2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Substituting x = Z in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12), multiplying by the scalar uT Z, and taking the expectation of the result gives E �� uT Z � r3(Z)2� = 1 (3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' )2 � c⊗2 3 , E �� uT Z � H3(Z)⊗2�� + 2 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E � (uT Z)r4(Z) ⟨c3, H3(Z)⟩ � + E [(uT Z)r4(Z)2] = 2 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E � (uT Z)r4(Z) ⟨c3, H3(Z)⟩ � + E �� uT Z � r4(Z)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13) For the term on the right-hand side of the first line of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13), note that we have chosen to move the scalar uT Z onto the second tensor H⊗2 3 in the tensor dot product, and we take the Z expectation only after doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This term drops out in the second line because each entry of (uT Z)H3(Z)⊗2 is a polynomial containing only odd powers of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To see why, see the primer on Hermite polynomials in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next, let g(x) = (uT x) ⟨c3, H3(x)⟩, so that E � (uT Z)r4(Z) ⟨c3, H3(Z)⟩ � = E [g(Z)r4(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since E [g(Z)2] < ∞ and r4 is the tail of a convergent Hermite series, we have E [g(Z)r4(Z)] = E � ∞ � k=4 g(Z) 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ⟨ck, Hk(Z)⟩ � = ∞ � k=4 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E [g(Z) ⟨ck, Hk(Z)⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, g is a fourth order polynomial, and is therefore orthogonal to all Hermite polynomials of order greater than four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' As a result, the above sum simplifies to E [g(Z)r4(Z)] = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E [g(Z) ⟨c4, H4(Z)⟩] = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E [(uT Z) ⟨c3, H3(Z)⟩ ⟨c4, H4(Z)⟩] = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='14) Combining these calculations and applying Cauchy-Schwarz to the term E [(uT Z)r4(Z)2] gives the prelimi- nary bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 Combining the bounds In the following sections, we bound each of the terms appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For convenience, we compile these bounds below, letting τ = d3/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 gives |⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3 ⊗ H4]⟩| ≲ d 7 2 N 3 2 ≤ τ 3 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15) Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 applied with Y = Z gives � E [r3(Z)4] ≲ d3 N = τ � E [r3(Z)6] ≲ �d3 N � 3 2 = τ 3 2 � E [r4(Z)4] ≲ d4 N 2 ≤ τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='16) 16 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 applied with Y = ¯X, together with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2, give � E [r3( ¯X)6] ≲ e √ d3/N �d3 N � 3 2 = e √ττ 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Finally, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 gives sup ∥u∥=1 � E [(uT ¯X)2] ≲ e √ d3/N = e √τ, sup ∥u∥=1 � E [((uT ¯X)2 − 1)2] ≲ e √ d3/N = e √τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='17) Substituting all of these bounds into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) finishes the proof of Theorem 1-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 Hermite-related Bounds In this section we bound ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩ as well as E [rk(Z)p] for k = 3, 4, p = 4, 6 and E [r3( ¯X)6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We take all of the assumptions to be true, either in the W ∈ C3 case or W ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If v ∈ C4 then |⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩| ≲ d7/2N −3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We use Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1, which shows that ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩ = ⟨u ⊗ c3, c4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) Writing c3 = �d i,j,k=1 cijk 3 ei ⊗ ej ⊗ ek and noting that |cijk 3 | ≤ ∥c3∥, we get |⟨u ⊗ c3, c4⟩| ≤ d � i,j,k=1 |cijk 3 | |⟨u ⊗ ei ⊗ ej ⊗ ek, c4⟩| ≤ d3∥c3∥∥c4∥ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='20) As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4, since v ∈ C4 we have ck = ck( ¯W) = E [∇k ¯W(Z)], k = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence ∥c3∥ ≤ E ∥∇3 ¯W(Z)∥ ≤ ∥σ∥3E ∥∇3W(m + σZ)∥ ≲ N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21) To get the last inequality, we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='16) to bound ∥m∥, ∥σ∥ by a constant, and we applied Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 with Y = m + σZ, p = 1, k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that E [∥(m + σZ)/ √ d∥s] ≲ 1 for any s ≥ 0, so the bound in the lemma reduces to N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The lemma applies since E [∥m + σZ∥s] ≲ √ d s for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Analogously, ∥c4∥ ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='22) Substituting the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='22) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='20) and using the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) gives the bound in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We now compute bounds on expectations of the form E [|rk(Z)|p], k = 3, 4, and on E [r3( ¯X)6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using the exact formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='29) for rk, we obtain the following bound: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 (Corollary (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let k = 3 if W ∈ C3 and k = 4 if W ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let Y ∈ Rd be a random variable such that E [∥Y ∥s] < ∞ for all 0 ≤ s ≤ 2pk + 2pq, where q is from Assumption W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then E [|rk(Y )|p] ≲ � dk N k−2 � p 2 �� E ∥Y/ √ d∥2kp + � E ∥Y/ √ d∥2(k−1)p + 1 � × � 1 + � E � ∥Y/ √ d∥2pq�� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) 17 Taking Y = Z, the expectations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) are all bounded by constants, so we immediately obtain E [|rk(Z)|p] ≲ � dk N k−2 � p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The corollary also applies to Y = ¯X, k = 3, p = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This is because, as we show in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 in the next section, E [∥X/ √ d∥s] ≲ exp(2 � d3/N) < ∞ for all s ≤ 36 + 12q = 2pk + 2pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since ¯X = σ−1(X − m), using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='16) we conclude that also E [∥ ¯X/ √ d∥s] ≲ exp(2 � d3/N) < ∞ for all s ≤ 36 + 12q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) gives E [r3( ¯X)6] ≲ exp � 2 � d3/N � �d3 N �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 Bounds on X Moments In this section we bound expectations of the form E [(aT X)p], ∥a∥ ≲ 1, and E [∥X∥p], both of which take the form E [f(X)] = � Rd f(x)e−W (x)dx � Rd e−W (x)dx , 0 ≤ f(x) ≲ ∥x∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='24) To evaluate this integral, we break up the numerator into inner, middle, and outer regions I = � ∥x∥ ≤ 2 √ 2 √ d � , M = � 2 √ 2 √ d ≤ ∥x∥ ≤ √ N � , O = � ∥x∥ ≥ √ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then bound E [f(X)] as E [f(X)] = E � f(X)1I(X) � + E � f(X)1M(X) � + E � f(X)1O(X) � ≲ 1 � I e−W (x)dx �� I f(x)e−W (x)dx + � M ∥x∥pe−W (x)dx + � O ∥x∥pe−W (x)dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='25) The inner region I is chosen so that (1) for x ∈ I, we can approximate e−W (x) by e−∥x∥2/2 and (2) the standard Gaussian density places O(1) mass on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This will allow us to show that � I f(x)e−W (x)dx � I e−W (x)dx ≲ E [f(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The middle region M is chosen so that (1) e−W (x) is bounded by another, greater variance Gaussian density, namely e−∥x∥2/4, and (2) this density places exponentially little mass on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The bound on � M ∥x∥pe−W (x)dx/ � I e−W (x)dx therefore involves a ratio of Gaussian normalization constants that grows exponentially in d, but this growth is neutralized by the exponentially decaying Gaussian tail probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Finally, in O we use Assumption W3 to bound the integral � O ∥x∥pe−W (x) by a number decaying expo- nentially in n times the tail integral of a function ∥x∥−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The following four short lemmas carry out this program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We let τ = d3/N in the statements and proofs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have � ∥x∥≤2 √ 2 √ d e−W (x)dx ≳ e−2√τ√ 2π d, where τ = d3/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 with C = 2 √ 2, we have e−W (x) ≥ e−∥x∥2/2e− 4 √ 2 3 d √ d/ √ N ≥ e−∥x∥2/2e−2√τ, ∥x∥ ≤ 2 √ 2 √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, � ∥x∥≤2 √ 2 √ d e−W (x)dx ≥ e−2√τ � ∥x∥≤2 √ 2 √ d e− 1 2 ∥x∥2dx = e−2√τ√ 2π dP(∥Z∥ ≤ 2 √ 2 √ d) ≳ e−2√τ√ 2π d, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='26) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 18 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then E [f(X)1I(X)] ≲ e2√τE [f(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In particular, E [∥X∥p1I(X)] ≲ e2√τdp/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, E [f(X){X ∈ I}] ≤ � I f(x)e−W (x)dx � I e−W (x)dx ≲ e2√τ √ 2π d � I f(x)e−W (x)dx ≲ e2√τ √ 2π d � I f(x)e− 1 2 ∥x∥2dx ≤ e2√τE [f(Z)], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='27) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have E [∥X∥p1M(X)] ≲ e2√τ for all p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, E [∥X∥p1M(X)] ≤ � M ∥x∥pe−W (x)dx � I e−W (x)dx ≲ e2√τ √ 2π d � M ∥x∥pe−∥x∥2/4dx ≤ e2√τ √ 2π d � ∥x∥≥2 √ 2 √ d ∥x∥pe−∥x∥2/4dx (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='28) We now change variables as x = √ 2y, so that ∥x∥pdx is bounded above by √ 2 d+p∥y∥pdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence E [∥X∥p1M(X)] ≲ √ 2 d+p� e2√τ √ 2π d � ∥y∥≥2 √ d ∥y∥pe− 1 2 ∥y∥2dz � ≲ e2√τ√ 2 d+pE � ∥Z∥p{∥Z∥ ≥ 2 √ d} � ≲ e2√τ �√ 2 d+pdp/2e−d/4� ≲ e2√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='29) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For all p ≤ 12q + 36 we have E [∥X∥p1O(X)] ≲ e2√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 and Assumption W3, we get E [∥X∥p1O(X)] ≤ � ∥x∥≥ √ N ∥x∥pe−W (x)dx � I e−W (x)dx ≲ e2√τ � ∥x∥≥ √ N ∥x∥p−d−12q−36dx ≲ e2√τ � ∞ √ N rp−12q−36−1dr ≲ e2√τ√ N p−12q−36 ≲ e2√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='30) In the third line, we left out the surface area of the (d − 1)-sphere, which is an at most O(1) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The above three lemmas immediately imply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For all p ≤ 12q + 36 we have E [∥X∥p] ≲ dp/2e2√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We also have 19 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let ¯X = σ−1(X − m), where ∥σ−1∥, ∥m∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If ∥u∥ = 1 then E [(uT ¯X)2] ≲ e2√τ, E [((uT ¯X)2 − 1)2] ≲ e2√τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have E [((uT ¯X)2 − 1)2] ≲ E [(uT ¯X)4] + 1, so it suffices to show E [(uT ¯X)k] ≲ e2√τ for k = 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since ¯X = σ−1(X − m), we have E [(uT ¯X)k] ≲ E [(uT σ−1X)k] + ∥σ−1m∥k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By the assumptions on σ and m, the term ∥σ−1m∥k is bounded by a constant, so it remains to show E [(aT X)k] ≲ e2√τ, where a = σ−1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6 and noting that ∥a∥ ≲ 1, we get E [(aT X)k] ≲ E [(aT X)k1I(X)] + E [∥X∥k1M(X)] + E [∥X∥k1O(X)] ≲ e2√τ(E [(aT Z)k] + 1) ≲ e2√τ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='31) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 5 Proof of Lemma 1-W In this section, we use m ∈ Rd, σ ∈ Rd×d to denote generic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Consider the equations (EW ), which we rewrite in the following form: E [∇W(m + σZ)] = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) E [∇2W(m + σZ)] = (σσT )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) Note that these equations are well-defined for all (σ, m) ∈ Rd×d×Rd, although we can only expect uniqueness of solutions in a subset of Sd ++ × Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' indeed, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) only depend on σ through S = σσT , which has multiple solutions σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We now restate Lemma 1-W using the following notation: Br(0, 0) = {(σ, m) ∈ Rd×d × Rd : ∥σ∥2 + ∥m∥2 ≤ r2}, Br = {σ ∈ Rd×d : ∥σ∥ ≤ r}, Sc1,c2 = {σ ∈ Sd + : c1Id ⪯ σ ⪯ c2Id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) In particular, note that S0,r ⊂ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let W satisfy Assumptions W0, W1, W2, and W3, and assume √ N/d ≥ 40 √ 2( √ 3+ � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let r = 2 √ 2, c1 = � 2/3, and c2 = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' There exists a unique pair (σ, m) ∈ Br(0, 0) ∩ S0,r/2 × Rd satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2), and this pair is such that σ ∈ Sc1,c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let us sketch the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f : Rd×d × Rd → Rd be given by f(σ, m) = E [∇W(m + σZ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that f(0, 0) = 0, so by the Implicit Function Theorem, there exists a map m(σ) defined in a neighborhood of σ = 0 such that f(σ, m(σ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, we make this statement quantitative, showing that for r = 2 √ 2 we have the following result: for every σ ∈ Br/2 there is a unique m = m(σ) such that (σ, m) ∈ Br(0, 0) and f(σ, m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since S0,r/2 ⊂ Br/2, we have in particular that any solution (σ, m) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) in the region Br(0, 0) ∩ S0,r/2 × Rd is of the form (σ, m(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus it remains to prove there exists a unique solution σ ∈ S0,r/2 to the equation E [∇2W(m(σ) + σZ)] = (σσT )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To do so, we rewrite this equation as F(σ) = σ, where F(σ) = E [∇2W(m(σ) + σZ)]−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We show in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 that F is well-defined on S0,r/2, a contraction, and satisfies F(S0,r/2) ⊂ Sc1,c2 ⊂ S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus by the Contraction Mapping Theorem, there is a unique σ ∈ S0,r/2 satisfying F(σ) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' But since F maps S0,r/2 to Sc1,c2, the fixed point σ necessarily lies in Sc1,c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using a quantitative statement of the Inverse Function Theorem given in [Lan93], the following lemma determines the size of the neighborhood in which the map m(σ) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 20 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , fd) : Rd×d ×Rd → Rd be C3, where Rd×d is the set of d×d matrices, endowed with the standard matrix operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Suppose f(0, 0) = 0, ∇σf(0, 0) = 0, ∇mf(σ, m) is symmetric for all m, σ, and ∇mf(0, 0) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let r > 0 be such that sup (σ,m)∈Br(0,0) ∥∇f(σ, m) − ∇f(0, m∗)∥op ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) Then for each σ ∈ Rd×d such that ∥σ∥ ≤ r/2 there exists a unique m = m(σ) ∈ Rd such that f(σ, m(σ)) = 0 and (σ, m(σ)) ∈ Br(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, the map σ �→ m(σ) is C2, with 1 2Id ⪯ ∇mf(σ, m) �� m=m(σ) ⪯ 3 2Id, ∥∇σm(σ)∥op ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) See Appendix E for careful definitions of the norms appearing above, as well as the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f : Rd×d × Rd → Rd be given by f(σ, m) = E [∇W(σZ + m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then all the conditions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 are satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' in particular, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) is satisfied with r = 2 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus the conclusions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 hold with this choice of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let r = 2 √ 2 and σ ∈ S0,r/2 �→ m(σ) ∈ Rd be the restriction of the map furnished by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 to symmetric nonnegative matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then the function F given by F(σ) = E [∇2W(m(σ) + σZ)]−1/2 is well-defined and a strict contraction on S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Moreover, F(S0,r/2) ⊆ Sc1,c2 ⊆ S0,r/2, where c1 = � 2/3, c2 = √ 2 = r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This lemma concludes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 since by the Contraction Mapping Theorem there is a unique fixed point σ ∈ S0,r/2 of F, and F(σ) = σ is simply a reformulation of the second optimality equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We know σ must lie in Sc1,c2 since F maps S0,r/2 to this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' See Appendix E for the proofs of the above lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Acknowledgments A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Katsevich is supported by NSF grant DMS-2202963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Rigollet is supported by NSF grants IIS-1838071, DMS-2022448, and CCF-2106377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' A Hermite Series Remainder A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 Brief Primer Here is a brief primer on Hermite polynomials, polynomials, and series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We let Hk : R → R, k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' be the kth order probabilist’s Hermite polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have H0(x) = 1, H1(x) = x, H2(x) = x2 − 1, H3(x) = x3 − 3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For all k ≥ 1, we can generate Hk+1 from the recurrence relation Hk+1(x) = xHk(x) − kHk−1(x), k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) In particular, Hk(x) is an order k polynomial given by a sum of monomials of the same parity as k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The Hk are orthogonal with respect to the Gaussian measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' namely, we have E [Hk(Z)Hj(Z)] = k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We also note for future reference that E [ZHk(Z)Hk+1(Z)] = E [(Hk+1(Z) + kHk−1(Z)) Hk+1(Z)] = (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=', (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) using the recurrence relation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 21 The Hermite polynomials are given by products of Hermite polynomials, and are indexed by γ ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , γd), with γj ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then Hγ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , xd) = d � j=1 Hγj(xj), which has order |γ| := �d j=1 γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that if |γ| = k then Hγ(x) is given by a sum of monomials of the same parity as k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, each Hγj(xj) is a linear combination of xγj−2p j , p ≤ ⌊γj/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus Hγ(x) is a linear combination of monomials of the form �d j=1 xγj−2pj j , which has total order k − 2 � j pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using the independence of the entries of Z = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , Zd), we have E [Hγ(Z)Hγ′(Z)] = γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' d � j=1 δγj,γ′ j, where γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' := �d j=1 γj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='. The Hγ can also be defined explicitly as follows: e−∥x∥2/2Hγ(x) = (−1)|γ|∂γ � e−∥x∥2/2� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) where ∂γf(x) = ∂γ1 x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ∂γd xdf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This leads to the useful Gaussian integration by parts identity, E [f(Z)Hγ(Z)] = E [∂γf(Z)], if f ∈ C|γ|(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The Hermite polynomials Hγ, γ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }d, form a complete orthogonal basis of the Hilbert space of functions f : Rd → R with inner product ⟨f, g⟩ = E [f(Z)g(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In particular, if f : Rd → R satisfies E [f(Z)2] < ∞, then f has the following Hermite expansion: f(x) = � γ∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }d 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='cγ(f)Hγ(x), cγ(f) := E [f(Z)Hγ(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) Let rk(x) = f(x) − � |γ|≤k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='cγ(f)Hγ(x) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) be the remainder of the Hermite series expansion of f after taking out the order ≤ k − 1 polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We can write rk as an integral of f against a kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Namely, define K(x, y) = � |γ|≤k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='Hγ(x)Hγ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) Note that E [f(Z)K(x, Z)] = � |γ|≤k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='cγ(f)Hγ(x) is the truncated Hermite series expansion of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, the remainder rk can be written as rk(x) = f(x) − E [f(Z)K(x, Z)] = E [(f(x) − f(Z))K(x, Z)], using that E [K(x, Z)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' B Exact Expression for the Remainder Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let k ≥ 1 and rk, K, be as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assume that f ∈ C1, and that ∥∇f(x)∥ ≲ ec∥x∥2 for some 0 ≤ c < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then rk(x) = E [(f(x) − f(Z))K(x, Z)] = � 1 0 d � i=1 � |γ|=k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E [∂if((1 − t)Z + tx) (Hγ+ei(x)Hγ(Z) − Hγ+ei(Z)Hγ(x))]dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) 22 The proof relies on the following identity: Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , d, it holds that K(x, y) = 1 xi − yi � |γ|=k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (Hγ+ei(x)Hγ(y) − Hγ+ei(y)Hγ(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) The proof of this identity is given at the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Write f(x) − f(Z) = � 1 0 (x − Z)T ∇f((1 − t)Z + tx)dt = d � i=1 � 1 0 (xi − Zi)∂if((1 − t)Z + tx)dt, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) so that, using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2), we have E [(f(x) − f(Z))K(x, Z)] = d � i=1 � |γ|=k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E �� 1 0 ∂if((1 − t)Z + tx) (Hγ+ei(x)Hγ(Z) − Hγ+ei(Z)Hγ(x)) dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) By assumption, sup t∈[0,1] |∂if((1 − t)Z + tx)| (|Hγ(Z)| + |Hγ+ei(Z)|) ≲ exp � c∥Z∥2 + 2c∥Z∥∥x∥ � (|Hγ(Z)| + |Hγ+ei(Z)|) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) for some 0 ≤ c < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The right-hand side is integrable with respect to the Gaussian measure, and therefore we can interchange the expectation and the integral in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, E [(f(x) − f(Z))K(x, Z)] = � 1 0 d � i=1 � |γ|=k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E [∂if((1 − t)Z + tx) (Hγ+ei(x)Hγ(Z) − Hγ+ei(Z)Hγ(x))]dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Without loss of generality, assume i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To simplify the proof, we will also assume d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The reader can check that the proof goes through in the same way for general d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By the recursion relation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) for 1d Hermite polynomials, we get that Hγ1+1,γ2(x) = x1Hγ1,γ2(x) − γ1Hγ1−1,γ2(x), where x = (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Multiply this equation by Hγ(y) (where y = (y1, y2)) and swap x and y, to get the two equations Hγ1+1,γ2(x)Hγ1,γ2(y) = x1Hγ1,γ2(x)Hγ1,γ2(y) − γ1Hγ1−1,γ2(x)Hγ1,γ2(y), Hγ1+1,γ2(y)Hγ1,γ2(x) = y1Hγ1,γ2(x)Hγ1,γ2(y) − γ1Hγ1−1,γ2(y)Hγ1,γ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) Let Sγ1,γ2 = Hγ1+1,γ2(x)Hγ1,γ2(y) − Hγ1+1,γ2(y)Hγ1,γ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Subtracting the second equation of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) from the first one, and using the Sγ1,γ2 notation, gives Sγ1,γ2 = (x1 − y1)Hγ1,γ2(x)Hγ1,γ2(y) + γ1Sγ1−1,γ2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) 23 and hence Sγ1,γ2 γ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (x1 − y1)Hγ1,γ2(x)Hγ1,γ2(y) γ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2 + Sγ1−1,γ2 (γ1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='. Iterating this recursive relationship γ1 − 1 times, we get Sγ1,γ2 γ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (x1 − y1) γ1−1 � j=0 Hγ1−j,γ2(x)Hγ1−j,γ2(y) (γ1 − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' + S0,γ2 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) Now, we have S0,γ2 = H1,γ2(x)H0,γ2(y) − H1,γ2(y)H0,γ2(x) = H1(x1)Hγ2(x2)Hγ2(y2) − H1(y1)Hγ2(y2)Hγ2(x2) = (x1 − y1)Hγ2(x2)Hγ2(y2) = (x1 − y1)H0,γ2(x)H0,γ2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) Therefore, S0,γ2 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (x1 − y1)Hγ1−j,γ2(x)Hγ1−j,γ2(y) (γ1 − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , j = γ1 so (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) can be written as Sγ1,γ2 γ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (x1 − y1) γ1 � j=0 Hγ1−j,γ2(x)Hγ1−j,γ2(y) (γ1 − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' and hence 1 x1 − y1 � γ1+γ2=k−1 Sγ1,γ2 γ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = � γ1+γ2=k−1 γ1 � j=0 Hγ1−j,γ2(x)Hγ1−j,γ2(y) (γ1 − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = � γ1+γ2≤k−1 Hγ1,γ2(x)Hγ1,γ2(y) γ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='γ2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = K(x, y), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) using the observation that {(γ1 − j, γ2) : γ1 + γ2 = k − 1, 0 ≤ j ≤ γ1} = {(˜γ1, γ2) : ˜γ1 + γ2 ≤ k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12) Substituting back in the definition of Sγ1,γ2 gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 Hermite Series Remainder in Tensor Form Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1), it is difficult to obtain an upper bound on |rk(x)|, since we need to sum over all γ of order k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In this section, we obtain a more compact representation of rk in terms of a scalar product of k-tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then take advantage of a very useful representation of the tensor of order-k Hermite polynomials, as an expectation of a vector outer product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' This allows us to bound the scalar product in the rk formula in terms of an operator norm rather than a Frobenius norm (the latter would incur larger d dependence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First, let us put all the unique kth order Hermite polynomials into a tensor of dk entries, some of which are repeating, enumerated by multi-indices α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , αk) ∈ [d]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Here, [d] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We do so as follows: given α ∈ [d]k, define γ(α) = (γ1(α), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , γd(α)) by γj(α) = k � ℓ=1 1{αℓ = j}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' γj(α) counts how many times index j appears in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For this reason, we use the term counting index to denote indices of the form γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , γd) ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }d, whereas we use the standard term “multi-index” 24 to refer to the α’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that we automatically have |γ(α)| = k if α ∈ [d]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, for x ∈ Rd, define H0(x) = 1 and Hk(x), k ≥ 1 as the tensor Hk(x) = {Hγ(α)(x)}α∈[d]k, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' When enumerating the entries of Hk, we write H(α) k to denote Hγ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that for each γ with |γ| = k, there are �k γ � α’s such that γ(α) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Consider the α = (i, j, j, k, k, k) entry of the tensor H6(x), where i, j, k ∈ [d] are all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We count that i occurs once, j occurs twice, and k occurs thrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus H(i,j,j,k,k,k) 6 (x) = H1(xi)H2(xj)H3(xk) = xi(x2 j − 1)(x3 k − 3xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The first two tensors H1, H2 can be written down explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For the entries of H1, we simply have H(i) 1 (x) = H1(xi) = xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' H1(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For the entries of H2, we have H(i,i) 2 (x) = H2(xi) = x2 i − 1 and H(i,j) 2 (x) = H1(xi)H1(xj) = xixj, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus H2(x) = xxT − Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We now group the terms in the Hermite series expansion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) based on the order |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Consider all γ in the sum such that |γ| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We claim that � |γ|=k 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='cγ(f)Hγ(x) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' � α∈[d]k cγ(α)(f)H(α) k (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13) Indeed, for a fixed γ such that |γ| = k, there are �k γ � α’s in [d]k for which γ(α) = γ, and the summands in the right-hand sum corresponding to these α’s are all identical, equalling cγ(f)Hγ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus we obtain �k γ � copies of cγ(f)Hγ(x), and it remains to note that �k γ � /k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = 1/γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='. Analogously to Hk(x), define the tensor ck ∈ (Rd)⊗k, whose α’th entry is c(α) k = cγ(α) = E [f(Z)Hγ(α)(Z)] = E [f(Z)H(α) k (Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then see that the sum (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13) can be written as 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨ck, Hk(x)⟩, and hence the series expansion of f can be written as f(x) = ∞ � k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨ck(f), Hk(x)⟩, ck(f) := E [f(Z)Hk(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='14) The main result of this section is Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 below, in which we express rk in terms of a tensor scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, let us first prove the following lemma, which is needed to bound the term ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3 ⊗ H4]⟩ in the preliminary bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let cp, cp+1 be symmetric tensors in (Rd)p and (Rd)p+1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='Then ⟨u ⊗ cp ⊗ cp+1, E [Z ⊗ Hp(Z) ⊗ Hp+1(Z)]⟩ = (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='⟨u ⊗ cp, cp+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let T = E [Z ⊗ Hp ⊗ Hp+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First ,we characterize the non-zero entries of T using the counting index notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In counting index notation, a typical entry of T takes the form E [ZiHγ(Z)Hγ′(Z)], where i ∈ [d], |γ| = p, and |γ′| = p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, E [ZiHγ(Z)Hγ′(Z)] = E [ZiHγi(Zi)Hγ′ i(Zi)] � j̸=i E [Hγj(Zj)Hγ′ j(Zj)] = E [ZiHγi(Zi)Hγ′ i(Zi)] � j̸=i δγj,γj′ γj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='16) For this to be nonzero, we must have γj = γj′ for all j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' But since |γ| = p and |γ′| = p + 1, it follows that we must have γ′ i = γi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='Hence γ′ = γ + ei, where ei is the ith unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To summarize, Ti,γ,γ′ is only 25 nonzero if γ′ = γ + ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In this case, we have E [ZiHγ(Z)Hγ+ei(Z)] = E [ZiHγi(Zi)Hγi+1(Zi)] � j̸=i γj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (γi + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' � j̸=i γj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (γ + ei)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='. (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='17) To get the second line we used the following recurrence relation for 1-d Hermite polynomials: xHk(x) = Hk+1(x) + kHk−1(x) for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, we take the inner product (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15) using counting index notation, recalling that each γ such that |γ| = p shows up in the tensor Hp exactly p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='/γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' times: � u ⊗ cp ⊗ cp+1, E [Z ⊗ Hp(Z) ⊗ Hp+1(Z)] � = d � i=1 � |γ|=p � |γ′|=p+1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (γ′)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' uicγcγ′E [ZiHγ(Z)Hγ′(Z)] = d � i=1 � |γ|=p p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (γ + ei)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='uicγcγ+ei(γ + ei)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' = (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' d � i=1 � |γ|=p p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='uicγcγ+ei = (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' d � i=1 d � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',jp=1 uicγ(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',jp)cγ(i,j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',jp) = (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' d � i=1 d � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',jp=1 uic(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',jp) p c(i,j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=',jp) p+1 = ⟨u ⊗ cp, cp+1⟩ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='18) Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f satisfy the assumptions of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1, and additionally, assume f ∈ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then the remainder rk, given as (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1, can also be written in the form rk(x) = � 1 0 (1 − t)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' E �� ∇kf ((1 − t)Z + tx) , Hk(x) − Z ⊗ Hk−1(x) �� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Recall that ∂γ := ∂γ1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ∂γd d , and that Hγ(z)e−∥z∥2/2 = (−1)|γ|∂γ(e−∥z∥2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then have for |γ| = k − 1, E [∂if((1 − t)Z + tx)Hγ(Z)] = (1 − t)k−1E [∂γ+eif], E [∂if((1 − t)Z + tx)Hγ+ei(Z)] = (1 − t)k−1E [∂γ+eifZi], (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='20) using the fact that f ∈ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We omitted the argument (1 − t)Z + tx from the right-hand side for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To get the second equation, we moved only γ of the γ + ei derivatives from e−∥z∥2/2 onto ∂if, leaving −∂i(e−∥z∥2/2) = zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Substituting these two equations into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1), we get E [(f(x) − f(Z))K(x, Z)] = � 1 0 (1 − t)k−1 d � i=1 � |γ|=k−1 1 γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='E [(∂γ+eif) (Hγ+ei(x) − ZiHγ(x))]dt = 1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' � 1 0 (1 − t)k−1 d � i=1 � |γ|=k−1 �k − 1 γ � E [(∂γ+eif) (Hγ+ei(x) − ZiHγ(x))]dt (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21) 26 Now, define the sets A = {(i, γ + ei) : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , d, γ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }d, |γ| = k − 1}, B = {(i, ˜γ) : ˜γ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' }d, |˜γ| = k, ˜γi ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='22) It is straightforward to see that A = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, d � i=1 � |γ|=k−1 �k − 1 γ � E [∂γ+eif]Hγ+ei(x) = � |˜γ|=k � i: ˜γi≥1 � k − 1 ˜γ − ei � E [∂˜γf]H˜γ(x) = � |˜γ|=k � i: ˜γi≥1 �k ˜γ � ˜γi k E [∂˜γf]H˜γ(x) = � |˜γ|=k �k ˜γ � E [∂˜γf]H˜γ(x) = ⟨E [∇kf], Hk(x)⟩ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) Next, note that � |γ|=k−1 �k − 1 γ � ∂γ∂ifHγ(x) = ⟨∇k−1∂if, Hk−1(x)⟩, and therefore d � i=1 � |γ|=k−1 �k − 1 γ � E [(∂γ+eif)Zi]Hγ(x) = E [⟨∇kf, Z ⊗ Hk−1(x)⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='24) Substituting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='24) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21) gives rk(x) = E [(f(x) − f(Z))K(x, Z)] = � 1 0 (1 − t)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' E �� ∇kf ((1 − t)Z + tx) , Hk(x) − Z ⊗ Hk−1(x) �� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='25) In the next section, we obtain a pointwise upper bound on |rk(x)| in the case f = ¯W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In order for this bound to be tight in its dependence on d, we need a supplementary result on inner products with Hermite tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To motivate this supplementary result, consider bounding the inner product in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) by the product of the Frobenius norms of the tensors on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' As a rough heuristic, ∥∇kf∥F ∼ dk/2∥∇kf∥, where recall that ∥∇kf∥ is the operator norm of ∇kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, we would prefer to bound the inner product in terms of ∥∇kf∥ to get a tighter dependence on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Apriori, however, this seems impossible, since Hk(x) is not given by an outer product of k vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' But the following representation of the order k Hermite polynomials will make this possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hk(x) = E [(x + iZ)⊗k], (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='26) where Z ∼ N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='26), we can bound scalar products of the form ⟨∇kf, Hk(x)⟩ and ⟨∇kf, Z ⊗ Hk−1(x)⟩ in terms of the operator norm of ∇kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' More generally, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let T ∈ (Rd)⊗k be a k-tensor, and v ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then for all 0 ≤ ℓ ≤ k, we have |⟨T, v⊗ℓ ⊗ Hk−ℓ(x)⟩| ≲ ∥T∥∥v∥ℓ(∥x∥k−ℓ + d k−ℓ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='26), we have ⟨T, v⊗ℓ ⊗ Hk−ℓ(x)⟩ = E [⟨T, v⊗ℓ ⊗ (x + iZ)⊗(k−ℓ)⟩] 27 and hence |⟨T, v⊗ℓ ⊗ Hk−ℓ(x)⟩| ≤ E |⟨T, v⊗ℓ ⊗ (x + iZ)⊗k−ℓ⟩| ≤ ∥T∥∥v∥ℓE � ∥x + iZ∥k−ℓ� ≲ ∥T∥∥v∥ℓ(∥x∥k−ℓ + √ d k−ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='27) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 Hermite-Related Proofs from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 In this section, we return to the setting in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We let W satisfy all the assumptions from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2, m ∈ Rd, σ ∈ Rd×d be such that ∥m∥, ∥σ∥ ≲ 1, and ¯W(x) = W(m + σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Also, let rk(x) = ¯W(x) − k−1 � j=0 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' � cj( ¯W), Hj(x) � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='28) where cj( ¯W) = E [ ¯W(Z)Hj(Z)] as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Combining Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5 allows us to upper bound quantities of the form E [|rk(Y )|p] in terms of the operator norm of ∇kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let rk be as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='28), the remainder of the Hermite series expansion of ¯W, where k = 3 if W ∈ C3 and k = 4 if W ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let Y ∈ Rd be a random variable such that E [∥Y ∥s] < ∞ for all 0 ≤ s ≤ 2pk + 2pq, where q is from Assumption W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then E [|rk(Y )|p] ≲ � dk N k−2 � p 2 �� E ∥Y/ √ d∥2kp + � E ∥Y/ √ d∥2(k−1)p + 1 � × � 1 + � E � ∥Y/ √ d∥2pq�� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let ∇k ¯W be shorthand for ∇k ¯W((1 − t)Z + tY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) for f = ¯W, we have |rk(Y )| ≲ � 1 0 E Z ���� ∇k ¯W, Hk(Y ) − Z ⊗ Hk−1(Y ) ���� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='30) Raising this inequality to the pth power and applying Jensen’s inequality twice, we have |rk(Y )|p ≲ � 1 0 E Z ���� ∇k ¯W, Hk(Y ) − Z ⊗ Hk−1(Y ) ���p� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='31) We now take the Y -expectation of both sides, and we are free to assume Y is independent of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that the integrand on the right-hand side can be bounded by a∥Y ∥p(q+k) + b for some a and b, since ∥∇k ¯W((1 − t)Z + tY )∥ ≲ (1 + ∥Z∥ + ∥Y ∥)q by Assumption W2, and since the tensors Hk(Y ), Hk−1(Y ) are made up of at most order k polynomials of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since E [∥Y ∥pq+pk] < ∞ by assumption, we can bring the Y -expectation inside the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence E [|rk(Y )|p] ≲ � 1 0 E ���� ∇k ¯W, Hk(Y ) − Z ⊗ Hk−1(Y ) ���p� dt, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='32) where the expectation is over both Z and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next, using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5 we have ����⟨∇k ¯W, Hk(Y )−Z ⊗ Hk−1(Y )⟩ ���� p ≲ ∥∇k ¯W∥p � ∥Y ∥kp + d kp 2 + ∥Z∥p∥Y ∥(k−1)p + ∥Z∥pd (k−1)p 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='33) 28 Substituting this into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='32) we have E [|rk(Y )|p] ≲ � 1 0 E ���∇k ¯W ��p � ∥Y ∥kp + d kp 2 + ∥Z∥p∥Y ∥(k−1)p + ∥Z∥pd (k−1)p 2 �� dt ≲ � 1 0 E � ∥∇k ¯W∥2p� 1 2 �� E ∥Y ∥2kp + d p 2 � E ∥Y ∥2(k−1)p + d kp 2 � dt ≤ d kp 2 �� E ∥Y/ √ d∥2kp + � E ∥Y/ √ d∥2(k−1)p + 1 � × � 1 0 E ���∇k ¯W ��2p� 1 2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='34) We used Cauchy-Schwarz and the independence of Y and Z to get the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Finally, recall that ∇k ¯W = ∇k ¯W((1 − t)Z + tY ) and note that since ∥σ∥ ≲ 1, we have ∥∇k ¯W((1 − t)Z + tY )∥ ≲ ∥∇kW(m + (1 − t)σZ + tσY )∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4 with Y = m + (1 − t)σZ + tσY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that E ����m + (1 − t)σZ + tσY ��/ √ d �2pq� ≲ 1 + E � ∥Y/ √ d∥2pq� and hence E ���∇k ¯W ��2p� 1 2 ≲ E ���∇kW(m + (1 − t)σZ + tσY ) ��2p� 1 2 ≲ � 1 + � E � ∥Y/ √ d∥2pq�� N p(1−k/2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='35) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Combining this inequality with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='34) and noting that dkp/2N p(1−k/2) = (dk/N k−2)p/2 gives (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' C Proofs Related to Affine Invariance Recall the equations E [∇V (m + S1/2Z)] = 0, E [∇2V (m + S1/2Z)] = S−1 (EV ) and the definition of RV for a measure π ∝ e−V : RV = � (m, S) ∈ Rd×Sd ++ : S ⪯ 2H−1, ∥ √ H √ S∥2 + ∥ √ H(m − m∗)∥2 ≤ 8 � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) where m∗ = argminx∈Rd V (x) and H = ∇2V (m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let V2(x) = V1(Ax + b) for some A ∈ Rd×d invertible and b ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then the pair (m1, S1) is a unique solution to (EV1) in the set RV1 if and only if the pair (m2, S2) given by m2 = A−1(m1 − b), S2 = A−1S1A−T (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) is a unique solution to (EV2) in the set RV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It suffices to prove the following two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (1) If (m1, S1) ∈ RV1 solves (EV1) then (m2, S2) given by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) lies in RV2 and solves (EV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (2) If (m2, S2) ∈ RV2 solves (EV2) then (m1, S1) given by m1 = Am2 + b, S1 = AS2AT lies in RV1 and solves (EV1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 29 We prove the first statement, and the second follows by a symmetric argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' So let (m1, S1) ∈ RV1 solve (EV1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We first show (m2, S2) given by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) solves (EV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have ∇V2(x) = AT ∇V1(Ax + b), ∇2V2(x) = AT ∇2V1(Ax + b)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) Note also that if σ = A−1S1/2 1 then σσT = A−1S1A−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We therefore have E � ∇V2 � A−1(m1 − b) + � A−1S1A−T �1/2 Z � � = E � ∇V2 � A−1(m1 − b) + A−1S1/2 1 Z �� = AT E � ∇V1 � m1 + S1/2 1 Z �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) Similarly, E � ∇2V2 � A−1(m1 − b) + � A−1S1A−T �1/2 Z � � = E � ∇2V2 � A−1(m1 − b) + A−1S1/2 1 Z �� = AT E � ∇2V1 � m1 + S1/2 1 Z �� A = AT S−1 1 A = S−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) To conclude, we show (m2, S2) ∈ RV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let m∗i be the global minimizer of Vi and Hi = ∇2V (m∗i), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then m∗2 = A−1(m∗1 − b) and H2 = AT H1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since S1 ⪯ 2H−1 1 , it follows that S2 = A−1S1A−T ⪯ 2A−1H−1 1 A−T = 2H−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, direct substitution shows that ∥ √ H2(m2 − m∗2)∥2 = (m2 − m∗2)T H2(m2 − m∗2) = (m1 − m∗1)T H1(m1 − m∗1) = ∥ √ H1(m1 − m∗1)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) Finally, note that ∥ � H2 � S2∥2 = ∥ � S2H2 � S2∥ = sup u̸=0 uT √S2H2 √S2u ∥u∥2 = sup u̸=0 uT H2u ∥√S2 −1u∥2 = sup u̸=0 uT H2u uT S−1 2 u = sup u̸=0 uT H1u uT S−1 1 u = ∥ � H1 � S1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) Therefore, ∥ � H2 � S2∥2 + ∥ √ H2(m2 − m∗2)∥2 = ∥ � H1 � S1∥2 + ∥ √ H1(m1 − m∗1)∥2 ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Recall that W(x) = nv � m∗ + √ nH −1x � , H = ∇2v(m∗), 30 and that N = nr, where r is from Assumption V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The following preliminary calculation will be useful for showing Assumptions V1, V2 imply Assumptions W1, W2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Given x ∈ Rd, let y = √α2 √ H −1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have √ N∥∇3W( √ Nx)∥ ≤ √ N √nα2 3 ���∇3(nv) � m∗ + √ nH −1√ Nx ���� = √r α2√α2 ���∇3v � m∗ + √r √ H −1x ���� = √r α2√α2 ����∇3v � m∗ + � r α2 y ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) Analogously, N∥∇4W( √ Nx)∥ ≤ N √nα2 4 ���∇4(nv) � m∗ + √ nH −1√ Nx ���� = r α2 2 ���∇4v � m∗ + √r √ H −1x ���� = r α2 2 ����∇4v � m∗ + � r α2 y ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assumptions V1, V2, and V3 imply Assumptions W1, W2, and W3 with N = nr, where r is from Assumption V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let y = √α2 √ H −1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that ∥y∥ ≤ ∥x∥ and in particular, if ∥x∥ ≤ 1 then ∥y∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To show that V1 implies W1, note that by the above calculation we have √ N sup ∥x∥≤1 ∥∇3W( √ Nx)∥ ≤ √r α2√α2 sup ∥y∥≤1 ����∇3v � m∗ + � r α2 y ����� ≤ 1 2, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To show that W2 implies V2, fix x ∈ Rd and note that √ N∥∇3W( √ Nx)∥ ≤ √r α2√α2 ����∇3v � m∗ + � r α2 y ����� ≤ 1 + ∥y∥q ≤ 1 + ∥x∥q, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The calculation for the fourth derivative is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To show that Assumption V3 implies W3, fix ∥x∥ ≥ √ N and let y = √ nH −1x, so that W(x) = nv(y + m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that ∥y∥ ≥ ∥x∥/√nβ2 ≥ � N/(nβ2) = � r/β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence we can apply Assumption V3 to conclude that W(x) = nv(m∗ + y) ≥ (d + 12q + 36) log(∥ � nβ2y∥) ≥ (d + 12q + 36) log(∥ √ nHy∥) = (d + 12q + 36) log ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' D Logistic Regression Example Details of Numerical Simulation For the numerical simulation displayed in Figure 1, we take d = 2 and n = 100, 200, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For each n, we draw ten sets of covariates xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , n from N(0, λ2Id) with λ = √ 5, yielding ten posterior 31 distributions πn(· | x1:n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For each πn we compute the ground truth mean and covariance by directly evaluating the integrals, using a regularly spaced grid (this is feasible in two dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The mode m∗ of πn is found by a standard optimization procedure, and the Gaussian VI estimates ˆm, ˆS are computed using the procedure described in [LCB+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We used the authors’ implementation of this algorithm, found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='com/marc-h-lambert/W-VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then compute the Laplace and VI mean and covariance approximation errors for each n and each of the ten posteriors at a given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The solid lines in Figure 1 depict the average approximation errors over the ten distributions at each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The shaded regions depict the spread of the middle eight out of ten approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Verifying the Assumptions As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, we make the approximation v(z) ≈ v∞(z) = −E [Y log s(XT z) + (1 − Y ) log(1 − s(XT z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Here, X ∼ N(0, λ2Id) and Y | X ∼ Bernoulli(s(X1)), since X1 = eT 1 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Recall that s is the sigmoid, s(a) = (1 + ea)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Below, the parameters α2, β2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' are all computed for the function v∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that z = e1 is the global minimizer of v∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have ∇2v∞(z) = E [s′(XT z)XXT ] and in particular, ∇2v∞(e1) = E [s′(X1)XXT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Also, s′(a) = s(a)(1 − σ(a)) = 1 2(1 + cosh(a)) ∈ (0, 1/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To lower bound λmin(∇2v∞(e1)), note that for ∥u∥ = 1 we have uT ∇2v∞(e1)u = E [s′(X1)(XT u)2] = u2 1E [s′(X1)X2 1] + λ2 d � j=2 u2 jE [s′(X1)] ≥ s′(λ) � �u2 1E [X2 1{|X1| ≤ λ}] + λ2 d � j=2 u2 jP(|X1| ≤ λ) � � ≳ λ2s′(λ), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) and hence α2 ≳ λ2s′(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using that s′ ≤ 1/4, we also have the upper bound λmax(∇2v∞(e1)) ≤ λ2 4 = β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) Next, we need to upper bound ∥∇3v∞∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have ∇3v∞(z) = E [s′′(XT z)X⊗3], s′′(a) = s(a)(1 − s(a))(1 − 2s(a)), so that ∥∇3v∞(z)∥ = sup ∥u1∥=∥u2∥=∥u3∥=1 E � s′′(XT z) 3 � k=1 (uT k X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' One can show that s′′(a) ∈ [−1, 1] for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence E � s′′(XT z) 3 � k=1 (uT k X) � ≤ E � 3 � k=1 |uT k X| � ≤ 3 � k=1 E � |uT k X|3�1/3 ≤ 2λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) Here, we used that uT k X d= N(0, λ2), whose third absolute moment is bounded by 2λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We therefore get the bound ∥∇3v∞(z)∥ ≤ β3 := 2λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) 32 Note that this constant bound holds for all z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Next, we need to find r such that sup ∥z−m∗∥≤√ r/α2 ∥∇3v∞(z)∥ ≤ α3/2 2 2√r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using the uniform bound (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) on ∥∇3v∞∥, it suffices to take r such that β3 = α3/2 2 2√r =⇒ r = α3 2 4β2 3 ≳ s′(λ)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) Finally, we verify Assumption V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To do so, recall that v∞ is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, if y lies on the line segment between 0 and z, with ∥y∥ = � r/β2 < ∥z∥, then v∞(m∗ + z) − v∞(m∗) ≥ ∥z∥ � r/β2 (v∞(m∗ + y) − v∞(m∗)) ≥ � β2/r inf ∥y∥=√ r/β2 [v∞(m∗ + y) − v∞(m∗)] ∥z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) It is clear that if λ is a constant then the parameters in this inequality, as well as the infimum, are lower bounded by absolute constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, since ∥z∥ ≥ log ∥z∥, Assumption V3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' E Proofs from Section 5 The proofs in this section rely on tensor-matrix and tensor-vector scalar products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let us review the rules of such scalar products, and how to bound the operator norms of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let v ∈ Rd, A ∈ Rd×d, and T ∈ Rd×d×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We define the vector ⟨T, A⟩ ∈ Rd and the matrix ⟨T, v⟩ ∈ Rd×d by ⟨T, A⟩i = d � j,k=1 TijkAjk, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , d, ⟨T, v⟩ij = d � k=1 Tijkvk, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1) We will always sum over the last two or last one indices of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that the norm of the matrix ⟨T, v⟩ is given by ∥⟨T, v⟩∥ = sup∥u∥=∥w∥=1 uT ⟨T, v⟩w, and we have uT ⟨T, v⟩w = d � i,j=1 uiwj d � k=1 Tijkvk = ⟨T, u ⊗ w ⊗ v⟩ ≤ ∥T∥∥v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Therefore, ∥⟨T, v⟩∥ ≤ ∥T∥∥v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We also review the notion of operator norm for derivatives of a function, and note the distinction between this kind of operator norm and the standard tensor operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Specifically, consider a C2 function f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , fd) : Rd×d ×Rd → Rd, where Rd×d is endowed with the standard matrix norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then ∇σf(σ, m) is a linear functional from Rd×d to Rd, and we let ⟨∇σf(σ, m), A⟩ ∈ Rd denote the application of ∇σf(σ, m) to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that we can represent ∇σf by the d × d × d tensor (∇σjkfi)d i,j,k=1, so that ⟨∇σf(σ, m), A⟩ coincides with the definition given above of tensor-matrix scalar products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' However, ∥∇σf∥op is not the standard tensor operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Rather, ∥∇σf∥op = sup A∈Rd×d,∥A∥=1 ∥⟨∇σf, A⟩∥ = sup A∈Rd×d,∥A∥=1, u∈Rd,∥u∥=1 ⟨∇σf, A ⊗ u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We continue to write ∥∇σf∥ to denote the standard tensor operator norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ∥∇σf∥ = sup u,v,w∈Rd, ∥u∥=∥v∥=∥w∥=1 ⟨∇σf, u ⊗ v ⊗ w⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 33 Note also that ∇mf ∈ Rd×d is a matrix, and that max � ∥∇σf(σ, m)∥op , ∥∇mf(σ, m)∥op � ≤ ∥∇f(σ, m)∥op ≤ ∥∇σf(σ, m)∥op + ∥∇mf(σ, m)∥op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2) Finally, recall the notation Br(0, 0) = {(σ, m) ∈ Rd×d × Rd : ∥σ∥2 + ∥m∥2 ≤ r2}, Br = {σ ∈ Rd×d : ∥σ∥ ≤ r}, Sc1,c2 = {σ ∈ Sd + : c1Id ⪯ σ ⪯ c2Id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3) Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' , fd) : Rd×d×Rd → Rd be C3, where Rd×d is the set of d×d matrices, endowed with the standard matrix operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Suppose f(0, 0) = 0, ∇σf(0, 0) = 0, ∇mf(σ, m) is symmetric for all m, and ∇mf(0, 0) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let r > 0 be such that sup (σ,m)∈Br(0,0) ∥∇f(σ, m) − ∇f(0, 0)∥op ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) Then for each σ ∈ Rd×d such that ∥σ∥ ≤ r/2 there exists a unique m = m(σ) ∈ Rd such that f(σ, m(σ)) = 0 and (σ, m(σ)) ∈ Br(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Furthermore, the map σ �→ m(σ) is C2, with 1 2Id ⪯ ∇mf(σ, m) �� m=m(σ) ⪯ 3 2Id, ∥∇σm(σ)∥op ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) The proof uses the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 in Chapter XIV of [Lan93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let U be open in a Banach space E, and let f : U → E be of class C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assume that f(0) = 0 and f ′(0) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let r > 0 be such that ¯Br(0) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' If |f ′(z) − f ′(x)| ≤ s, ∀z, x ∈ ¯Br(0) for some s ∈ (0, 1), then f maps ¯Br(0) bijectively onto ¯B(1−s)r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let φ : Rd×d × Rd → Rd×d × Rd be given by φ(σ, m) = (σ, f(σ, m)), so that φ(0, 0) = (0, 0), and ∇φ(σ, m) = � Id×d 0 ∇σf(σ, m) ∇mf(σ, m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6) For each (σ, m), (σ′, m′) ∈ Br(0, 0), we have ∥∇φ(σ, m) − ∇φ(σ′, m′)∥op = ∥∇f(σ, m) − ∇f(σ′, m′)∥op ≤ 2 sup (σ,m)∈Br(0,0) ∥∇f(σ, m) − ∇f(0, 0)∥op ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='7) Note also that ∇φ(0, 0) is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2, we have that φ is a bijection from Br(0, 0) to Br/2(φ(0, 0)) = Br/2(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, fix any σ ∈ Rd×d such that ∥σ∥ ≤ r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then (σ, 0) ∈ Br/2(0, 0), and hence there exists a unique (σ′, m) ∈ Br(0, 0) such that (σ, 0) = φ(σ′, m) = (σ′, f(σ′, m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus σ = σ′ and f(σ, m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, for each σ such that ∥σ∥ ≤ r/2 there exists a unique m = m(σ) such that (σ, m(σ)) ∈ Br(0, 0) and such that 0 = f(σ, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The map σ �→ m(σ) is C2 by standard Implicit Function Theorem arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To show that the first inequality of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) holds, note that we have ∥∇mf(σ, m(σ)) − ∇mf(0, 0)∥op ≤ ∥∇f(σ, m(σ)) − ∇f(0, 0)∥op ≤ 1/4 ≤ 1/2 34 by (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) since we know that (σ, m(σ)) ∈ Br(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus, Id = ∇2W(0) = ∇mf(0, 0) =⇒ 1 2Id ⪯ ∇mf(σ, m(σ)) ⪯ 3 2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='8) To show the second inequality of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5), we first need the supplementary bound ∥∇σf(σ, m(σ))∥op = ∥∇σf(σ, m(σ)) − ∇σf(0, 0)∥op ≤ 1/2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) which holds by the same reasoning as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, ∂σjkm = −∇mf(σ, m)−1∂σjkf(σ, m) ∈ Rd by standard Implicit Function Theorem arguments, where ∇mf(σ, m) is a matrix, ∂σjkf(σ, m) is a vector, and ∇σm, ∇σf are linear maps from Rd×d to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence by the first inequality in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) combined with (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='9) we have ∥∇σm(σ)∥op = sup ∥A∥=1 ∥⟨∇σm(σ), A⟩∥ = sup ∥A∥=1 ∥∇mf(σ, m)−1 d � j,k=1 ∂σjkf(σ, m)Ajk∥ = sup ∥A∥=1 ∥∇mf(σ, m)−1⟨∇σf, A⟩∥ ≤ ∥∇mf(σ, m)−1∥∥∇σf∥op ≤ 2 × 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='10) Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let f : Rd×d × Rd → Rd be given by f(σ, m) = E [∇W(σZ + m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then all the conditions of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 are satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' in particular, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4) is satisfied with r = 2 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Thus the conclusions of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 hold with this choice of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that f is C2 thanks to the fact that W is C3 and ∇W grows polynomially by Assumption W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then immediately have f(0, 0) = ∇W(0) = 0, ∇mf(σ, m) = E [∇2W(m + σZ)] is symmetric for all m, σ, and ∇mf(0, 0) = ∇2W(0) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To show ∇σf(0, 0) = 0, we compute the i, j, k term of this tensor: ∂σjkfi = ∂σjkE [∂iW(m + σZ)] = E [∂2 ijW(m + σZ)Zk], so that ∂σjkfi(0, 0) = E [∂2 i,jW(0)Zk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It remains to show that for r = 2 √ 2 we have sup (σ,m)∈Br(0,0) ∥∇f(σ, m) − ∇f(0, 0)∥op ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='11) First, note that sup (σ,m)∈Br(0,0) ∥∇f(σ, m) − ∇f(0, 0)∥op ≤ r sup (σ,m)∈Br(0,0) ∥∇2f(σ, m)∥op, where ∇2f(σ, m) is a bilinear form on (Rd×d × Rd)2, and we have ∥∇2f(σ, m)∥op ≤ ∥∇2 σf(σ, m)∥op + 2∥∇σ∇mf(σ, m)∥op + ∥∇2 mf(σ, m)∥op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' For f(σ, m) = E [∇W(σZ + m)], these second order derivatives are given by ∂2 mi,mjf(σ, m) = E [∂2 i,j∇W(m + σZ)], ∂mi∂σjkf(σ, m) = E [∂2 i,j∇W(m + σZ)Zk], ∂2 σjk,σℓpf(σ, m) = E [∂2 j,ℓ∇W(m + σZ)ZkZp], (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='12) 35 each a vector in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' From the first line, we get that ∥∇2 mf(σ, m)∥op ≤ E ∥∇3W(m + σZ)∥, where ∥∇3W∥ is the standard tensor norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' From the second line, we get ∥∇m∇σf(σ, m)∥op = sup ∥A∥=1,∥x∥=1 ����E � d � i,j,k=1 ∂2 i,j∇W(m + σZ)ZkxiAjk ����� = sup ∥A∥=1,∥x∥=1 ����E � d � i,j=1 ∂2 i,j∇W(m + σZ)xi(AZ)j ����� = sup ∥A∥=1,∥x∥=1 ����E �� ∇3W(m + σZ), x ⊗ AZ ������ ≤ sup ∥A∥=1,∥x∥=1 E � ∥x∥∥AZ∥∥∇3W(m + σZ)∥ � ≤ √ d � E [∥∇3W(m + σZ)∥2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='13) A similar computation gives ∥∇2 σf(σ, m)∥op ≤ sup ∥A∥=1,∥B∥=1 E [∥AZ∥∥BZ∥∥∇3W(m + σZ)∥] ≤ 2d � E [∥∇3W(m + σZ)∥2] ≲ d/ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='14) Thus overall we have ∥∇2f(σ, m)∥op ≤ (2d + 2 √ d + 1) � E [∥∇3W(m + σZ)∥2] ≤ 5d sup (σ,m)∈Br(0,0) � E [∥∇3W(m + σZ)∥2] (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='15) and hence sup (σ,m)∈Br(0,0) ∥∇f(σ, m) − ∇f(0, 0)∥op ≤ 5rd sup (σ,m)∈Br(0,0) � E [∥∇3W(m + σZ)∥2] ≤ 10 √ 2d √ N ( √ 3 + � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='16) where in the last line we applied Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5 and substituted r = 2 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' To conclude, recall that ( √ 3 + � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' )/ √ N ≤ 1/(40 √ 2d) by the assumption in the statement of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let r = 2 √ 2 and σ ∈ S0,r/2 �→ m(σ) ∈ Rd be the restriction to symmetric nonnegative matrices of the map furnished by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then the function F given by F(σ) = E [∇2W(m(σ) + σZ)]−1/2 is well-defined and a strict contraction on S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Moreover, F(S0,r/2) ⊆ Sc1,c2 ⊆ S0,r/2, where c1 = � 2/3, c2 = √ 2 = r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First, let G(σ) = E [∇2W(m(σ) + σZ)] and f(σ, m) = E [∇W(m + σZ)] as in Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Note that ∇mf(σ, m) = E [∇2W(σZ + m)], so that G(σ) = ∇mf(σ, m)|m=m(σ) and hence by (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5) of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1 we have 1 2Id ⪯ G(σ) ⪯ 3 2Id, ∀σ ∈ S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='17) 36 But then G(σ) has a unique invertible symmetric positive definite square root, and we define F(σ) = G(σ)−1/2 to be the inverse of this square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Moreover, using (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='17), it follows that c1Id ⪯ F(σ) ⪯ c2Id, ∀σ ∈ S0,r/2, where c1 = � 2/3 and c2 = √ 2 = r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In other words, F(S0,r/2) ⊆ Sc1,c2 ⊆ S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' It remains to show F is a contraction on S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let σ1, σ2 ∈ S0,r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We will first bound ∥G(σ1) − G(σ2)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We have ∥G(σ1) − G(σ2)∥ ≤ ∥σ1 − σ2∥ sup σ∈S0,r/2 ∥∇σG(σ)∥op, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='18) and ∥∇σG(σ)∥op = sup ∥A∥=1 ��� ∇σG(σ), A ��� = sup ∥A∥=1 ����E � ∇3W, � A, ∇σ (m(σ) + σZ) ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='19) Here, the quantities inside of the ∥ · ∥ on the right are matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Indeed, ⟨∇σG, A⟩ denotes the application of ∇σG to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Since G sends matrices to matrices, ∇σG is a linear functional which also sends matrices to matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' In the third line, ∇σ(m(σ) + σZ) should be interpreted as a linear functional from Rd×d to Rd, so ⟨A, ∇σ(m(σ) + σZ)⟩ is a vector in Rd, and the inner product of this vector with the d × d × d tensor ∇3W is a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using that ∥⟨T, x⟩∥ ≤ ∥T∥∥x∥, as explained at the beginning of this section, we have ���� � ∇3W, � A, ∇σ (m(σ) + σZ) ������ ≤ ��∇3W �� ∥⟨A, ∇σ (m(σ) + σZ)⟩∥ ≤ ∥∇3W∥∥∇σ(m(σ) + σZ)∥op = ∥∇3W∥∥∇σm(σ) + Z ⊗ Id∥op ≤ ∥∇3W∥(1 + ∥Z∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='20) To get the last bound, we used that ∥∇σm(σ)∥op ≤ 1, shown in Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We also use the fact that ∥Z ⊗ Id∥op = sup∥A∥=1 ∥ ⟨A, Z ⊗ Id⟩ ∥ = sup∥A∥=1 ∥AZ∥ = ∥Z∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (Recall that since Z ⊗ Id is part of ∇σm, we are considering Z ⊗ Id as an operator on matrices rather than as a d × d × d tensor, and this is why we take the supremum over matrices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=') Substituting (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='20) back into (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='18), we get ∥G(σ1) − G(σ2)∥ ≤ ∥σ1 − σ2∥ sup σ∈S0,r/2 E ���∇3W (m(σ) + σZ) �� (1 + ∥Z∥) � ≤ ∥σ1 − σ2∥ √ 2(1 + √ d) √ 3 + � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' √ N ≤ ∥σ1 − σ2∥1 + √ d 40d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='21) The second inequality is by Cauchy-Schwarz and Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' The third inequality uses that ( √ 3 + � (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' )/ √ N ≤ 1/(40 √ 2d), by the assumption in the statement of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, note that thanks to Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='3, both λmin(G(σ1)) and λmin(G(σ2)) are bounded below by 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Using Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6, we therefore have ∥F(σ1) − F(σ2)∥ ≤ √ 2∥G(σ1) − G(σ2)∥ ≤ 1 + √ d 20 √ 2d ∥σ1 − σ2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='22) Hence F is a strict contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assume 4 √ 2 ≤ � N/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then sup (σ,m)∈B2 √ 2(0,0) E [∥∇3W(m + σZ)∥2] ≤ (3 + (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' )/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' 37 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Fix ∥m∥, ∥σ∥ ≤ 2 √ 2, so that 2∥m∥ ≤ √ N and 2∥σ∥ √ d ≤ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' By Assumption W2, we have NE [∥∇3W(m + σZ)∥2] ≤ 2 + 2E � ∥(m + σZ)/ √ N∥2q� ≤ 2 + 22q ��∥m∥ √ N �2q + � ∥σ∥ √ N �2q E [∥Z∥2q] � ≤ 2 + �2∥m∥ √ N �2q + � 2∥σ∥ √ d √ N �2q (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' ≤ 3 + (2q)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='. (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='23) Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Let A0 and A1 be psd, and A1/2 0 , A1/2 1 their unique psd square roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Assume without loss of generality that λmin(A0) ≤ λmin(A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Then ∥A−1/2 1 − A−1/2 0 ∥ ≤ ∥A1 − A0∥ 2λmin(A0)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' First note that A−1/2 1 − A−1/2 0 = A−1/2 1 (A1/2 0 − A1/2 1 )A−1/2 0 and hence ∥A−1/2 1 − A−1/2 0 ∥ ≤ ∥A−1/2 1 ∥∥A−1/2 1 ∥∥A1/2 1 − A1/2 0 ∥ ≤ ∥A1/2 1 − A1/2 0 ∥ λmin(A0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now, define At = A0 + t(A1 − A0) and let Bt = A1/2 t , where Bt is the unique psd square root of At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We then have ∥A1/2 1 − A1/2 0 ∥ ≤ supt∈[0,1] ∥ ˙Bt∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' We will now express ˙Bt in terms of ˙At and Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Differentiating B2 t = At, we get Bt ˙Bt + ˙BtBt = ˙At = A1 − A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content='24) Now, one can check that the solution ˙Bt to this equation is given by ˙Bt = � ∞ 0 e−sBt(A1 − A0)e−sBtds and hence ∥ ˙Bt∥ ≤ ∥A1 − A0∥ � ∞ 0 ∥e−sBt∥2dt = ∥A1 − A0∥ 2λmin(Bt) = ∥A1 − A0∥ 2 � λmin(At) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Now note that λmin(At) ≥ λmin(A0), since At is just a convex combination of A0 and A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Hence ∥ ˙Bt∥ ≤ ∥A1 − A0∥/2 � λmin(A0) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Combining all of the above estimates gives ∥A−1/2 1 − A−1/2 0 ∥ ≤ ∥A1 − A0∥ 2λmin(A0)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' References [AR20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Alquier and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Ridgway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dA0T4oBgHgl3EQfO_9E/content/2301.02168v1.pdf'} +page_content=' Concentration of tempered posteriors and of their variational approx- imations.' metadata={'source': 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