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+On the Approximation Accuracy of
+Gaussian Variational Inference
+Anya Katsevich
+akatsevi@mit.edu
+Philippe Rigollet
+rigollet@math.mit.edu
+January 6, 2023
+Abstract
+The main quantities of interest in Bayesian inference are arguably the first two moments of the
+posterior distribution. 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. However, little is known about the approximation accuracy of VI. In this
+work, we bound the mean and covariance approximation error of Gaussian VI in terms of dimension
+and sample size. Our results indicate that Gaussian VI outperforms significantly the classical Gaussian
+approximation obtained from the ubiquitous Laplace method. 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.
+1
+Introduction
+A central challenge in Bayesian inference is to sample from, or compute summary statistics of, a posterior
+distribution π on Rd. 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. However, MCMC can
+be expensive, and it is notoriously difficult to identify clear-cut stopping criteria for the algorithm [CC96].
+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. An alternative, often computationally
+cheaper, approach is variational inference (VI) [BKM17]. The idea of VI is to find, among all measures in
+a certain parameterized family P, the closest measure to π. 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. Typically, statistics of interest, chiefly its first two moments, for measures in the
+family P are either readily available or else easily computable. In this work, we consider the family of normal
+distributions, which are directly parameterized by their mean and covariance. We define
+ˆπ = N( ˆm, ˆS) ∈ argmin
+p∈PGauss
+KL( p ∥ π),
+(1.1)
+and take ˆm, ˆS as our estimates of the true mean mπ and covariance Sπ of π. PGauss denotes the family of
+non-degenerate Gaussian distributions on Rd.
+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. On the other hand, the output of a perfect VI algorithm is
+ˆπ, which is itself only an approximation to π. Therefore, a fundamental question in VI is to understand the
+quality of the approximation ˆπ ≈ π, particularly in terms of the statistics of interest. In this work, we bound
+the mean and covariance estimation errors ∥ ˆm − mπ∥ and ∥ ˆS − Sπ∥ for the Gaussian VI estimate (1.1).
+Of course, we cannot expect an arbitrary, potentially multimodal π to be well-approximated by a Gaussian
+distribution. 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]. To understand why this is the case, consider a generic posterior
+1
+arXiv:2301.02168v1  [math.ST]  5 Jan 2023
+
+π = πn with density of the form
+πn(θ | x1:n) ∝ ν(θ)
+n
+�
+i=1
+pθ(xi)
+(1.2)
+Here, ν is the prior, pθ is the model density, and x1:n = x1, . . . , xn are i.i.d observations. 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).
+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∗. In other words, πn is effectively unimodal, and hence a Gaussian approximation is
+reasonable in this case. This reasoning drives a second, so-called Laplace approximation to πn, which is a
+Gaussian centered at m∗. Hence, the mode m∗ can also serve as an approximation to the true mean mπn.
+However, as we discuss below, Gaussian VI yields a more accurate estimate of mπn.
+Main contributions.
+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. In line with the above
+reasoning, the key assumption is that vn has a unique global minimizer.
+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. It is easy to see that πn ∝ e−nv has variance of
+order 1/n. To account for this vanishing variance, we rescale the approximation errors appropriately in the
+statement of the following theorem.
+Theorem. Let πn ∝ exp(−nv) have mean and covariance mπn, Sπn respectively. Assume that d3 ≤ n and
+that v ∈ C4(Rd) has a unique strict minimum at m∗. 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 (1.1) to π satisfy
+√n∥ ˆmn − mπn∥ ≲
+�d3
+n
+�3/2
+n∥ ˆSn − Sπn∥ ≲ d3
+n ,
+(1.3)
+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∗. This v-dependent
+factor is made explicit in Section 2.
+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. As such, it is a compelling
+endorsement of Gaussian VI for estimating the posterior mean and covariance in the finite sample regime.
+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].
+As mentioned above, the Laplace method is a competing Gaussian approximation to πn that is widespread
+in practice for its computational simplicity. We use it as a benchmark to put the above error bounds into
+context. 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. This approximation simply replaces v by its second order Taylor
+expansion around m∗. The recent works [Spo22] and [KGB22] derive error bounds for the Laplace approxima-
+tion. 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.
+For the covariance approximation, an explicit error bound is stated only in [KGB22]; the authors show that
+n∥(n∇2v(m∗))−1 − Sπn∥ ≲ 1/√n. Meanwhile, Spokoiny states lemmas in the appendix from which one can
+derive a d-dependent covariance error bound.
+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.
+(1.4)
+2
+
+This n dependence can also be obtained using the approach in [Spo22].
+Let us summarize the n-dependence of these bounds, incorporating the 1/√n and 1/n scaling of the
+mean and covariance errors. 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. The covariance approximation error
+is the same for both methods (using the tighter covariance bound (1.4)): n−1 × n−1. VI’s improved mean
+approximation accuracy is confirmed in our simulations of a simple Bayesian logistic regression example in
+d = 2; see Figure 1, and Section 2.3 for more details.
+Figure 1: Gaussian VI yields a more accurate mean estimate than does Laplace, while the two covariance
+estimates are on the same order.
+Here, πn is the likelihood of logistic regression given n observations
+in dimension d = 2.
+For the left-hand plot, the slopes of the best-fit lines are −1.04 for the Laplace
+approximation and −2.02 for Gaussian VI. For covariance: the slopes of the best-fit lines are -2.09 for
+Laplace, -2.12 for VI.
+We note that the Laplace approximation error bounds in the companion work [Kat23] are also tighter in
+their dimension dependence.
+First-order optimality conditions and Hermite series expansions.
+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.
+Hereafter, we replace θ by x and let V = nv, π ∝ e−V . The focal point of this work are the first order
+optimality equations for the minimization (1.1):
+∇m,SKL( N(m, S) ∥ π)
+��
+(m,S)=( ˆm, ˆS) = 0.
+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]. 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; see [LCB+22] for this
+calculation. 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. Rather, the true reason has to do with properties of solutions to the
+fixed point equations (EV ).
+To see why, consider the function ¯V (x) = V ( ˆm + ˆS1/2x). 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). In other words, we should have that ¯V (x) ≈
+const. + ∥x∥2/2. This is ensured by the first order optimality equations (EV ). Indeed, note that (EV ) can
+be written in terms of ¯V as
+E [∇ ¯V (Z)] = 0,
+E [∇2 ¯V (Z)] = Id.
+(1.5)
+3
+
+Mean approx. error m - mπ
+(
+10-2
+10-3
+m = m* (Laplace)
+m = mn (Gaussian VI)
+10-
+10-6
+102
+103
+nCovariance approx. 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 3.4, the equations (1.5) set the first and second order coefficients in the Hermite
+series expansion of ¯V to 0 and Id, respectively. As a result, ¯V (x) − ∥x∥2/2 = const. + r3(x), where r3 is a
+Hermite series containing only third and higher order Hermite polynomials. 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.
+See Section 3.4 for a high-level summary of this Hermite series based error analysis.
+Related Work.
+The literature on VI can be roughly divided into statistical and algorithmic works. 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.
+(We use “variational posterior” as an abbreviation for
+variational approximation to the posterior.) 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.
+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.
+We are only aware of one other work studying the problem of quantifying posterior approximation accuracy.
+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. They show that the the mean ˆm of their
+approximation satisfies √n∥ ˆm − mπ∥ ≲ 1/n1/4.
+Since the algorithmic side of VI is not our focus here, we simply refer the reader to the work [LCB+22] and
+references therein. This work complements our analysis in that it provides rigorous convergence guarantees
+for an algorithm that solves the optimization problem (1.1).
+Organization of the paper.
+The rest of the paper is organized as follows. In Section 2, we first redefine
+( ˆm, ˆS) as a certain “canonical” solution to the first order optimality conditions (EV ). 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. In Section 3, we give an overview of the proof,
+and in Section 4 we flesh out the details. Section 5 outlines the proof of the existence and uniqueness of
+the aforementioned “canonical” solution ( ˆm, ˆS) to (EV ). 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.
+Notation. For two k-tensors T, Q ∈ (Rd)⊗k, we define
+⟨T, Q⟩ =
+d
+�
+i1,...,ik=1
+Ti1...ikQi1...ik,
+and let ∥T∥F = ⟨T, T⟩1/2 be the Frobenius norm of T. We will more often make use of the operator norm
+of T, denoted simply by ∥ · ∥:
+∥T∥ =
+sup
+∥u1∥≤1,...,∥uk∥≤1
+⟨u1 ⊗ · · · ⊗ uk, T⟩,
+(1.6)
+where the supremum is over vectors u1, . . . , uk ∈ Rd. 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 (1.3) above, in which ≲ also
+incorporated a v dependent factor). We let
+mπ = E π[X],
+Sπ = Covπ(X) = E π[(X − mπ)(X − mπ)T ].
+Finally, for a function V with a unique global minimizer m∗, we let HV denote ∇2V (m∗).
+2
+Statement of Main Result
+Throughout the rest of the paper, we write π ∝ e−nv. Note that v may depend on n in a mild fashion as is
+often the case for Bayesian posteriors. We also define V = nv.
+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 (1.1). These definitions diverge
+only in the case that V is not strongly convex. Indeed, if V is strongly convex then KL( · ∥ π) is strongly
+geodesically convex in the submanifold of normal distributions; see, e.g., [LCB+22]. 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 ). In general, however, if (m, S) solve (EV ) this does not guarantee that m is a good estimator of
+mπ. 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.
+(2.1)
+Let x ̸= m∗ be a critical point of v, that is, ∇v(x) = 0, and consider the pair (m, S) = (x, 0). For this (m, S)
+we have
+E [∇v(m + S1/2Z)] = ∇v(x) = 0,
+S E [∇2v(m + S1/2Z)] = 0 ≈ 1
+nId.
+Thus (x, 0) is an approximate solution to (2.1), and by continuity, we expect that there is an exact solution
+nearby. In other words, to each critical point x of v is associated a solution (m, S) ≈ (x, 0) of (2.1). The
+solution (m, S) of (2.1) which we are interested in, then, is the one near (m∗, 0). Lemma 1 below formalizes
+this intuition; 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
+�
+,
+(2.2)
+where H = ∇2V (m∗) = n∇2v(m∗).
+Note that due to the scaling of H with n, the set RV is a small
+neighborhood of (m∗, 0). We call this unique solution (m, S) in RV the “canonical” solution of (EV ).
+We expect the Gaussian distribution corresponding to this canonical solution to be the minimizer of (1.1),
+although we have not proved this. Regardless of whether it is true, we will redefine ( ˆm, ˆS) to denote the
+canonical solution. 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.
+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.
+2.1
+Assumptions
+Our main theorem rests on rather mild assumptions on the regularity of the potential v.
+Assumption V0. The function v is at least C3 and has a unique global minimizer x = m∗.
+Let α2 be a lower bound on λmin(∇2v(m∗)) and β2 be an upper bound on λmax(∇2v(m∗)).
+Assumption V1. There exists r > 0 such that N := nr ≥ d3 and
+√r
+α2√α2
+sup
+∥y∥≤1
+���∇3v
+�
+m∗ +
+�
+r/α2 y
+���� ≤ 1
+2.
+(2.3)
+Note that the left-hand side of (2.3) is monotonically increasing with r. 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.
+Furthermore, the left-hand side equals zero when r = 0. 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.
+Remark 2.1. Define β3 = 1
+2α2√α2/√r, so that r = 1
+4α3
+2/β2
+3. By Assumption V1, we have
+sup
+∥y∥≤1
+∥∇3v(m∗ +
+�
+r/α2 y)∥ ≤ β3.
+5
+
+Hence, we can also think of β3 as an upper bound on ∥∇3v∥. For future reference, we also define
+C2,3 := 1/r = 4β2
+3
+α3
+2
+.
+(2.4)
+Assumption V2 (Polynomial growth of ∥∇kv∥, k = 3, 4). For some 0 < q ≲ 1 we have
+√r
+α2√α2
+���∇3v
+�
+m∗ +
+�
+r/α2 y
+���� ≤ 1 + ∥y∥q,
+∀y ∈ Rd.
+(2.5)
+Here, r is from Assumption V1. 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.
+Note that Assumption V1 guarantees that (2.5) is satisfied inside the unit ball {∥y∥ ≤ 1}. Therefore,
+(2.5) simply states that we can extend the constant bound 1/2 to a polynomial bound outside the unit ball.
+Also, note that if (2.5) is satisfied for some q only up to a constant factor (i.e. ≲) in the region {∥y∥ ≥ 1}
+then we can always increase q to ensure the inequality is satisfied exactly.
+Assumption V3 (Growth of v away from the minimum). Let q be as in Assumption V2. Then
+v (m∗ + x) ≥ d + 12q + 36
+n
+log(
+�
+nβ2∥x∥),
+∀∥x∥ ≥
+�
+r/β2.
+(2.6)
+See Section 3 below for further explanation of the intuition behind and consequences of the above as-
+sumptions.
+2.2
+Main result
+We are now ready to state our main results. First, we characterize the Gaussian VI parameters ( ˆmπ, ˆSπ):
+Lemma 1. Let Assumptions V0, V1, V2 be satisfied and assume √nr/d ≥ 40
+√
+2(
+√
+3 +
+�
+(2q)!), where r, q
+are from Assumptions V1, V2, respectively. Define H = ∇2V (m∗) = n∇2v(m∗). 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 ). Moreover, ˆSπ satisfies
+2/3
+nβ2
+Id ⪯ ˆSπ ⪯
+2
+nα2
+Id.
+(2.7)
+We now state our bounds on the mean and covariance errors. For simplicity, we restrict ourselves to the
+case v ∈ C4. See Theorem 1-W for results in the case v ∈ C3 \ C4.
+Theorem 1 (Accuracy of Gaussian VI). Let Assumption V3 and the assumptions from Lemma 1 be satisfied,
+and ˆmπ, ˆSπ be as in this lemma. Recall the definition of C2,3 from (2.4). If v ∈ C4, then
+∥ ˆmπ − mπ∥ ≲
+1
+√nα2
+�C2,3d3
+n
+�3/2
+∥ ˆSπ − Sπ∥ ≲
+1
+nα2
+C2,3d3
+n
+.
+(2.8)
+In Section 3.3, we prove that Lemma 1 and Theorem 1 are a consequence of analogous statements for a
+certain affine invariant density. See that subsection, and Section 3 more generally, for proof overviews.
+6
+
+2.3
+An example: Logistic Regression
+As noted in the introduction, our results show that Gaussian VI yields very accurate mean and covariance
+approximations; in fact, the mean estimate is a full factor of 1/n more accurate than the mean estimate given
+by the Laplace approximation. 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. We also show how to check the assumptions for this example.
+In logistic regression, we observe n covariates xi ∈ Rd and corresponding labels yi ∈ {0, 1}. 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. In other words, yi ∼ Bern(s(xT
+i z)). We take the ground truth z
+to be z = e1 = (1, 0, . . . , 0), and we generate the xi, i = 1, . . . , n i.i.d. from N(0, λ2Id), so in particular the
+covariates themselves do not depend on z. 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))
+�
+.
+(2.9)
+Numerical Simulation
+For the numerical simulation displayed in Figure 1, we take d = 2 and n = 100, 200, . . . , 1000. For each n, we
+draw ten sets of covariates xi, i = 1, . . . , n from N(0, λ2Id) with λ =
+√
+5, yielding ten posterior distributions
+πn(· | x1:n). 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. The solid lines in Figure 1 depict the average approximation
+errors over the ten distributions at each n. The shaded regions depict the spread of the middle six out of
+ten approximation errors. See Appendix D for details about the simulation.
+In the left panel of Figure 1 depicting the mean error, the slopes of the best fit lines are −1.04 and −2.02
+for Laplace and Gaussian VI, respectively. For the covariance error in the righthand panel, the slopes of the
+best fit lines are −2.09 and −2.12 for Laplace and Gaussian VI. This confirms that our bounds, the mean
+bound of [KGB22] and the bound (1.4) (also implied by results in [Spo22]) are tight in their n dependence.
+Verification of Assumptions
+It is well known that the likelihood (2.9) is convex, and has a finite global minimizer z = m∗ (the MLE)
+provided the data xi, i = 1, . . . , n are not linearly separable. Assumption V0 is satisfied in this case. 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. Using
+this approximation, we show in Appendix D that
+α2 ≳ λ2s′(λ),
+β2 ≤ λ2
+4 ,
+and
+∥∇3v∞(z)∥ ≤ β3 := 2λ3,
+∀z ∈ Rd.
+(2.10)
+To verify Assumption V1, we need to find r such that
+sup
+∥z−m∗∥≤√
+r/α2
+∥∇3v(z)∥ ≤ α3/2
+2
+2√r .
+Using the uniform bound (2.10) 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,
+(2.11)
+7
+
+using that s′(λ) = s(λ)(1 − s(λ)) = 1
+2(1 + cosh(λ))−1. 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. Next, we can use (2.10) and a
+similar bound on ∥∇4v∥ (which is also bounded uniformly over Rd) to show that Assumption V2 is satisfied
+with q = 0. It remains to check Assumption V3, which we do in Appendix D using the convexity of v.
+Indeed, convexity immediately implies at least linear growth away from any point. We conclude that the
+conditions of Theorem 1 are met.
+3
+Proof Overview: Affine Invariant Rescaling and Hermite Ex-
+pansion
+In this section, we overview the proof of Theorem 1. We start in Section 3.1 by explaining the affine invariance
+inherent to this problem. This motivates us to rescale V = nv to obtain a new affine invariant function W.
+In Section 3.2, we state Assumptions W0-W3 on W, which include the definition of a scale-free parameter
+N intrinsic to W. We then state our main results for W: Lemma 1-W and Theorem 1-W. In Section 3.3,
+we deduce Lemma 1 and Theorem 1 for V from the lemma and theorem for W. We outline the proof of
+Theorem 1-W in Section 3.4. The proof of Lemma 1-W is of a different flavor, and is postponed to Section 5.
+3.1
+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∥.
+(3.1)
+As shown in Section 3.3, combining the bounds on (3.1) with bounds on ∥ ˆSπ∥ will give the desired estimates
+in Theorems 1. The reason for considering (3.1) rather than directly bounding the quantities in the theorems
+is explained in Section 3.4.
+In the following lemma, we show that the quantities (3.1) are affine invariant. We discuss the implications
+of this fact at the end of the subsection. First, define
+Definition 3.1. Let f be a C2 function with unique global minimizer m∗f, and let Hf = ∇2f(m∗f). Then
+Rf =
+�
+(m, S) ∈ Rd×Sd
+++ : S ⪯ 2H−1
+f ,
+∥
+√
+Hf
+√
+S∥2 + ∥
+√
+Hf(m − m∗f)∥2 ≤ 8
+�
+.
+(3.2)
+Lemma 3.1. Let V1, V2 ∈ C2(Rd), where V2(x) = V1(Ax + b) for some b ∈ Rd and invertible A ∈ Rd×d. Let
+πi ∝ e−Vi, i = 1, 2. 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
+(3.3)
+is a unique solution to (EV2) in the set RV2. 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∥.
+(3.4)
+See Lemma C.1 of Appendix C for the proof of the first statement. The proof of (3.4) follows from (3.3),
+the fact that
+mπ2 = A−1(mπ1 − b),
+Sπ2 = A−1Sπ1A−T ,
+(3.5)
+and the following lemma
+Lemma 3.2. Let C, D ∈ Sd
+++ be symmetric positive definite matrices and x ∈ Rd. Then ∥C−1/2x∥ =
+√
+xT C−1x and
+∥C−1/2DC−1/2 − Id∥ = sup
+u̸=0
+uT Du
+uT Cu − 1.
+This is a simple linear algebra result, so we omit the proof.
+8
+
+Discussion.
+Lemma 3.1 shows that our bounds on the quantities (3.1) should themselves be affine invariant, i.e. 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}.
+This motivates us to identify an affine-invariant large parameter N. 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. Another natural candidate
+for N, which removes this degree of freedom, is N = λmin(∇2V (m∗)). However, λmin(∇2V (m∗)) is not
+affine-invariant. Indeed, replacing V (x) by V (cx), for example, changes λmin by a factor of c2. 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 ).
+(3.6)
+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).
+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. This function is the “correct” object of study, and any bounds we obtain must follow
+from properties intrinsic to W.
+3.2
+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.
+This parameter is such that as N increases, the measure ρ ∝ e−W is more closely
+approximated by a Gaussian. 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.
+Assumption W0. Let W be at least C3, with unique global minimizer x = 0, and ∇2W(0) = Id. Moreover,
+assume without loss of generality that W(0) = 0.
+Next, we identify N as a parameter quantifying the size of ∥∇3W∥ in a certain neighborhood of zero:
+Assumption W1. There exists N ≥ d3 such that
+√
+N sup
+∥x∥≤1
+∥∇3W(
+√
+Nx)∥ ≤ 1
+2.
+(3.7)
+This definition ensures that N scales proportionally to n. Indeed, suppose W1 is the affine invariant
+function corresponding to the equivalence class containing n1v, and let W1 satisfy (3.7) with N = N1.
+Then the affine invariant W2 corresponding to the equivalence class containing n2v is given by W2(x) =
+n2
+n1 W1(
+√n1
+√n2 x). From here it is straightforward to see that W2 satisfies (3.7) with N = N2, where N2/N1 =
+n2/n1. To further understand the intuition behind this assumption, consider the following lemma.
+Lemma 3.3. Let W satisfy Assumptions W0 and W1 and let C ≤
+�
+N/d. Then
+����W(x) − ∥x∥2
+2
+���� ≤ C3
+12
+d
+√
+d
+√
+N
+,
+∀∥x∥ ≤ C
+√
+d,
+W(x) ≥ ∥x∥2
+4
+,
+∀∥x∥ ≤
+√
+N.
+(3.8)
+The lemma shows that N quantifies how close W is to a quadratic, and therefore how close ρ ∝ e−W is
+to being Gaussian.
+Proof. Taylor expanding W(x) to second order for ∥x∥ ≤ C
+√
+d and using (3.7), we have
+|W(x) − ∥x∥2/2| ≤ 1
+3!
+sup
+∥x∥≤C
+√
+d
+∥x∥3∥∇3W(x)∥ ≤ C3
+12
+d
+√
+d
+√
+N
+.
+(3.9)
+9
+
+The second inequality in (3.8) follows from the fact that ∇2W(x) ⪰ 1
+2Id for all ∥x∥ ≤
+√
+N, as we now show.
+Taylor expanding ∇2W(x) to zeroth order, we get that
+∥∇2W(x) − ∇2W(0)∥ ≤
+sup
+∥x∥≤
+√
+N
+∥x∥∥∇3W(x)∥ ≤ 1
+2.
+Since ∇2W(0) = Id it follows that ∇2W(x) ⪰ 1
+2Id.
+Assumption W2 (Polynomial growth of ∥∇kW∥, k = 3, 4). There exists 0 < q ≲ 1 such that
+√
+N
+���∇3W
+�√
+Nx
+���� ≤ 1 + ∥x∥q
+∀x ∈ Rd.
+(3.10)
+If W is C4, then the following bound also holds with the same q:
+N
+���∇4W
+�√
+Nx
+���� ≲ 1 + ∥x∥q,
+∀x ∈ Rd.
+(3.11)
+The N 1 in (3.11) is also chosen to respect the proportional scaling of N with n: if the affine invariant
+W1 corresponding to n1v satisfies (3.11) with N = N1, then the affine invariant W2 corresponding to n2v
+satisfies (3.11) with the same q and N = N2, where N2/N1 = n2/n1.
+Note that Assumption W1 guarantees that (3.10) is satisfied inside the unit ball; therefore, (3.10) simply
+states that we can extend the constant bound 1/2 to a polynomial bound outside of this ball. Also, note
+that if the inequality is satisfied for some q only up to a constant factor (i.e. ≲) in the region {∥x∥ ≥ 1},
+then we can always increase q to ensure the inequality is satisfied exactly.
+Assumption W2 implies that expectations of the form E [∥∇kW(Y )∥p], k = 3, 4, decay with N. Indeed,
+we have
+Lemma 3.4. Let p ≥ 0 and Y ∈ Rd be a random variable such that E [∥Y ∥pq] < ∞, where q is from
+Assumption W2. Let k = 3 or 4, corresponding to the cases W ∈ C3 or W ∈ C4, respectively. Then
+E [∥∇kW(Y )∥p] ≲ N −p( k
+2 −1) �
+1 + E
+�
+∥Y/
+√
+d∥pq��
+.
+Proof. 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��
+.
+(3.12)
+In the second line we used that d ≤ N.
+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).
+Assumption W3 (Separation from Zero; Growth at Infinity). We have
+W(x) ≥ (d + 12q + 36) log ∥x∥,
+∀∥x∥ ≥
+√
+N,
+where q is from Assumption W2.
+Remark 3.1. 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.
+(3.13)
+Recall that inside the unit ball, W(
+√
+Nx) is no less than N∥x∥2/4, by Lemma 3.3. Therefore, the value of
+W(
+√
+Nx) increases up to at least N/4 as x approaches unit norm. We can interpret (3.13) 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∥ → ∞.
+10
+
+We now state the existence and uniqueness of solutions to (EW ) in the region RW .
+Lemma 1-W. Take Assumptions W0, W1, and W2 to be true, and assume
+√
+N/d ≥ 40
+√
+2(
+√
+3 +
+�
+(2q)!),
+where q is from Assumption W2. Then there exists a unique (m, S) = ( ˆmρ, ˆSρ) ∈ RW ,
+RW = {(m, S) ∈ Rd × Sd
+++ : S ⪯ 2Id, ∥S∥ + ∥m∥2 ≤ 8},
+(3.14)
+solving (EW ). The matrix ˆSρ furthermore satisfies
+2
+3Id ⪯ ˆSρ ⪯ 2Id.
+(3.15)
+See Section 5 for the proof. Note that RW as defined here is the same as in Definition 3.1, since m∗W = 0
+and HW = ∇2W(0) = Id. 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.
+(3.16)
+Theorem 1-W. Take Assumptions W0, W1, W2, and W3 to be true, and let ( ˆmρ, ˆSρ) be as in the above
+lemma. Then
+∥ ˆS−1/2
+ρ
+( ˆmρ − mρ)∥ ≲
+�
+�
+�
+d3
+N
+if W ∈ C3,
+�
+d3
+N
+�3/2
+,
+if W ∈ C4.
+,
+∥ ˆS−1/2
+ρ
+Sρ ˆS−1/2
+ρ
+− Id∥ ≲ d3
+N .
+3.3
+From V to W and back
+In the following sections, we prove Lemma 1-W and Theorem 1-W. In Lemma C.2 in the appendix, we show
+that Assumptions V0-V3 imply Assumptions W0-W3 with N = nr and the same q. From these results,
+Lemma 1 and Theorem 1 easily follow.
+Proof of Lemma 1. Let ρ ∝ e−W , where W is defined as in (3.6). By Lemma C.2, the assumptions on V
+imply the assumptions on W. Hence, we can apply Lemma 1-W to conclude there is a unique ( ˆmρ, ˆSρ) ∈ RW
+solving (EW ), with 2
+3Id ⪯ ˆSρ ⪯ 2Id. Since W is an affine transformation of V , it follows by Lemma 3.1 that
+there exists a unique ( ˆmπ, ˆSπ) ∈ RV solving (EV ), with ˆSπ = H−1/2
+V
+ˆSρH−1/2
+V
+. The inequality (2.7) for π
+can be deduced from the corresponding inequality (3.15) for ˆSρ.
+Proof of Theorem 1. 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∥ = ∥ ��S1/2
+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∥,
+(3.17)
+using the bound on ˆSπ from Lemma 1. Next note that by Lemma 3.1 (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∥.
+(3.18)
+Apply Theorem 1-W to conclude, recalling that N = nr and C2,3 = 1/r so that d3/N = d3/nr = C2,3d3/n.
+11
+
+3.4
+Overview of Theorem 1-W proof
+For brevity let m = ˆmρ, S = ˆSρ, and σ = S1/2. We continue to denote the mean and covariance of ρ by mρ
+and Sρ, respectively. 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.
+(3.19)
+The proof of Theorem 1-W is based on several key observations.
+1) The optimality conditions (3.19) imply that the Hermite series expansion of ¯W is given by ¯W(x) =
+const. + 1
+2∥x∥2 + r3(x), where
+r3(x) =
+�
+k≥3
+1
+k!⟨ck( ¯W), Hk(x)⟩.
+(3.20)
+2) The assumptions on W imply that r3 ∼ N −1/2.
+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,
+(3.21)
+where
+f1,u(x) = uT x,
+f2,u(x) = (uT x)2 − 1.
+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 + . . . )]
+E [e−r3(Z)]
+(3.22)
+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.
+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 (3.21) the quantities of interest are no larger than N −1. This is the essence of the proof when
+W ∈ C3.
+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.
+1) We can write W(x) = const. + 1
+2∥x∥2 + r3(x), where r3 is the third order Hermite series
+remainder. The Hermite series expansion of ¯W is defined as
+¯W(x) =
+∞
+�
+k=0
+1
+k!⟨ck( ¯W), Hk(x)⟩,
+ck( ¯W) := E [ ¯W(Z)Hk(Z)].
+(3.23)
+Here, the ck and Hk(x) are tensors in (Rd)⊗k. Specifically, Hk(x) is the tensor of all order k Hermite
+polynomials, enumerated as H(α)
+k
+, α ∈ [d]k with some entries repeating. For k = 0, 1, 2, the Hermite tensors
+are given by
+H0(x) = 1,
+H1(x) = x,
+H2(x) = xxT − Id.
+See Appendix A.1 and B.1 for further details on Hermite series. Distinct Hermite polynomials are orthogonal
+to each other with respect to the Gaussian weight. In particular, if f is an order k polynomial and ℓ > k
+then
+E [f(Z)H(α)
+ℓ
+(Z)] = 0,
+∀α ∈ [d]ℓ.
+12
+
+In general, the Hk are given by
+Hk(x)e−∥x∥2/2 = (−1)k∇ke−∥x∥2/2.
+(3.24)
+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)].
+(3.25)
+This is a generalization of Stein’s identity, E [Zif(Z)] = E [∂xif(Z)].
+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,
+(3.26)
+where the last equality in each line comes from the optimality conditions (3.19). 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. + 1
+2∥x∥2 + r3(x),
+(3.27)
+where r3 is the third order remainder.
+2) The assumptions imply r3 ∼ 1/
+√
+N. 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)].
+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. Therefore each ck ∼ 1/
+√
+N for k ≥ 3, so r3 ∼ 1/
+√
+N as well.
+Now suppose W ∈ C4, and write r3 as r3(x) =
+1
+3!⟨c3, H3(x)⟩ + r4(x). 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.
+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). 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. This is why the mean error is smaller when W ∈ C4.
+We will prove 3) in the next section, and 4) follows directly from the representation (3.27). 5) follows
+from the definition of r3 as a sum of third and higher order Hermite polynomials. This discussion explains
+how the N −1 and N −3/2 scalings arise in Theorem 1-W. Obtaining the correct scaling with dimension d
+requires a bit more work. 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 4.1 below for further discussion
+of the bound’s d-dependence).
+We show that E [rk(Z)p] ∼ E [
+�
+∥Z∥k�p] ∼ dpk/2 using the following explicit formula for rk. This result
+is known in one dimension; see Section 4.15 in [Leb72]. However, we could not find the multidimensional
+version in the literature, so we have proved it here.
+Proposition 3.1. Assume ¯W ∈ Ck for k = 3 or k = 4. Let ¯W(x) = �∞
+j=0
+1
+j!⟨cj( ¯W), Hj(x)⟩ be the Hermite
+series expansion of ¯W, and define
+rk(x) = ¯W(x) −
+k−1
+�
+j=0
+1
+j!⟨cj( ¯W), Hj(x)⟩.
+(3.28)
+Then
+rk(x) =
+� 1
+0
+(1 − t)k−1
+(k − 1)! E
+��
+∇k ¯W ((1 − t)Z + tx) , Hk(x) − Z ⊗ Hk−1(x)
+��
+dt.
+(3.29)
+13
+
+Note that (3.29) is analogous to the integral form of the remainder of a Taylor series. We state and prove
+this proposition in greater generality in Appendix B.1 below. 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. 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∥. This explains the
+dpk/2 scaling of E [|rk(Z)|p].
+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. Also, let
+r3(x) =
+�
+k≥3
+1
+k!⟨ck( ¯W), Hk(x)⟩
+be the remainder of the Hermite expansion of ¯W.
+Lemma 4.1 (Preliminary Bound). 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
+(4.1)
+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
+(4.2)
+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.
+(4.3)
+Remark 4.1. From the discussion in the previous section, we know c3, r3 ∼ N −1/2 and c4, r4 ∼ N −1.
+Therefore, we can easily read off the N-dependence of the overall error bound from (4.1) and (4.3). 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. Furthermore, simple Laplace-type integral bounds in Section 4.3 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. Finally, the
+d-dependence of ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3 ⊗ H4]⟩ can be estimated using the structure of the Hermite tensors;
+in particular, we show at most O(d4) of the d8 entries of E [Z ⊗ H3 ⊗ H4] are nonzero.
+Proof. First, we prove point 3) from the above proof overview. Recall that f1,u(x) = uT x and f2,u(x) =
+(uT x)2 − 1. Note that we can write ¯X = σ−1(X − m), where X ∼ ρ ∝ e−W . 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)].
+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.
+(4.4)
+14
+
+Now, recalling that ¯W(x) = const. + ∥x∥2/2 + r3(x), note that
+E [f( ¯X)] = E [f(Z)e−r3(Z)]
+E [e−r3(Z)]
+.
+(4.5)
+Write
+e−r3(Z) = 1 − r3(Z) + 1
+2r3(Z)2 − 1
+3!r3(Z)3eξ(Z),
+where ξ(Z) lies on the interval between 0 and −r3(Z). 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. Therefore,
+E
+�
+f(Z)e−r3(Z)�
+= E
+�
+f(Z)
+�
+1 − r3(Z) + 1
+2r3(Z)2 − 1
+3!r3(Z)3eξ(Z)
+��
+= E
+�
+f(Z)
+�1
+2r3(Z)2 − 1
+3!r3(Z)3eξ(Z)
+��
+.
+(4.6)
+Combining (4.6) with (4.5), we get
+E[f(Z)] = 1
+2
+E
+�
+f(Z)r3(Z)2�
+E
+�
+e−r3(Z)�
+− 1
+3!
+E [f(Z)r3(Z)3eξ(Z)]
+E
+�
+e−r3(Z)�
+=: I1 + I2.
+(4.7)
+Using Jensen’s inequality and that E [r3(Z)] = 0, we have E [e−r3(Z)] ≥ 1. Hence,
+|I1| ≲
+��E
+�
+f(Z)r3(Z)2��� .
+(4.8)
+To bound I2, note that eξ ≤ 1 + e−r3, since ξ ≤ 0 if r3 ≥ 0 and ξ ≤ −r3 if r3 ≤ 0. 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)�
+,
+(4.9)
+again using that E [e−r3(Z)] ≥ 1. Furthermore, using the conversion between Z and ¯X expectations (4.5),
+observe that
+E
+�
+|f(Z)| |r3(Z)|3 e−r3(Z)�
+E
+�
+e−r3(Z)�
+= E
+���f( ¯X)
+�� ��r3( ¯X)
+��3�
+.
+Incorporating this into the above bound on |I2| we get
+|I2| ≤ E
+�
+|f(Z)| |r3(Z)|3�
++ E
+���f( ¯X)
+�� ��r3( ¯X)
+��3�
+.
+(4.10)
+Applying Cauchy-Schwarz to (4.10) we get
+|I2| ≤
+�
+E [r3(Z)6]
+�
+E f(Z)2 +
+�
+E r3( ¯X)6
+�
+E f( ¯X)2.
+Adding this inequality to (4.8), 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�
+.
+(4.11)
+Taking f(x) = uT x and f(x) = (uT x)2−1 and applying Cauchy-Schwarz to the first term in (4.11) gives (4.1)
+and (4.2), respectively. If v ∈ C4 and f(x) = uT x, we can refine the bound (4.11), specifically the first term
+E [(uT Z)r3(Z)2]. Write
+r3(x) = 1
+3! ⟨c3, H3(x)⟩ + r4(x).
+15
+
+Then
+r3(x)2 =
+1
+(3!)2
+�
+c⊗2
+3 , H3(x)⊗2�
++ 2
+3!r4(x) ⟨c3, H3(x)⟩ + r4(x)2.
+(4.12)
+To get the first summand on the right we use the fact that ⟨T, S⟩2 = ⟨T ⊗2, S⊗2⟩. Substituting x = Z
+in (4.12), multiplying by the scalar uT Z, and taking the expectation of the result gives
+E
+��
+uT Z
+�
+r3(Z)2�
+=
+1
+(3!)2
+�
+c⊗2
+3 , E
+��
+uT Z
+�
+H3(Z)⊗2��
++ 2
+3!E
+�
+(uT Z)r4(Z) ⟨c3, H3(Z)⟩
+�
++ E [(uT Z)r4(Z)2]
+= 2
+3!E
+�
+(uT Z)r4(Z) ⟨c3, H3(Z)⟩
+�
++ E
+��
+uT Z
+�
+r4(Z)2�
+.
+(4.13)
+For the term on the right-hand side of the first line of (4.13), 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. 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. To see why, see the primer on Hermite polynomials in Section A.1.
+Next, let g(x) = (uT x) ⟨c3, H3(x)⟩, so that
+E
+�
+(uT Z)r4(Z) ⟨c3, H3(Z)⟩
+�
+= E [g(Z)r4(Z)].
+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! ⟨ck, Hk(Z)⟩
+�
+=
+∞
+�
+k=4
+1
+k!E [g(Z) ⟨ck, Hk(Z)⟩].
+Furthermore, g is a fourth order polynomial, and is therefore orthogonal to all Hermite polynomials of order
+greater than four. As a result, the above sum simplifies to
+E [g(Z)r4(Z)] = 1
+4!E [g(Z) ⟨c4, H4(Z)⟩]
+= 1
+4!E [(uT Z) ⟨c3, H3(Z)⟩ ⟨c4, H4(Z)⟩]
+= 1
+4! ⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩ .
+(4.14)
+Combining these calculations and applying Cauchy-Schwarz to the term E [(uT Z)r4(Z)2] gives the prelimi-
+nary bound (4.3).
+4.1
+Combining the bounds
+In the following sections, we bound each of the terms appearing in (4.1), (4.2), (4.3). For convenience, we
+compile these bounds below, letting τ = d3/N.
+Lemma 4.2 gives
+|⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3 ⊗ H4]⟩| ≲ d
+7
+2
+N
+3
+2 ≤ τ
+3
+2
+(4.15)
+Corollary 4.1 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.
+(4.16)
+16
+
+Corollary 4.1 applied with Y = ¯X, together with Corollary 4.2, give
+�
+E [r3( ¯X)6] ≲ e
+√
+d3/N
+�d3
+N
+� 3
+2
+= e
+√ττ
+3
+2 .
+Finally, Corollary 4.3 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.
+(4.17)
+Substituting all of these bounds into (4.1), (4.2), and (4.3) finishes the proof of Theorem 1-W.
+4.2
+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]. We take all of the assumptions to be true, either in the W ∈ C3 case or W ∈ C4.
+Lemma 4.2. If v ∈ C4 then
+|⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩| ≲ d7/2N −3/2.
+(4.18)
+Proof. We use Lemma B.3 in Appendix B.1, which shows that
+⟨u ⊗ c3 ⊗ c4, E [Z ⊗ H3(Z) ⊗ H4(Z)]⟩ = ⟨u ⊗ c3, c4⟩.
+(4.19)
+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∥
+(4.20)
+As explained in Section 3.4, since v ∈ C4 we have ck = ck( ¯W) = E [∇k ¯W(Z)], k = 3, 4. Hence
+∥c3∥ ≤ E ∥∇3 ¯W(Z)∥ ≤ ∥σ∥3E ∥∇3W(m + σZ)∥ ≲ N −1/2.
+(4.21)
+To get the last inequality, we used (3.16) to bound ∥m∥, ∥σ∥ by a constant, and we applied Lemma 3.4 with
+Y = m + σZ, p = 1, k = 3. Note that E [∥(m + σZ)/
+√
+d∥s] ≲ 1 for any s ≥ 0, so the bound in the lemma
+reduces to N −1/2. The lemma applies since E [∥m + σZ∥s] ≲
+√
+d
+s for all s �� 0. Analogously,
+∥c4∥ ≲ N −1.
+(4.22)
+Substituting the bounds (4.21), (4.22) into (4.20) and using the equality (4.19) gives the bound in the
+statement of the lemma.
+We now compute bounds on expectations of the form E [|rk(Z)|p], k = 3, 4, and on E [r3( ¯X)6]. Using the
+exact formula (3.29) for rk, we obtain the following bound:
+Corollary 4.1 (Corollary (B.1) in Appendix B.2). Let k = 3 if W ∈ C3 and k = 4 if W ∈ C4. Let Y ∈ Rd
+be a random variable such that E [∥Y ∥s] < ∞ for all 0 ≤ s ≤ 2pk + 2pq, where q is from Assumption W2.
+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��
+(4.23)
+17
+
+Taking Y = Z, the expectations in (4.23) are all bounded by constants, so we immediately obtain
+E [|rk(Z)|p] ≲
+�
+dk
+N k−2
+� p
+2
+.
+The corollary also applies to Y = ¯X, k = 3, p = 6. This is because, as we show in Corollary 4.2 in the
+next section, E [∥X/
+√
+d∥s] ≲ exp(2
+�
+d3/N) < ∞ for all s ≤ 36 + 12q = 2pk + 2pq. Since ¯X = σ−1(X − m),
+using (3.16) we conclude that also E [∥ ¯X/
+√
+d∥s] ≲ exp(2
+�
+d3/N) < ∞ for all s ≤ 36 + 12q. Hence (4.23)
+gives
+E [r3( ¯X)6] ≲ exp
+�
+2
+�
+d3/N
+� �d3
+N
+�3
+.
+4.3
+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.
+(4.24)
+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
+�
+.
+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
+�
+.
+(4.25)
+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. This will allow us to show that
+�
+I f(x)e−W (x)dx
+�
+I e−W (x)dx
+≲ E [f(Z)].
+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.
+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.
+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. The following four short lemmas carry out this
+program. We let τ = d3/N in the statements and proofs below.
+Lemma 4.3. We have
+�
+∥x∥≤2
+√
+2
+√
+d
+e−W (x)dx ≳ e−2√τ√
+2π
+d,
+where τ = d3/N.
+Proof. By Lemma 3.3 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.
+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,
+(4.26)
+as desired.
+18
+
+Lemma 4.4. Let f ≥ 0. Then E [f(X)1I(X)] ≲ e2√τE [f(Z)]. In particular, E [∥X∥p1I(X)] ≲ e2√τdp/2.
+Proof. Using Lemma 4.3 and Lemma 3.3,
+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)],
+(4.27)
+as desired.
+Lemma 4.5. We have E [∥X∥p1M(X)] ≲ e2√τ for all p ≥ 0.
+Proof. Using Lemma 4.3 and Lemma 3.3,
+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
+(4.28)
+We now change variables as x =
+√
+2y, so that ∥x∥pdx is bounded above by
+√
+2
+d+p∥y∥pdy. 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.
+(4.29)
+Lemma 4.6. For all p ≤ 12q + 36 we have E [∥X∥p1O(X)] ≲ e2√τ.
+Proof. Using Lemma 4.3 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.
+(4.30)
+In the third line, we left out the surface area of the (d − 1)-sphere, which is an at most O(1) factor.
+The above three lemmas immediately imply
+Corollary 4.2. For all p ≤ 12q + 36 we have E [∥X∥p] ≲ dp/2e2√τ.
+We also have
+19
+
+Corollary 4.3. Let ¯X = σ−1(X − m), where ∥σ−1∥, ∥m∥ ≲ 1.
+If ∥u∥ = 1 then E [(uT ¯X)2] ≲ e2√τ,
+E [((uT ¯X)2 − 1)2] ≲ e2√τ.
+Proof. 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.
+Since ¯X = σ−1(X − m), we have
+E [(uT ¯X)k] ≲ E [(uT σ−1X)k] + ∥σ−1m∥k.
+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. Using Lemmas 4.4, 4.5, and 4.6 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,
+(4.31)
+as desired.
+5
+Proof of Lemma 1-W
+In this section, we use m ∈ Rd, σ ∈ Rd×d to denote generic arguments. Consider the equations (EW ), which
+we rewrite in the following form:
+E [∇W(m + σZ)] = 0,
+(5.1)
+E [∇2W(m + σZ)] = (σσT )−1.
+(5.2)
+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; indeed, (5.1) and (5.2) only depend on σ through S = σσT , which has
+multiple solutions σ. 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}.
+(5.3)
+In particular, note that S0,r ⊂ Br.
+Lemma 5.1. Let W satisfy Assumptions W0, W1, W2, and W3, and assume
+√
+N/d ≥ 40
+√
+2(
+√
+3+
+�
+(2q)!).
+Let r = 2
+√
+2, c1 =
+�
+2/3, and c2 =
+√
+2. There exists a unique pair (σ, m) ∈ Br(0, 0) ∩ S0,r/2 × Rd satisfying
+(5.1) and (5.2), and this pair is such that σ ∈ Sc1,c2.
+Let us sketch the proof of the lemma. Let f : Rd×d × Rd → Rd be given by f(σ, m) = E [∇W(m +
+σZ)]. 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. In Lemma 5.3, 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. Since S0,r/2 ⊂ Br/2, we have in particular that any solution (σ, m)
+to (5.1) in the region Br(0, 0) ∩ S0,r/2 × Rd is of the form (σ, m(σ)). Thus it remains to prove there exists
+a unique solution σ ∈ S0,r/2 to the equation E [∇2W(m(σ) + σZ)] = (σσT )−1. To do so, we rewrite this
+equation as F(σ) = σ, where
+F(σ) = E [∇2W(m(σ) + σZ)]−1/2.
+We show in Lemma 5.4 that F is well-defined on S0,r/2, a contraction, and satisfies F(S0,r/2) ⊂ Sc1,c2 ⊂
+S0,r/2. Thus by the Contraction Mapping Theorem, there is a unique σ ∈ S0,r/2 satisfying F(σ) = σ. But
+since F maps S0,r/2 to Sc1,c2, the fixed point σ necessarily lies in Sc1,c2. This finishes the proof.
+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.
+20
+
+Lemma 5.2. Let f = (f1, . . . , fd) : Rd×d ×Rd → Rd be C3, where Rd×d is the set of d×d matrices, endowed
+with the standard matrix operator norm. Suppose f(0, 0) = 0, ∇σf(0, 0) = 0, ∇mf(σ, m) is symmetric for
+all m, σ, and ∇mf(0, 0) = Id. Let r > 0 be such that
+sup
+(σ,m)∈Br(0,0)
+∥∇f(σ, m) − ∇f(0, m∗)∥op ≤ 1
+4.
+(5.4)
+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). Furthermore, the map σ �→ m(σ) is C2, with
+1
+2Id ⪯ ∇mf(σ, m)
+��
+m=m(σ) ⪯ 3
+2Id,
+∥∇σm(σ)∥op ≤ 1.
+(5.5)
+See Appendix E for careful definitions of the norms appearing above, as well as the proof of the lemma.
+Lemma 5.3. Let f : Rd×d × Rd → Rd be given by f(σ, m) = E [∇W(σZ + m)]. Then all the conditions of
+Lemma 5.2 are satisfied; in particular, (5.4) is satisfied with r = 2
+√
+2. Thus the conclusions of Lemma 5.2
+hold with this choice of r.
+Lemma 5.4. Let r = 2
+√
+2 and σ ∈ S0,r/2 �→ m(σ) ∈ Rd be the restriction of the map furnished by
+Lemmas 5.2 and 5.3 to symmetric nonnegative matrices. 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. Moreover,
+F(S0,r/2) ⊆ Sc1,c2 ⊆ S0,r/2,
+where c1 =
+�
+2/3, c2 =
+√
+2 = r/2.
+This lemma concludes the proof of Lemma 5.1 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 (5.2). We know σ must lie in Sc1,c2 since F maps S0,r/2 to this set. See Appendix E for the proofs
+of the above lemmas.
+Acknowledgments
+A. Katsevich is supported by NSF grant DMS-2202963. P. Rigollet is supported by NSF grants IIS-1838071,
+DMS-2022448, and CCF-2106377.
+A
+Hermite Series Remainder
+A.1
+Brief Primer
+Here is a brief primer on Hermite polynomials, polynomials, and series expansions. We let Hk : R → R,
+k = 0, 1, 2, . . . be the kth order probabilist’s Hermite polynomial. We have H0(x) = 1, H1(x) = x, H2(x) =
+x2 − 1, H3(x) = x3 − 3x. For all k ≥ 1, we can generate Hk+1 from the recurrence relation
+Hk+1(x) = xHk(x) − kHk−1(x),
+k ≥ 1.
+(A.1)
+In particular, Hk(x) is an order k polynomial given by a sum of monomials of the same parity as k. The Hk
+are orthogonal with respect to the Gaussian measure; namely, we have E [Hk(Z)Hj(Z)] = k!δjk. 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)!,
+(A.2)
+using the recurrence relation (A.1).
+21
+
+The Hermite polynomials are given by products of Hermite polynomials, and are indexed by γ ∈
+{0, 1, 2, . . . }d. Let γ = (γ1, . . . , γd), with γj ∈ {0, 1, 2, . . . }. Then
+Hγ(x1, . . . , xd) =
+d
+�
+j=1
+Hγj(xj),
+which has order |γ| := �d
+j=1 γj. Note that if |γ| = k then Hγ(x) is given by a sum of monomials of the
+same parity as k. Indeed, each Hγj(xj) is a linear combination of xγj−2p
+j
+, p ≤ ⌊γj/2⌋. 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. Using the
+independence of the entries of Z = (Z1, . . . , Zd), we have
+E [Hγ(Z)Hγ′(Z)] = γ!
+d
+�
+j=1
+δγj,γ′
+j,
+where γ! := �d
+j=1 γj!. The Hγ can also be defined explicitly as follows:
+e−∥x∥2/2Hγ(x) = (−1)|γ|∂γ �
+e−∥x∥2/2�
+,
+(A.3)
+where ∂γf(x) = ∂γ1
+x1 . . . ∂γd
+xdf(x). This leads to the useful Gaussian integration by parts identity,
+E [f(Z)Hγ(Z)] = E [∂γf(Z)],
+if f ∈ C|γ|(Rd).
+The Hermite polynomials Hγ, γ ∈ {0, 1, . . . }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)]. In particular, if f : Rd → R satisfies
+E [f(Z)2] < ∞, then f has the following Hermite expansion:
+f(x) =
+�
+γ∈{0,1,... }d
+1
+γ!cγ(f)Hγ(x),
+cγ(f) := E [f(Z)Hγ(Z)].
+(A.4)
+Let
+rk(x) = f(x) −
+�
+|γ|≤k−1
+1
+γ!cγ(f)Hγ(x)
+(A.5)
+be the remainder of the Hermite series expansion of f after taking out the order ≤ k − 1 polynomials. We
+can write rk as an integral of f against a kernel. Namely, define
+K(x, y) =
+�
+|γ|≤k−1
+1
+γ!Hγ(x)Hγ(y).
+(A.6)
+Note that
+E [f(Z)K(x, Z)] =
+�
+|γ|≤k−1
+1
+γ!cγ(f)Hγ(x)
+is the truncated Hermite series expansion of f. 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.
+B
+Exact Expression for the Remainder
+Lemma B.1. Let k ≥ 1 and rk, K, be as in (A.5), (A.6), respectively. Assume that f ∈ C1, and that
+∥∇f(x)∥ ≲ ec∥x∥2 for some 0 ≤ c < 1/2. Then
+rk(x) = E [(f(x) − f(Z))K(x, Z)]
+=
+� 1
+0
+d
+�
+i=1
+�
+|γ|=k−1
+1
+γ!E [∂if((1 − t)Z + tx) (Hγ+ei(x)Hγ(Z) − Hγ+ei(Z)Hγ(x))]dt.
+(B.1)
+22
+
+The proof relies on the following identity:
+Lemma B.2. For each i = 1, . . . , d, it holds that
+K(x, y) =
+1
+xi − yi
+�
+|γ|=k−1
+1
+γ! (Hγ+ei(x)Hγ(y) − Hγ+ei(y)Hγ(x)) .
+(B.2)
+The proof of this identity is given at the end of the section.
+Proof of Lemma B.1. 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.3)
+so that, using (B.2), we have
+E [(f(x) − f(Z))K(x, Z)]
+=
+d
+�
+i=1
+�
+|γ|=k−1
+1
+γ!E
+�� 1
+0
+∂if((1 − t)Z + tx) (Hγ+ei(x)Hγ(Z) − Hγ+ei(Z)Hγ(x)) dt
+�
+.
+(B.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.5)
+for some 0 ≤ c < 1/2. The right-hand side is integrable with respect to the Gaussian measure, and therefore
+we can interchange the expectation and the integral in (B.4). Therefore,
+E [(f(x) − f(Z))K(x, Z)]
+=
+� 1
+0
+d
+�
+i=1
+�
+|γ|=k−1
+1
+γ!E [∂if((1 − t)Z + tx) (Hγ+ei(x)Hγ(Z) − Hγ+ei(Z)Hγ(x))]dt.
+(B.6)
+Proof of Lemma B.2. Without loss of generality, assume i = 1. To simplify the proof, we will also assume
+d = 2. The reader can check that the proof goes through in the same way for general d. By the recursion
+relation (A.1) for 1d Hermite polynomials, we get that
+Hγ1+1,γ2(x) = x1Hγ1,γ2(x) − γ1Hγ1−1,γ2(x),
+where x = (x1, x2). 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).
+(B.7)
+Let
+Sγ1,γ2 = Hγ1+1,γ2(x)Hγ1,γ2(y) − Hγ1+1,γ2(y)Hγ1,γ2(x).
+Subtracting the second equation of (B.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.8)
+23
+
+and hence
+Sγ1,γ2
+γ1!γ2! = (x1 − y1)Hγ1,γ2(x)Hγ1,γ2(y)
+γ1!γ2
++
+Sγ1−1,γ2
+(γ1 − 1)!γ2!.
+Iterating this recursive relationship γ1 − 1 times, we get
+Sγ1,γ2
+γ1!γ2! = (x1 − y1)
+γ1−1
+�
+j=0
+Hγ1−j,γ2(x)Hγ1−j,γ2(y)
+(γ1 − j)!γ2!
++ S0,γ2
+0!γ2! .
+(B.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).
+(B.10)
+Therefore,
+S0,γ2
+0!γ2! = (x1 − y1)Hγ1−j,γ2(x)Hγ1−j,γ2(y)
+(γ1 − j)!γ2!
+,
+j = γ1
+so (B.9) can be written as
+Sγ1,γ2
+γ1!γ2! = (x1 − y1)
+γ1
+�
+j=0
+Hγ1−j,γ2(x)Hγ1−j,γ2(y)
+(γ1 − j)!γ2!
+and hence
+1
+x1 − y1
+�
+γ1+γ2=k−1
+Sγ1,γ2
+γ1!γ2! =
+�
+γ1+γ2=k−1
+γ1
+�
+j=0
+Hγ1−j,γ2(x)Hγ1−j,γ2(y)
+(γ1 − j)!γ2!
+=
+�
+γ1+γ2≤k−1
+Hγ1,γ2(x)Hγ1,γ2(y)
+γ1!γ2!
+= K(x, y),
+(B.11)
+using the observation that
+{(γ1 − j, γ2) : γ1 + γ2 = k − 1, 0 ≤ j ≤ γ1}
+= {(˜γ1, γ2) : ˜γ1 + γ2 ≤ k − 1}.
+(B.12)
+Substituting back in the definition of Sγ1,γ2 gives the desired result.
+B.1
+Hermite Series Remainder in Tensor Form
+Using (B.1), it is difficult to obtain an upper bound on |rk(x)|, since we need to sum over all γ of order k −1.
+In this section, we obtain a more compact representation of rk in terms of a scalar product of k-tensors.
+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. 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).
+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, . . . , αk) ∈ [d]k. Here, [d] = {1, . . . , d}. We do so as
+follows: given α ∈ [d]k, define γ(α) = (γ1(α), . . . , γd(α)) by
+γj(α) =
+k
+�
+ℓ=1
+1{αℓ = j},
+i.e. γj(α) counts how many times index j appears in α. For this reason, we use the term counting index to
+denote indices of the form γ = (γ1, . . . , γd) ∈ {0, 1, 2, . . . }d, whereas we use the standard term “multi-index”
+24
+
+to refer to the α’s. Note that we automatically have |γ(α)| = k if α ∈ [d]k. 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.
+When enumerating the entries of Hk, we write H(α)
+k
+to denote Hγ(α). Note that for each γ with |γ| = k,
+there are
+�k
+γ
+�
+α’s such that γ(α) = γ.
+Example B.1. Consider the α = (i, j, j, k, k, k) entry of the tensor H6(x), where i, j, k ∈ [d] are all distinct.
+We count that i occurs once, j occurs twice, and k occurs thrice. Thus
+H(i,j,j,k,k,k)
+6
+(x) = H1(xi)H2(xj)H3(xk) = xi(x2
+j − 1)(x3
+k − 3xk).
+The first two tensors H1, H2 can be written down explicitly. For the entries of H1, we simply have H(i)
+1 (x) =
+H1(xi) = xi, i.e. H1(x) = x. 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. Thus H2(x) = xxT − Id.
+We now group the terms in the Hermite series expansion (A.4) based on the order |γ|. Consider all γ in
+the sum such that |γ| = k. We claim that
+�
+|γ|=k
+1
+γ!cγ(f)Hγ(x) = 1
+k!
+�
+α∈[d]k
+cγ(α)(f)H(α)
+k
+(x).
+(B.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). Thus we obtain
+�k
+γ
+�
+copies of cγ(f)Hγ(x), and it remains to note that
+�k
+γ
+�
+/k! = 1/γ!.
+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)].
+We then see that the sum (B.13) can be written as
+1
+k!⟨ck, Hk(x)⟩, and hence the series expansion of f can
+be written as
+f(x) =
+∞
+�
+k=0
+1
+k!⟨ck(f), Hk(x)⟩,
+ck(f) := E [f(Z)Hk(Z)].
+(B.14)
+The main result of this section is Lemma B.4 below, in which we express rk in terms of a tensor scalar
+product. 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 (4.3).
+Lemma B.3. Let cp, cp+1 be symmetric tensors in (Rd)p and (Rd)p+1, respectively.Then
+⟨u ⊗ cp ⊗ cp+1, E [Z ⊗ Hp(Z) ⊗ Hp+1(Z)]⟩ = (p + 1)!⟨u ⊗ cp, cp+1⟩.
+(B.15)
+Proof. Let T = E [Z ⊗ Hp ⊗ Hp+1]. First ,we characterize the non-zero entries of T using the counting index
+notation. 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. 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!
+(B.16)
+For this to be nonzero, we must have γj = γj′ for all j ̸= i. But since |γ| = p and |γ′| = p + 1, it follows that
+we must have γ′
+i = γi + 1.Hence γ′ = γ + ei, where ei is the ith unit vector. To summarize, Ti,γ,γ′ is only
+25
+
+nonzero if γ′ = γ + ei. In this case, we have
+E [ZiHγ(Z)Hγ+ei(Z)] = E [ZiHγi(Zi)Hγi+1(Zi)]
+�
+j̸=i
+γj!
+= (γi + 1)!
+�
+j̸=i
+γj! = (γ + ei)!.
+(B.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. Now, we take the inner product (B.15) using counting index notation,
+recalling that each γ such that |γ| = p shows up in the tensor Hp exactly p!/γ! times:
+�
+u ⊗ cp ⊗ cp+1, E [Z ⊗ Hp(Z) ⊗ Hp+1(Z)]
+�
+=
+d
+�
+i=1
+�
+|γ|=p
+�
+|γ′|=p+1
+p!
+γ!
+(p + 1)!
+(γ′)!
+uicγcγ′E [ZiHγ(Z)Hγ′(Z)]
+=
+d
+�
+i=1
+�
+|γ|=p
+p!
+γ!
+(p + 1)!
+(γ + ei)!uicγcγ+ei(γ + ei)!
+= (p + 1)!
+d
+�
+i=1
+�
+|γ|=p
+p!
+γ!uicγcγ+ei
+= (p + 1)!
+d
+�
+i=1
+d
+�
+j1,...,jp=1
+uicγ(j1,...,jp)cγ(i,j1,...,jp)
+= (p + 1)!
+d
+�
+i=1
+d
+�
+j1,...,jp=1
+uic(j1,...,jp)
+p
+c(i,j1,...,jp)
+p+1
+= ⟨u ⊗ cp, cp+1⟩
+(B.18)
+Lemma B.4. Let f satisfy the assumptions of Lemma B.1, and additionally, assume f ∈ Ck. Then the
+remainder rk, given as (B.1) in Lemma B.1, can also be written in the form
+rk(x) =
+� 1
+0
+(1 − t)k−1
+(k − 1)! E
+��
+∇kf ((1 − t)Z + tx) , Hk(x) − Z ⊗ Hk−1(x)
+��
+(B.19)
+Proof. Recall that ∂γ := ∂γ1
+1 . . . ∂γd
+d , and that
+Hγ(z)e−∥z∥2/2 = (−1)|γ|∂γ(e−∥z∥2/2).
+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.20)
+using the fact that f ∈ Ck. We omitted the argument (1 − t)Z + tx from the right-hand side for brevity.
+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. Substituting these two equations into (B.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!E [(∂γ+eif) (Hγ+ei(x) − ZiHγ(x))]dt
+=
+1
+(k − 1)!
+� 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.21)
+26
+
+Now, define the sets
+A = {(i, γ + ei) : i = 1, . . . , d, γ ∈ {0, 1, . . . }d, |γ| = k − 1},
+B = {(i, ˜γ) : ˜γ ∈ {0, 1, . . . }d, |˜γ| = k, ˜γi ≥ 1}.
+(B.22)
+It is straightforward to see that A = B. 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.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)⟩].
+(B.24)
+Substituting (B.23) and (B.24) into (B.21) gives
+rk(x) = E [(f(x) − f(Z))K(x, Z)]
+=
+� 1
+0
+(1 − t)k−1
+(k − 1)! E
+��
+∇kf ((1 − t)Z + tx) , Hk(x) − Z ⊗ Hk−1(x)
+��
+(B.25)
+In the next section, we obtain a pointwise upper bound on |rk(x)| in the case f = ¯W. In order for this
+bound to be tight in its dependence on d, we need a supplementary result on inner products with Hermite
+tensors. To motivate this supplementary result, consider bounding the inner product in (B.19) by the product
+of the Frobenius norms of the tensors on either side. As a rough heuristic, ∥∇kf∥F ∼ dk/2∥∇kf∥, where
+recall that ∥∇kf∥ is the operator norm of ∇kf. Therefore, we would prefer to bound the inner product in
+terms of ∥∇kf∥ to get a tighter dependence on d. Apriori, however, this seems impossible, since Hk(x) is not
+given by an outer product of k vectors. But the following representation of the order k Hermite polynomials
+will make this possible.
+Hk(x) = E [(x + iZ)⊗k],
+(B.26)
+where Z ∼ N(0, Id). Using (B.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. More generally, we have the following lemma.
+Lemma B.5. Let T ∈ (Rd)⊗k be a k-tensor, and v ∈ Rd. Then for all 0 ≤ ℓ ≤ k, we have
+|⟨T, v⊗ℓ ⊗ Hk−ℓ(x)⟩| ≲ ∥T∥∥v∥ℓ(∥x∥k−ℓ + d
+k−ℓ
+2 ).
+Proof. Using (B.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).
+(B.27)
+B.2
+Hermite-Related Proofs from Section 4.2
+In this section, we return to the setting in the main text.
+We let W satisfy all the assumptions from
+Section 3.2, m ∈ Rd, σ ∈ Rd×d be such that ∥m∥, ∥σ∥ ≲ 1, and ¯W(x) = W(m + σx). Also, let
+rk(x) = ¯W(x) −
+k−1
+�
+j=0
+1
+j!
+�
+cj( ¯W), Hj(x)
+�
+,
+(B.28)
+where cj( ¯W) = E [ ¯W(Z)Hj(Z)] as usual.
+Combining Lemmas B.4 and B.5 allows us to upper bound quantities of the form E [|rk(Y )|p] in terms of
+the operator norm of ∇kW.
+Corollary B.1. Let rk be as in (B.28), the remainder of the Hermite series expansion of ¯W, where k = 3
+if W ∈ C3 and k = 4 if W ∈ C4. Let Y ∈ Rd be a random variable such that E [∥Y ∥s] < ∞ for all
+0 ≤ s ≤ 2pk + 2pq, where q is from Assumption W2. 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.29)
+Proof. Let ∇k ¯W be shorthand for ∇k ¯W((1 − t)Z + tY ). Using (B.19) for f = ¯W, we have
+|rk(Y )| ≲
+� 1
+0
+E Z
+����
+∇k ¯W, Hk(Y ) − Z ⊗ Hk−1(Y )
+����
+dt.
+(B.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.
+(B.31)
+We now take the Y -expectation of both sides, and we are free to assume Y is independent of Z. 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 . Since E [∥Y ∥pq+pk] < ∞ by assumption, we can bring the
+Y -expectation inside the integral. Hence
+E [|rk(Y )|p] ≲
+� 1
+0
+E
+����
+∇k ¯W, Hk(Y ) − Z ⊗ Hk−1(Y )
+���p�
+dt,
+(B.32)
+where the expectation is over both Z and Y . Next, using Lemma B.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
+�
+.
+(B.33)
+28
+
+Substituting this into (B.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.
+(B.34)
+We used Cauchy-Schwarz and the independence of Y and Z to get the second line. 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 )∥.
+We now apply Lemma 3.4 with Y = m + (1 − t)σZ + tσY . 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.35)
+for all t ∈ [0, 1]. Combining this inequality with (B.34) and noting that dkp/2N p(1−k/2) = (dk/N k−2)p/2
+gives (B.29).
+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.1)
+where m∗ = argminx∈Rd V (x) and H = ∇2V (m∗).
+Lemma C.1. Let V2(x) = V1(Ax + b) for some A ∈ Rd×d invertible and b ∈ Rd. 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.2)
+is a unique solution to (EV2) in the set RV2.
+Proof. It suffices to prove the following two statements. (1) If (m1, S1) ∈ RV1 solves (EV1) then (m2, S2)
+given by (C.2) lies in RV2 and solves (EV2). (2) If (m2, S2) ∈ RV2 solves (EV2) then (m1, S1) given by
+m1 = Am2 + b, S1 = AS2AT lies in RV1 and solves (EV1).
+29
+
+We prove the first statement, and the second follows by a symmetric argument. So let (m1, S1) ∈ RV1
+solve (EV1). We first show (m2, S2) given by (C.2) solves (EV2). We have
+∇V2(x) = AT ∇V1(Ax + b),
+∇2V2(x) = AT ∇2V1(Ax + b)A.
+(C.3)
+Note also that if σ = A−1S1/2
+1
+then σσT = A−1S1A−T . 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.
+(C.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 .
+(C.5)
+To conclude, we show (m2, S2) ∈ RV2. Let m∗i be the global minimizer of Vi and Hi = ∇2V (m∗i), i = 1, 2.
+Then m∗2 = A−1(m∗1 − b) and H2 = AT H1A. Since S1 ⪯ 2H−1
+1 , it follows that
+S2 = A−1S1A−T ⪯ 2A−1H−1
+1 A−T = 2H−1
+2 .
+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.
+(C.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.
+(C.7)
+Therefore,
+∥
+�
+H2
+�
+S2∥2 + ∥
+√
+H2(m2 − m∗2)∥2 = ∥
+�
+H1
+�
+S1∥2 + ∥
+√
+H1(m1 − m∗1)∥2 ≤ 8.
+Recall that
+W(x) = nv
+�
+m∗ +
+√
+nH
+−1x
+�
+,
+H = ∇2v(m∗),
+30
+
+and that N = nr, where r is from Assumption V1. The following preliminary calculation will be useful for
+showing Assumptions V1, V2 imply Assumptions W1, W2, respectively. Given x ∈ Rd, let
+y = √α2
+√
+H
+−1x.
+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
+����� .
+(C.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
+����� .
+(C.9)
+Lemma C.2. Assumptions V1, V2, and V3 imply Assumptions W1, W2, and W3 with N = nr, where r is
+from Assumption V1.
+Proof. Let y = √α2
+√
+H
+−1x. Note that ∥y∥ ≤ ∥x∥ and in particular, if ∥x∥ ≤ 1 then ∥y∥ ≤ 1. 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.10)
+as desired. 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.11)
+as desired. The calculation for the fourth derivative is analogous.
+To show that Assumption V3 implies W3, fix ∥x∥ ≥
+√
+N and let y =
+√
+nH
+−1x, so that W(x) =
+nv(y + m∗). Note that ∥y∥ ≥ ∥x∥/√nβ2 ≥
+�
+N/(nβ2) =
+�
+r/β2. 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∥.
+(C.12)
+as desired.
+D
+Logistic Regression Example
+Details of Numerical Simulation
+For the numerical simulation displayed in Figure 1, we take d = 2 and n = 100, 200, . . . , 1000. For each
+n, we draw ten sets of covariates xi, i = 1, . . . , n from N(0, λ2Id) with λ =
+√
+5, yielding ten posterior
+31
+
+distributions πn(· | x1:n).
+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). 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]. We used the authors’ implementation of this algorithm, found at
+https://github.com/marc-h-lambert/W-VI. 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. The solid lines in Figure 1 depict
+the average approximation errors over the ten distributions at each n. The shaded regions depict the spread
+of the middle eight out of ten approximation errors.
+Verifying the Assumptions
+As discussed in Section 2.3, we make the approximation
+v(z) ≈ v∞(z) = −E [Y log s(XT z) + (1 − Y ) log(1 − s(XT z)].
+Here, X ∼ N(0, λ2Id) and Y | X ∼ Bernoulli(s(X1)), since X1 = eT
+1 X.
+Recall that s is the sigmoid,
+s(a) = (1 + ea)−1. Below, the parameters α2, β2, etc. are all computed for the function v∞.
+Note that z = e1 is the global minimizer of v∞. We have ∇2v∞(z) = E [s′(XT z)XXT ] and in particular,
+∇2v∞(e1) = E [s′(X1)XXT ]. Also,
+s′(a) = s(a)(1 − σ(a)) =
+1
+2(1 + cosh(a)) ∈ (0, 1/4].
+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.1)
+and hence α2 ≳ λ2s′(λ). Using that s′ ≤ 1/4, we also have the upper bound
+λmax(∇2v∞(e1)) ≤ λ2
+4 = β2.
+(D.2)
+Next, we need to upper bound ∥∇3v∞∥. 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)
+�
+.
+One can show that s′′(a) ∈ [−1, 1] for all a ∈ R. 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.
+(D.3)
+Here, we used that uT
+k X
+d= N(0, λ2), whose third absolute moment is bounded by 2λ3. We therefore get the
+bound
+∥∇3v∞(z)∥ ≤ β3 := 2λ3.
+(D.4)
+32
+
+Note that this constant bound holds for all z ∈ Rd. Next, we need to find r such that
+sup
+∥z−m∗∥≤√
+r/α2
+∥∇3v∞(z)∥ ≤ α3/2
+2
+2√r .
+Using the uniform bound (D.4) on ∥∇3v∞∥, it suffices to take r such that
+β3 = α3/2
+2
+2√r =⇒ r = α3
+2
+4β2
+3
+≳ s′(λ)3.
+(D.5)
+Finally, we verify Assumption V3. To do so, recall that v∞ is convex. 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∥.
+(D.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. Therefore, since ∥z∥ ≥ log ∥z∥, Assumption V3 is satisfied.
+E
+Proofs from Section 5
+The proofs in this section rely on tensor-matrix and tensor-vector scalar products. Let us review the rules
+of such scalar products, and how to bound the operator norms of these quantities. Let v ∈ Rd, A ∈ Rd×d,
+and T ∈ Rd×d×d. 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, . . . , d,
+⟨T, v⟩ij =
+d
+�
+k=1
+Tijkvk,
+i, j = 1, . . . , d.
+(E.1)
+We will always sum over the last two or last one indices of the tensor. 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∥.
+Therefore, ∥⟨T, v⟩∥ ≤ ∥T∥∥v∥.
+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. Specifically, consider a C2 function
+f = (f1, . . . , fd) : Rd×d ×Rd → Rd, where Rd×d is endowed with the standard matrix norm. 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. 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. However, ∥∇σf∥op is not the
+standard tensor operator norm. Rather,
+∥∇σf∥op =
+sup
+A∈Rd×d,∥A∥=1
+∥⟨∇σf, A⟩∥ =
+sup
+A∈Rd×d,∥A∥=1,
+u∈Rd,∥u∥=1
+⟨∇σf, A ⊗ u⟩.
+We continue to write ∥∇σf∥ to denote the standard tensor operator norm, i.e.
+∥∇σf∥ =
+sup
+u,v,w∈Rd,
+∥u∥=∥v∥=∥w∥=1
+⟨∇σf, u ⊗ v ⊗ w⟩.
+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.
+(E.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}.
+(E.3)
+Lemma E.1. Let f = (f1, . . . , fd) : Rd×d×Rd → Rd be C3, where Rd×d is the set of d×d matrices, endowed
+with the standard matrix operator norm. Suppose f(0, 0) = 0, ∇σf(0, 0) = 0, ∇mf(σ, m) is symmetric for
+all m, and ∇mf(0, 0) = Id. Let r > 0 be such that
+sup
+(σ,m)∈Br(0,0)
+∥∇f(σ, m) − ∇f(0, 0)∥op ≤ 1
+4.
+(E.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). Furthermore, the map σ �→ m(σ) is C2, with
+1
+2Id ⪯ ∇mf(σ, m)
+��
+m=m(σ) ⪯ 3
+2Id,
+∥∇σm(σ)∥op ≤ 1.
+(E.5)
+The proof uses the following lemma.
+Lemma E.2 (Lemma 1.3 in Chapter XIV of [Lan93]). Let U be open in a Banach space E, and let f : U → E
+be of class C1. Assume that f(0) = 0 and f ′(0) = I. Let r > 0 be such that ¯Br(0) ⊂ U. 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).
+Proof of Lemma E.1. 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)
+�
+.
+(E.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.
+(E.7)
+Note also that ∇φ(0, 0) is the identity. Thus by Lemma E.2, we have that φ is a bijection from Br(0, 0)
+to Br/2(φ(0, 0)) = Br/2(0, 0). Now, fix any σ ∈ Rd×d such that ∥σ∥ ≤ r/2. 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)). Thus σ = σ′ and
+f(σ, m) = 0. 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).
+The map σ �→ m(σ) is C2 by standard Implicit Function Theorem arguments. To show that the first
+inequality of (E.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.4) since we know that (σ, m(σ)) ∈ Br(0, 0). Thus,
+Id = ∇2W(0) = ∇mf(0, 0)
+=⇒ 1
+2Id ⪯ ∇mf(σ, m(σ)) ⪯ 3
+2Id.
+(E.8)
+To show the second inequality of (E.5), we first need the supplementary bound
+∥∇σf(σ, m(σ))∥op = ∥∇σf(σ, m(σ)) − ∇σf(0, 0)∥op ≤ 1/2
+(E.9)
+which holds by the same reasoning as above. 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. Hence by the first inequality in (E.5) combined with (E.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.
+(E.10)
+Lemma E.3. Let f : Rd×d × Rd → Rd be given by f(σ, m) = E [∇W(σZ + m)]. Then all the conditions of
+Lemma E.1 are satisfied; in particular, (E.4) is satisfied with r = 2
+√
+2. Thus the conclusions of Lemma E.1
+hold with this choice of r.
+Proof. Note that f is C2 thanks to the fact that W is C3 and ∇W grows polynomially by Assumption W2.
+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. 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. 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.
+(E.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.
+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.12)
+35
+
+each a vector in Rd. From the first line, we get that ∥∇2
+mf(σ, m)∥op ≤ E ∥∇3W(m + σZ)∥, where ∥∇3W∥
+is the standard tensor norm. 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].
+(E.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.
+(E.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.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)!),
+(E.16)
+where in the last line we applied Lemma E.5 and substituted r = 2
+√
+2. To conclude, recall that (
+√
+3 +
+�
+(2q)!)/
+√
+N ≤ 1/(40
+√
+2d) by the assumption in the statement of Lemma 5.1.
+Lemma E.4. 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 5.2 and 5.3 . 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. Moreover,
+F(S0,r/2) ⊆ Sc1,c2 ⊆ S0,r/2,
+where c1 =
+�
+2/3, c2 =
+√
+2 = r/2.
+Proof. First, let G(σ) = E [∇2W(m(σ) + σZ)] and f(σ, m) = E [∇W(m + σZ)] as in Lemma E.3. Note that
+∇mf(σ, m) = E [∇2W(σZ + m)], so that G(σ) = ∇mf(σ, m)|m=m(σ) and hence by (E.5) of Lemma E.1 we
+have
+1
+2Id ⪯ G(σ) ⪯ 3
+2Id,
+∀σ ∈ S0,r/2.
+(E.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. Moreover, using (E.17), it follows that
+c1Id ⪯ F(σ) ⪯ c2Id,
+∀σ ∈ S0,r/2,
+where c1 =
+�
+2/3 and c2 =
+√
+2 = r/2. In other words, F(S0,r/2) ⊆ Sc1,c2 ⊆ S0,r/2. It remains to show F is
+a contraction on S0,r/2. Let σ1, σ2 ∈ S0,r/2. We will first bound ∥G(σ1) − G(σ2)∥. We have
+∥G(σ1) − G(σ2)∥ ≤ ∥σ1 − σ2∥
+sup
+σ∈S0,r/2
+∥∇σG(σ)∥op,
+(E.18)
+and
+∥∇σG(σ)∥op = sup
+∥A∥=1
+���
+∇σG(σ), A
+���
+= sup
+∥A∥=1
+����E
+�
+∇3W,
+�
+A, ∇σ (m(σ) + σZ)
+������.
+(E.19)
+Here, the quantities inside of the ∥ · ∥ on the right are matrices. Indeed, ⟨∇σG, A⟩ denotes the application
+of ∇σG to A. Since G sends matrices to matrices, ∇σG is a linear functional which also sends matrices to
+matrices. 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. 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∥).
+(E.20)
+To get the last bound, we used that ∥∇σm(σ)∥op ≤ 1, shown in Lemma E.3. We also use the fact that
+∥Z ⊗ Id∥op = sup∥A∥=1 ∥ ⟨A, Z ⊗ Id⟩ ∥ = sup∥A∥=1 ∥AZ∥ = ∥Z∥. (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.)
+Substituting (E.20) back into (E.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)!
+√
+N
+≤ ∥σ1 − σ2∥1 +
+√
+d
+40d
+.
+(E.21)
+The second inequality is by Cauchy-Schwarz and Lemma E.5 below. The third inequality uses that (
+√
+3 +
+�
+(2q)!)/
+√
+N ≤ 1/(40
+√
+2d), by the assumption in the statement of Lemma 5.1. Now, note that thanks to
+Lemma E.3, both λmin(G(σ1)) and λmin(G(σ2)) are bounded below by 1/2. Using Lemma E.6, we therefore
+have
+∥F(σ1) − F(σ2)∥ ≤
+√
+2∥G(σ1) − G(σ2)∥
+≤ 1 +
+√
+d
+20
+√
+2d ∥σ1 − σ2∥.
+(E.22)
+Hence F is a strict contraction.
+Lemma E.5. Assume 4
+√
+2 ≤
+�
+N/d. Then
+sup
+(σ,m)∈B2
+√
+2(0,0)
+E [∥∇3W(m + σZ)∥2] ≤ (3 + (2q)!)/N.
+37
+
+Proof. Fix ∥m∥, ∥σ∥ ≤ 2
+√
+2, so that 2∥m∥ ≤
+√
+N and 2∥σ∥
+√
+d ≤
+√
+N. 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)!
+≤ 3 + (2q)!.
+(E.23)
+Lemma E.6. Let A0 and A1 be psd, and A1/2
+0
+, A1/2
+1
+their unique psd square roots. Assume without loss of
+generality that λmin(A0) ≤ λmin(A1). Then
+∥A−1/2
+1
+− A−1/2
+0
+∥ ≤
+∥A1 − A0∥
+2λmin(A0)3/2 .
+Proof. 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)
+.
+Now, define At = A0 + t(A1 − A0) and let Bt = A1/2
+t
+, where Bt is the unique psd square root of At. We
+then have ∥A1/2
+1
+− A1/2
+0
+∥ ≤ supt∈[0,1] ∥ ˙Bt∥. We will now express ˙Bt in terms of ˙At and Bt. Differentiating
+B2
+t = At, we get
+Bt ˙Bt + ˙BtBt = ˙At = A1 − A0.
+(E.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)
+.
+Now note that λmin(At) ≥ λmin(A0), since At is just a convex combination of A0 and A1. Hence ∥ ˙Bt∥ ≤
+∥A1 − A0∥/2
+�
+λmin(A0) for all t ∈ [0, 1]. Combining all of the above estimates gives
+∥A−1/2
+1
+− A−1/2
+0
+∥ ≤
+∥A1 − A0∥
+2λmin(A0)3/2 .
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