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+Theoretical Characterization of How Neural Network
+Pruning Affects its Generalization
+Hongru Yang ∗
+Yingbin Liang †
+Xiaojie Guo‡
+Lingfei Wu§
+Zhangyang Wang ¶
+Abstract
+It has been observed in practice that applying pruning-at-initialization methods to neural
+networks and training the sparsified networks can not only retain the testing performance of the
+original dense models, but also sometimes even slightly boost the generalization performance.
+Theoretical understanding for such experimental observations are yet to be developed. This
+work makes the first attempt to study how different pruning fractions affect the model’s gradient
+descent dynamics and generalization. Specifically, this work considers a classification task for
+overparameterized two-layer neural networks, where the network is randomly pruned according
+to different rates at the initialization. It is shown that as long as the pruning fraction is below
+a certain threshold, gradient descent can drive the training loss toward zero and the network
+exhibits good generalization performance.
+More surprisingly, the generalization bound gets
+better as the pruning fraction gets larger. To complement this positive result, this work further
+shows a negative result: there exists a large pruning fraction such that while gradient descent
+is still able to drive the training loss toward zero (by memorizing noise), the generalization
+performance is no better than random guessing. This further suggests that pruning can change
+the feature learning process, which leads to the performance drop of the pruned neural network.
+Up to our knowledge, this is the first generalization result for pruned neural networks, suggesting
+that pruning can improve the neural network’s generalization.
+1
+Introduction
+Neural network pruning can be dated back to the early stage of the development of neural networks
+(LeCun et al., 1989). Since then, many research works have been focusing on using neural network
+pruning as a model compression technique, e.g. (Molchanov et al., 2019; Luo and Wu, 2017; Ye
+et al., 2020; Yang et al., 2021). However, all these work focused on pruning neural networks after
+training to reduce inference time, and, thus, the efficiency gain from pruning cannot be directly
+transferred to the training phase. It is not until the recent days that Frankle and Carbin (2018)
+showed a surprising phenomenon: a neural network pruned at the initialization can be trained to
+achieve competitive performance to the dense model. They called this phenomenon the lottery
+ticket hypothesis. The lottery ticket hypothesis states that there exists a sparse subnetwork inside
+∗Department of Computer Science, The University of Texas at Austin; e-mail: hy6385@utexas.edu
+†Department of Electrical and Computer Engineering, The Ohio State University; e-mail: liang.889@osu.edu
+‡IBM Thomas.J. Watson Research Center; e-mail: xguo7@gmu.edu
+§Pinterest; e-mail: lwu@email.wm.edu
+¶Department
+of
+Electrical
+and
+Computer
+Engineering,
+The
+University
+of
+Texas
+at
+Austin;
+e-mail:
+atlaswang@utexas.edu
+1
+arXiv:2301.00335v1  [cs.LG]  1 Jan 2023
+
+a dense network at the random initialization stage such that when trained in isolation, it can
+match the test accuracy of the original dense network after training for at most the same number
+of iterations. On the other hand, the algorithm Frankle and Carbin (2018) proposed to find the
+lottery ticket requires many rounds of pruning and retraining which is computationally expensive.
+Many subsequent works focused on developing new methods to reduce the cost of finding such a
+network at the initialization (Lee et al., 2018; Wang et al., 2019; Tanaka et al., 2020; Liu and Zenke,
+2020; Chen et al., 2021a). A further investigation by Frankle et al. (2020) showed that some of
+these methods merely discover the layer-wise pruning ratio instead of sparsity pattern.
+The discovery of the lottery ticket hypothesis sparkled further interest in understanding this
+phenomenon. Another line of research focused on finding a subnetwork inside a dense network
+at the random initialization such that the subnetwork can achieve good performance (Zhou et al.,
+2019; Ramanujan et al., 2020). Shortly after that, Malach et al. (2020) formalized this phenomenon
+which they called the strong lottery ticket hypothesis: under certain assumption on the weight ini-
+tialization distribution, a sufficiently overparameterized neural network at the initialization contains
+a subnetwork with roughly the same accuracy as the target network. Later, Pensia et al. (2020)
+improved the overparameterization parameters and Sreenivasan et al. (2021) showed that such a
+type of result holds even if the weight is binary. Unsurprisingly, as it was pointed out by Malach
+et al. (2020), finding such a subnetwork is computationally hard. Nonetheless, all of the analysis is
+from a function approximation perspective and none of the aforementioned works have considered
+the effect of pruning on gradient descent dynamics, let alone the neural networks’ generalization.
+Interestingly, via empirical experiments, people have found that sparsity can further improve
+generalization in certain scenarios (Chen et al., 2021b; Ding et al., 2021; He et al., 2022). There
+have also been empirical works showing that random pruning can be effective (Frankle et al., 2020;
+Su et al., 2020; Liu et al., 2021b). However, theoretical understanding of such benefit of pruning
+of neural networks is still limited. In this work, we take the first step to answer the following
+important open question from a theoretical perspective:
+How does pruning fraction affect the training dynamics and the model’s generalization,
+if the model is pruned at the initialization and trained by gradient descent?
+We study this question using random pruning. We consider a classification task where the input
+data consists of class-dependent sparse signal and random noise. We analyze the training dynamics
+of a two-layer convolutional neural network pruned at the initialization. Specifically, this work
+makes the following contributions:
+• Mild pruning. We prove that there indeed exists a range of pruning fraction where the
+pruning fraction is small and the generalization error bound gets better as pruning fraction
+gets larger. In this case, the signal in the feature is well-preserved and due to the effect of
+pruning purifying the feature, the effect from noise is reduced. We provide detailed explana-
+tion in Section 3. Up to our knowledge, this is the first theoretical result on generalization for
+pruned neural networks, which suggests that pruning can improve generalization under some
+setting. Further, we conduct experiments to verify our results.
+• Over pruning. To complement the above positive result, we also show a negative result: if
+the pruning fraction is larger than a certain threshold, then the generalization performance
+is no better than a simple random guessing, although gradient descent is still able to drive
+the training loss toward zero. This further suggests that the performance drop of the pruned
+2
+
+Probability Density
+Signal Strength
+μ
+Mild Pruning
+Full model
+Over Pruning
+Figure 1: A pictorial demonstration of our results. The bell-shaped curves model the distribution of
+the signal in the features, where the mean represents the signal strength and the width of the curve
+indicates the variance of noise. Our results show that mild pruning preserves the signal strength
+and reduces the noise variance (and hence yields better generalization), whereas over pruning lowers
+signal strength albeit reducing noise variance.
+neural network is not solely caused by the pruned network’s own lack of trainability or ex-
+pressiveness, but also by the change of gradient descent dynamics due to pruning.
+• Technically, we develop novel analysis to bound pruning effect to weight-noise and weight-
+signal correlation.
+Further, in contrast to many previous works that considered only the
+binary case, our analysis handles multi-class classification with general cross-entropy loss.
+Here, a key technical development is a gradient upper bound for multi-class cross-entropy
+loss, which might be of independent interest.
+Pictorially, our result is summarized in Figure 1. We point out that the neural network training we
+consider is in the feature learning regime, where the weight parameters can go far away from their
+initialization. This is fundamentally different from the popular neural tangent kernel regime,
+where the neural networks essentially behave similar to its linearization.
+1.1
+Related Works
+The Lottery Ticket Hypothesis and Sparse Training. The discovery of the lottery ticket
+hypothesis (Frankle and Carbin, 2018) has inspired further investigation and applications. One line
+of research has focused on developing computationally efficient methods to enable sparse training:
+the static sparse training methods are aiming at identifying a sparse mask at the initialization stage
+based on different criterion such as SNIP (loss-based) (Lee et al., 2018), GraSP (gradient-based)
+(Wang et al., 2019), SynFlow (synaptic strength-based) (Tanaka et al., 2020), neural tangent kernel
+based method (Liu and Zenke, 2020) and one-shot pruning (Chen et al., 2021a). Random pruning
+has also been considered in static sparse training such as uniform pruning (Mariet and Sra, 2015;
+He et al., 2017; Gale et al., 2019; Suau et al., 2018), non-uniform pruning (Mocanu et al., 2016),
+expander-graph-related techniques (Prabhu et al., 2018; Kepner and Robinett, 2019) Erd¨os-R´enyi
+(Mocanu et al., 2018) and Erd¨os-R´enyi-Kernel (Evci et al., 2020). On the other hand, dynamic
+sparse training allows the sparse mask to be updated (Mocanu et al., 2018; Mostafa and Wang,
+2019; Evci et al., 2020; Jayakumar et al., 2020; Liu et al., 2021c,d,a; Peste et al., 2021). The sparsity
+pattern can also be learned by using sparsity-inducing regularizer (Yang et al., 2020). Recently, He
+et al. (2022) discovered that pruning can exhibit a double descent phenomenon when the data-set
+labels are corrupted.
+Another line of research has focused on studying pruning the neural networks at its random
+initialization to achieve good performance (Zhou et al., 2019; Ramanujan et al., 2020). In particular,
+3
+
+Ramanujan et al. (2020) showed that it is possible to prune a randomly initialized wide ResNet-50
+to match the performance of a ResNet-34 trained on ImageNet. This phenomenon is named the
+strong lottery ticket hypothesis. Later, Malach et al. (2020) proved that under certain assumption
+on the initialization distribution, a target network of width d and depth l can be approximated by
+pruning a randomly initialized network that is of a polynomial factor (in d, l) wider and twice deeper
+even without any further training. However finding such a network is computationally hard, which
+can be shown by reducing the pruning problem to optimizing a neural network. Later, Pensia et al.
+(2020) improved the widening factor to being logarithmic and Sreenivasan et al. (2021) proved that
+with a polylogarithmic widening factor, such a result holds even if the network weight is binary. A
+follow-up work shows that it is possible to find a subnetwork achieving good performance at the
+initialization and then fine-tune (Sreenivasan et al., 2022). Our work, on the other hand, analyzes
+the gradient descent dynamics of a pruned neural network and its generalization after training.
+Analyses of Training Neural Networks by Gradient Descent. A series of work (Allen-
+Zhu et al., 2019; Du et al., 2019; Lee et al., 2019; Zou et al., 2020; Zou and Gu, 2019; Ji and
+Telgarsky, 2019; Chen et al., 2020b; Song and Yang, 2019; Oymak and Soltanolkotabi, 2020) has
+proved that if a deep neural network is wide enough, then (stochastic) gradient descent provably can
+drive the training loss toward zero in a fast rate based on neural tangent kernel (NTK) (Jacot et al.,
+2018). Further, under certain assumption on the data, the learned network is able to generalize
+(Cao and Gu, 2019; Arora et al., 2019). However, as it is pointed out by Chizat et al. (2019), in the
+NTK regime, the gradient descent dynamics of the neural network essentially behaves similarly to
+its linearization and the learned weight is not far away from the initialization, which prohibits the
+network from performing any useful feature learning. In order to go beyond NTK regime, one line
+of research has focused on the mean field limit (Song et al., 2018; Chizat and Bach, 2018; Rotskoff
+and Vanden-Eijnden, 2018; Wei et al., 2019; Chen et al., 2020a; Sirignano and Spiliopoulos, 2020;
+Fang et al., 2021). Recently, people have started to study the neural network training dynamics in
+the feature learning regime where data from different class is defined by a set of class-related signals
+which are low rank (Allen-Zhu and Li, 2020, 2022; Cao et al., 2022; Shi et al., 2021; Telgarsky,
+2022). However, all previous works did not consider the effect of pruning. Our work also focuses
+on the aforementioned feature learning regime, but for the first time characterizes the impact of
+pruning on the generalization performance of neural networks.
+2
+Preliminaries and Problem Formulation
+In this section, we introduce our notation, data generation process, neural network architecture
+and the optimization algorithm.
+Notations. We use lower case letters to denote scalars and boldface letters and symbols (e.g.
+x) to denote vectors and matrices. We use ⊙ to denote element-wise product. For an integer n, we
+use [n] to denote the set of integers {1, 2, . . . , n}. We use x = O(y), x = Ω(y), x = Θ(y) to denote
+that there exists a constant C such that x ≤ Cy, x ≥ Cy, x = Cy respectively. We use �O, �Ω and
+�Θ to hide polylogarithmic factor in these notations.
+Finally, we use x = poly(y) if x = O(yC) for
+some positive constant C, and x = poly log y if x = poly(log y).
+2.1
+Settings
+Definition 2.1 (Data distribution of K classes). Consider we are given the set of signal vectors
+{µei}K
+i=1, where µ > 0 denotes the strength of the signal, and ei denotes the i-th standard basis
+4
+
+vector with its i-th entry being 1 and all other coordinates being 0. Each data point (x, y) with
+x = [x⊤
+1 , x⊤
+2 ]⊤ ∈ R2d and y ∈ [K] is generated from the following distribution D:
+1. The label y is generated from a uniform distribution over [K].
+2. A noise vector ξ is generated from the Gaussian distribution N(0, σ2
+nI).
+3. With probability 1/2, assign x1 = µy, x2 = ξ; with probability 1/2, assign x2 = µy, x1 = ξ
+where µy = µey.
+The sparse signal model is motivated by the empirical observation that during the process of
+training neural networks, the output of each layer of ReLU is usually sparse instead of dense. This
+is partially due to the fact that in practice the bias term in the linear layer is used (Song et al.,
+2021). For samples from different classes, usually a different set of neurons fire. Our study can be
+seen as a formal analysis on pruning the second last layer of a deep neural network in the layer-
+peeled model as in Zhu et al. (2021); Zhou et al. (2022). We also point out that our assumption on
+the sparsity of the signal is necessary for our analysis. If we don’t have this sparsity assumption
+and only make assumption on the ℓ2 norm of the signal, then in the extreme case, the signal is
+uniformly distributed across all coordinate and the effect of pruning to the signal and the noise will
+be essentially the same: their ℓ2 norm will both be reduced by a factor of √p.
+Network architecture and random pruning. We consider a two-layer convolutional neural
+network model with polynomial ReLU activation σ(z) = (max{0, z})q, where we focus on the case
+when q = 3 1 The network is pruned at the initialization by mask M where each entry in the mask
+M is generated i.i.d. from Bernoulli(p). Let mj,r denotes the r-th row of Mj. Given the data (x, y),
+the output of the neural network can be written as F(W ⊙ M, x) = (F1(W1 ⊙ M1, x), F2(W2 ⊙
+M2, x), . . . , Fk(Wk ⊙ Mk, x)) where the j-th output is given by
+Fj(Wj ⊙ Mj, x) =
+m
+�
+r=1
+[σ(⟨wj,r ⊙ mj,r, x1⟩) + σ(⟨wj,r ⊙ mj,r, x2⟩)]
+=
+m
+�
+r=1
+[σ(⟨wj,r ⊙ mj,r, µ⟩) + σ(⟨wj,r ⊙ mj,r, ξ⟩)].
+The mask M is only sampled once at the initialization and remains fixed through the entire training
+process. From now on, we use tilde over a symbol to denote its masked version, e.g.,
+�
+W = W ⊙ M and �wj,r = wj,r ⊙ mj,r.
+Since µj ⊙ mj,r = 0 with probability 1 − p, some neurons will not receive the corresponding
+signal at all and will only learn noise. Therefore, for each class j ∈ [k], we split the neurons into
+two sets based on whether it receives its corresponding signal or not:
+Sj
+signal = {r ∈ [m] : µj ⊙ mj,r ̸= 0},
+Sj
+noise = {r ∈ [m] : µj ⊙ mj,r = 0}.
+Gradient descent algorithm. We consider the network is trained by cross-entropy loss with
+softmax. We denote by logiti(F, x) :=
+eFi(x)
+�
+j∈[k] eFj(x) and the cross-entropy loss can be written as
+1We point out that as many previous works (Allen-Zhu and Li, 2020; Zou et al., 2021; Cao et al., 2022), polynomial
+ReLU activation can help us simplify the analysis of gradient descent, because polynomial ReLU activation can give
+a much larger separation of signal and noise (thus, cleaner analysis) than ReLU. Our analysis can be generalized to
+ReLU activation by using the arguments in (Allen-Zhu and Li, 2022).
+5
+
+ℓ(F(x, y)) = − log logity(F, x). The convolutional neural network is trained by minimizing the
+empirical cross-entropy loss given by
+LS(W) = 1
+n
+n
+�
+i=1
+ℓ[F(W ⊙ M; xi, yi)] = E
+S ℓ[F(W ⊙ M; xi, yi)],
+where S = {(xi, yi)}n
+i=1 is the training data set. Similarly, we define the generalization loss as
+LD := E
+(x,y)[ℓ(F(W ⊙ M; x, y))].
+The model weights are initialized from a i.i.d. Gaussian N(0, σ2
+0). The gradient of the cross-entropy
+loss is given by ℓ′
+j,i := ℓ′
+j(xi, yi) = logitj(F, xi) − I(j = yi). Since
+∇wj,rLS(W ⊙ M) = ∇wj,r⊙mj,rLS(W ⊙ M) ⊙ mj,r = ∇ �wj,rLS(�
+W) ⊙ mj,r,
+we can write the full-batch gradient descent update of the weights as
+�w(t+1)
+j,r
+= �w(t)
+j,r − η∇ �wj,rLS(�
+W) ⊙ mj,r
+= �w(t)
+j,r − η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, ξi
+��
+· �ξj,r,i − η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i σ′ ��
+�w(t)
+j,r, µyi
+��
+µyi ⊙ mj,r,
+for j ∈ [K] and r ∈ [m], where �ξj,r,i = ξi ⊙ mj,r.
+Condition 2.2. We consider the parameter regime described as follows: (1) Number of classes
+K = O(log d). (2) Total number of training samples n = poly log d. (3) Dimension d ≥ Cd for
+some sufficiently large constant Cd. (4) Relationship between signal strength and noise strength:
+µ = Θ(σn
+√
+d log d) = Θ(1). (5) The number of neurons in the network m = Ω(poly log d). (6)
+Initialization variance: σ0 = �Θ(m−4n−1µ−1). (7) Learning rate: Ω(1/ poly(d)) ≤ η ≤ �O(1/µ2).
+(8) Target training loss: ϵ = Θ(1/ poly(d)).
+Conditions (1) and (2) ensure that there are enough samples in each class with high probability.
+Condition (3) ensures that our setting is in high-dimensional regime. Condition (4) ensures that
+the full model can be trained to exhibit good generalization. Condition (5), (6) and (7) ensures that
+the neural network is sufficiently overparameterized and can be optimized efficiently by gradient
+descent. Condition (7) and (8) further ensures that training time is polynomial in d. We further
+discuss the practical consideration of η and ϵ to justify their condition in Remark D.9.
+3
+Mild Pruning
+3.1
+Main result
+The first main result shows that there exists a threshold on the pruning fraction p such that pruning
+helps the neural network’s generalization.
+Theorem 3.1 (Main Theorem for Mild Pruning, Informal). Under Condition 2.2, if p ∈ [C1
+log d
+m , 1]
+for some constant C1, then with probability at least 1 − O(d−1) over the randomness in the data,
+network initialization and pruning, there exists T = �O(Kη−1σ2−q
+0
+µ−q +K2m4µ−2η−1ϵ−1) such that
+6
+
+1. The training loss is below ϵ: LS(�
+W(T)) ≤ ϵ.
+2. The generalization loss can be bounded by LD(�
+W(T)) ≤ O(Kϵ) + exp(−n2/p).
+Theorem 3.1 indicates that there exists a threshold in the order of Θ(log d
+m ) such that if p is
+above this threshold (i.e., the fraction of the pruned weights is small), gradient descent is able to
+drive the training loss towards zero (as item 1 claims) and the overparameterized network achieves
+good testing performance (as item 2 claims). In the next subsection, we explain why pruning can
+help generalization via an outline of our proof, and we defer all the detailed proofs in Appendix D.
+3.2
+Proof Outline
+Our proof contains the establishment of the following two properties:
+• First we show that after mild pruning the network is still able to learn the signal, and the
+magnitude of the signal in the feature is preserved.
+• Then we show that given a new sample, pruning reduces the noise effect in the feature which
+leads to the improvement of generalization.
+We first show the above properties for three stages of gradient descent: initialization, feature
+growing phase, and converging phase, and then establish the generalization property.
+Initialization. First of all, readers might wonder why pruning can even preserve signal at all.
+Intuitively, a network will achieve good performance if its weights are highly correlated with the
+signal (i.e., their inner product is large). Two intuitive but misleading heuristics are given by the
+following:
+• Consider a fixed neuron weight.
+At the random initialization, in expectation, the signal
+correlation with the weights is given by Ew,m[| ⟨w ⊙ m, µ⟩ |] ≤ pσ0µ and the noise correlation
+with the weights is given by Ew,m,ξ[| ⟨w ⊙ m, ξ⟩ |] ≤
+�
+Ew,m,ξ[⟨w ⊙ m, ξ⟩2] = σ0σn
+√pd by
+Jensen’s inequality. Based on this argument, taking a sum over all the neurons, pruning will
+hurt weight-signal correlation more than weight-noise correlation.
+• Since we are pruning with Bernoulli(p), a given neuron will not receive signal at all with
+probability 1 − p. Thus, there is roughly p fraction of the neurons receiving the signal and
+the rest 1 − p fraction will be purely learning from noise. Even though for every neuron,
+roughly √p portion of ℓ2 mass from the noise is reduced, at the same time, pruning also
+creates 1 − p fraction of neurons which do not receive signals at all and will purely output
+noise after training. Summing up the contributions from every neuron, the signal strength
+is reduced by a factor of p while the noise strength is reduced by a factor of √p. We again
+reach the conclusion of pruning under any rate will hurt the signal more than noise.
+The above analysis shows that under any pruning rate, it seems pruning can only hurt the signal
+more than noise at the initialization. Such analysis would be indicative if the network training is
+under the neural tangent kernel regime, where the weight of each neuron does not travel far from its
+initialization so that the above analysis can still hold approximately after training. However, when
+the neural network training is in the feature learning regime, this average type analysis becomes
+misleading. Namely, in such a regime, the weights with large correlation with the signal at the
+initialization will quickly evolve into singleton neurons and those weights with small correlation
+7
+
+will remain small. In our proof, we focus on the featuring learning regime, and analyze how the
+network weights change and what are the effect of pruning during various stages of gradient descent.
+We now analyze the effect of pruning on weight-signal correlation and weight-noise correlation at
+the initialization. Our first lemma leverages the sparsity of our signal and shows that if the pruning
+is mild, then it will not hurt the maximum weight-signal correlation much at the initialization. On
+the other hand, the maximum weight-noise correlation is reduced by a factor of √p.
+Lemma 3.2 (Initialization). With probability at least 1 − 2/d, for all i ∈ [n],
+σ0σn
+�
+pd ≤ max
+r
+�
+�w(0)
+j,r , ξi
+�
+≤
+�
+2 log(Kmd)σ0σn
+�
+pd.
+Further, suppose pm ≥ Ω(log(Kd)), with probability 1 − 2/d, for all j ∈ [K],
+σ0 ∥µj∥2 ≤
+max
+r∈Sj
+signal
+�
+�w(0)
+j,r , µj
+�
+≤
+�
+2 log(8pmKd)σ0 ∥µj∥2 .
+Given this lemma, we now prove that there exists at least one neuron that is heavily aligned
+with the signal after training. Similarly to previous works (Allen-Zhu and Li, 2020; Zou et al., 2021;
+Cao et al., 2022), the analysis is divided into two phases: feature growing phase and converging
+phase.
+Feature Growing Phase. In this phase, the gradient of the cross-entropy is large and the
+weight-signal correlation grows much more quickly than weight-noise correlation thanks to the
+polynomial ReLU. We show that the signal strength is relatively unaffected by pruning while the
+noise level is reduced by a factor of √p.
+Lemma 3.3 (Feature Growing Phase, Informal). Under Condition 2.2, there exists time T1 such
+that
+1. The max weight-signal correlation is large: maxr
+�
+�w(T1)
+j,r , µj
+�
+≥ m−1/q for j ∈ [K].
+2. The weight-noise and cross-class weight-signal correlations are small: if j ̸= yi, then maxj,r,i
+���
+�
+�w(T1)
+j,r , ξi
+���� ≤
+O(σ0σn
+√pd) and maxj,r,k
+���
+�
+�w(T1)
+j,r , µk
+���� ≤ �O(σ0µ).
+Converging Phase. We show that gradient descent can drive the training loss toward zero
+while the signal in the feature is still large. An important intermediate step in our argument is
+the development of the following gradient upper bound for multi-class cross-entropy loss which
+introduces an extra factor of K in the gradient upper bound.
+Lemma 3.4 (Gradient Upper Bound, Informal). Under Condition 2.2, we have
+���∇LS(�
+W(t)) ⊙ M
+���
+2
+F ≤ O(Km2/qµ2)LS(�
+W(t)).
+Proof Sketch. To prove this upper bound, note that for a given input (xi, yi), ℓ′(t)
+yi,i∇Fyi(xi) should
+make major contribution to
+���∇ℓ(�
+W; xi, yi)
+���
+F .
+Further note that |ℓ′(t)
+yi,i| = 1 − logityi(F; xi) =
+�
+j̸=yi eFj(xi)
+�
+j eFj(xi)
+≤
+�
+j̸=yi eFj(xi)
+eFyi (xi)
+. Now, apply the property that Fj(xi) is small for j ̸= yi (which we
+prove in the appendix), the numerator will contribute a factor of K. To bound the rest, we utilize
+8
+
+the special property of multi-class cross-entropy loss: |ℓ′(t)
+j,i | ≤ |ℓ′(t)
+yi,i| ≤ ℓ(t)
+i .
+However, a naive
+application of this inequality will result in a factor of K3 instead K in our bound. The trick is to
+further use the fact that �
+j̸=yi |ℓ′(t)
+j,i | = |ℓ′(t)
+yi,i|.
+Using the above gradient upper bound, we can show that the objective can be minimized.
+Lemma 3.5 (Converging Phase, Informal). Under Condition 2.2, there exists T2 such that for
+some time t ∈ [T1, T2] we have
+1. The results from the feature growing phase (Lemma 3.3) hold up to constant factors.
+2. The training loss is small LS(�
+W(t)) ≤ ϵ.
+Notice that the weight-noise correlation still remains reduced by a factor of √p after training.
+Lemma 3.5 proves the statement of the training loss in Theorem 3.1.
+Generalization Analysis. Finally, we show that pruning can purify the feature by reducing
+the variance of the noise by a factor of p when a new sample is given. The lemma below shows that
+the variance of weight-noise correlation for the trained weights is reduced by a factor of p.
+Lemma 3.6. The neural network weight �
+W⋆ after training satisfies that
+P
+ξ
+�
+max
+j,r
+����w⋆
+j,r, ξ
+��� ≥ (2m)−2/q
+�
+≤ 2Km exp
+�
+− (2m)−4/q
+O(σ2
+0σ2npd)
+�
+.
+Using this lemma, we can show that pruning yields better generalization bound (i.e., the bound
+on the generalization loss) claimed in Theorem 3.1.
+4
+Over Pruning
+Our second result shows that there exists a relatively large pruning fraction (i.e., small p) such that
+the learned model yields poor generalization, although gradient descent is still able to drive the
+training error toward zero. The full proof is defered to Appendix E.
+Theorem 4.1 (Main Theorem for Over Pruning, Informal). Under Condition 2.2 if p = Θ(
+1
+Km log d),
+then with probability at least 1−1/ poly log d over the randomness in the data, network initialization
+and pruning, there exists T = O(η−1nσq−2
+0
+σ−q
+n (pd)−q/2 + η−1ϵ−1m4nσ−2
+n (pd)−1) such that
+1. The training loss is below ϵ: LS(�
+W(T)) ≤ ϵ.
+2. The generalization loss is large: LD(�
+W(T)) ≥ Ω(log K).
+Remark 4.2. The above theorem indicates that in the over-pruning case, the training loss can still
+go to zero. However, the generalization loss of our neural network behaves no much better than
+random guessing, because given any sample, random guessing will assign each class with probability
+1/K, which yields a generalization loss of log K. The readers might wonder why the condition for
+this to happen is p = Θ(
+1
+Km log d) instead of O(
+1
+Km log d). Indeed, the generalization will still be bad
+if p is too small. However, now the neural network is not only unable to learn the signal but also
+cannot efficiently memorize the noise via gradient descent.
+Proof Outline. Now we analyze the over-pruning case. We first show that there is a good chance
+that the model will not receive any signal after pruning due to the sparse signal assumption and
+mild overparameterization of the neural network. Then, leveraging such a property, we bound the
+9
+
+weight-signal and weight-noise properties for the feature growing and converging phases of gradient
+descent, as stated in the following two lemmas, respectively. Our result indicates that the training
+loss can still be driven toward zero by letting the neural network memorize the noise, the proof of
+which further exploits the fact that high dimensional Gaussian noise are nearly orthogonal.
+Lemma 4.3 (Feature Growing Phase, Informal). Under Condition 2.2, there exists T1 such that
+• Some weights has large correlation with noise: maxr
+�
+�w(T1)
+yi,r , ξi
+�
+≥ m−1/q for all i ∈ [n].
+• The cross-class weight-noise and weight-signal correlations are small: if j ̸= yi, then maxj,r,i
+���
+�
+�w(T1)
+j,r , ξi
+���� =
+�O(σ0σn
+√pd) and maxj,r,k
+���
+�
+�w(T1)
+j,r , µk
+���� ≤ �O(σ0µ).
+Lemma 4.4 (Converging Phase, Informal). Under Condition 2.2, there exists a time T2 such that
+∃t ∈ [T1, T2], the results from phase 1 still holds (up to constant factors) and LS(�
+W(t)) ≤ ϵ.
+Finally, since the above lemmas show that the network is purely memorizing the noise, we
+further show that such a network yields poor generalization performance as stated in Theorem
+4.1.
+5
+Experiments
+5.1
+Simulations to Verify Our Results
+In this section, we conduct simulations to verify our results. We conduct our experiment using
+binary classification task and show that our result holds for ReLU networks.
+Our experiment
+settings are the follows: we choose input to be x = [x1, x2] = [ye1, ξ] ∈ R800 and x1, x2 ∈ R400,
+where ξi is sampled from a Gaussian distribution. The class labels y are {±1}. We use 100 training
+examples and 100 testing examples. The network has width 150 and is initialized with random
+Gaussian distribution with variance 0.01. Then, p fraction of the weights are randomly pruned.
+We use the learning rate of 0.001 and train the network over 1000 iterations by gradient descent.
+The observations are summarized as follows. In Figure 2a, when the noise level is σn = 0.5,
+the pruned network usually can perform at the similar level with the full model when p ≤ 0.5
+and noticably better when p = 0.3. When p > 0.5, the test error increases dramatically while
+the training accuracy still remains perfect. On the other hand, when the noise level becomes large
+σn = 1 (Figure 2b), the full model can no longer achieve good testing performance but mild pruning
+can improve the model’s generalization. Note that the training accuracy in this case is still perfect
+(omitted in the figure). We observe that in both settings when the model test error is large, the
+variance is also large. However, in Figure 2b, despite the large variance, the mean curve is already
+smooth. In particular, Figure 2c plots the testing error over the training iterations under p = 0.5
+pruning rate. This suggests that pruning can be beneficial even when the input noise is large.
+5.2
+On the Real World Dataset
+To further demonstrate the mild/over pruning phenomenon, we conduct experiments on MNIST
+(Deng, 2012) and CIFAR-10 (Krizhevsky et al., 2009) datasets. We consider neural network ar-
+chitectures including MLP with 2 hidden layers of width 1024, VGG, ResNets (He et al., 2016)
+and wide ResNet (Zagoruyko and Komodakis, 2016). In addition to random pruning, we also add
+10
+
+0.0
+0.2
+0.4
+0.6
+0.8
+Pruning rates
+0.00
+0.05
+0.10
+0.15
+0.20
+Error
+Training/Testing Error over Pruning Rates
+Testing error
+Training error
+(a)
+0.0
+0.2
+0.4
+0.6
+0.8
+Pruning rates
+0.20
+0.22
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+Error
+Training/Testing Error over Pruning Rates
+Testing error
+(b)
+0
+200
+400
+600
+800
+1000
+Iterations
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+Error
+Testing error
+full
+pruned
+(c)
+Figure 2: Figure (a) shows the relationship between pruning rates p and training/testing error
+under noise variance σn = 0.5. Figure (b) shows the relationship between pruning rates p and
+testing error under noise variance σn = 1. The training error is omitted since it stays effectively
+at zero across all pruning rates. Figure (c) shows a particular training curve under pruning rate
+p = 50% and noise variance σn = 1. Each data point is created by taking an average over 10
+independent runs.
+0.0 20.0 36.0 48.8 59.0 67.2 73.8 79.0 83.2 86.6 89.3 91.4 93.1 94.5 95.6 96.5 97.2
+Sparsity
+96.5
+97.0
+97.5
+98.0
+98.5
+99.0
+99.5
+100.0
+Accuracy
+MLP MNIST Accuracy vs Sparsity
+Random (Train)
+Random (Test)
+IMP (Train)
+IMP (Test)
+(a)
+0.0 20.0 36.0 48.8 59.0 67.2 73.8 79.0 83.2 86.6 89.3 91.4 93.1 94.5 95.6 96.5 97.2
+Sparsity
+88
+90
+92
+94
+96
+98
+100
+Accuracy
+VGG-16 CIFAR-10 Accuracy vs Sparsity
+Random (Train)
+Random (Test)
+IMP (Train)
+IMP (Test)
+(b)
+Figure 3: Figure (a) shows the result between pruning rates p and accuracy on MLP-1024-1024
+on MNIST. Figure (b) shows the result on VGG-16 on CIFAR-10. Each data point is created by
+taking an average over 3 independent runs.
+iterative-magnitude-based pruning Frankle and Carbin (2018) into our experiments. Both pruning
+methods are prune-at-initialization methods. Our implementation is based on Chen et al. (2021c).
+Under the real world setting, we do not expect our theorem to hold exactly. Instead, our
+theorem implies that (1) there exists a threshold such that the testing performance is no much
+worse than (or sometimes may slightly better than) its dense counter part; and (2) the training
+error decreases later than the testing error decreases. Our experiments on MLP (Figure 3a) and
+VGG-16 (Figure 3b) show that this is the case: for MLP the test accuracy is steady competitive to
+its dense counterpart when the sparsity is less than 79% and 36% for VGG-16. We further provide
+experiments on ResNet in the appendix for validation of our theoretical results.
+6
+Discussion and Future Direction
+In this work, we provide theory on the generalization performance of pruned neural networks trained
+by gradient descent under different pruning rates. Our results characterize the effect of pruning
+under different pruning rates: in the mild pruning case, the signal in the feature is well-preserved
+11
+
+and the noise level is reduced which leads to improvement in the trained network’s generalization;
+on the other hand, over pruning significantly destroys signal strength despite of reducing noise
+variance. One open problem on this topic still appears challenging. In this paper, we characterize
+two cases of pruning: in mild pruning the signal is preserved and in over pruning the signal is
+completely destroyed.
+However, the transition between these two cases is not well-understood.
+Further, it would be interesting to consider more general data distribution, and understand how
+pruning affects training multi-layer neural networks. We leave these interesting directions as future
+works.
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+17
+
+A
+Experiment Details
+The experiments of MLP, VGG and ResNet-32 are run on NVIDIA A5000 and ResNet-50 and
+ResNet-20-128 is run on 4 NIVIDIA V100s. We list the hyperparameters we used in training. All
+of our models are trained with SGD and the detailed settings are summarized below.
+Table 1: Summary of architectures, dataset and training hyperparameters
+Model
+Data
+Epoch
+Batch Size
+LR
+Momentum
+LR Decay, Epoch
+Weight Decay
+LeNet
+MNIST
+120
+128
+0.1
+0
+0
+0
+VGG
+CIFAR-10
+160
+128
+0.1
+0.9
+0.1 × [80, 120]
+0.0001
+ResNets
+CIFAR-10
+160
+128
+0.1
+0.9
+0.1 × [80, 120]
+0.0001
+B
+Further Experiment Results
+We plot the experiment result of ResNet-20-128 in Figure 4. This figure further verifies our results
+that there exists pruning rate threshold such that the testing performance of the pruned network
+is on par with the testing performance of the dense model while the training accuracy remains
+perfect.
+0.0
+20.0
+36.0
+48.8
+59.0
+67.2
+73.8
+79.0
+83.2
+86.6
+Sparsity
+95
+96
+97
+98
+99
+100
+Accuracy
+ResNet-20-128 CIFAR-10 Accuracy vs Sparsity
+Random (Train)
+Random (Test)
+IMP (Train)
+IMP (Test)
+Figure 4: The figure shows the experiment results of ResNet-20-128 under various sparsity by
+random pruning and IMP. Each data point is averaged over 2 runs.
+C
+Preliminary for Analysis
+In this section, we introduce the following signal-noise decomposition of each neuron weight from
+Cao et al. (2022), and some useful properties for the terms in such a decomposition, which are
+useful in our analysis.
+Definition C.1 (signal-noise decomposition). For each neuron weight j ∈ [K], r ∈ [m], there exist
+18
+
+coefficients γ(t)
+j,r,k, ζ(t)
+j,r,i, ω(t)
+j,r,i such that
+�w(t)
+j,r = �w(0)
+j,r +
+K
+�
+k=1
+γ(t)
+j,r,k · ∥µk∥−2
+2
+· µk ⊙ mj,r +
+n
+�
+i=1
+ζ(t)
+j,r,i ·
+����ξj,r,i
+���
+−2
+2
+· �ξj,r,i +
+n
+�
+i=1
+ω(t)
+j,r,i
+����ξj,r,i
+���
+−2
+2
+· �ξj,r,i,
+where γ(t)
+j,r,j ≥ 0, γ(t)
+j,r,k ≤ 0, ζ(t)
+j,r,i ≥ 0, ω(t)
+j,r,i ≤ 0.
+It is straightforward to see the following:
+γ(0)
+j,r,k, ζ(0)
+j,r,i, ω(0)
+j,r,i = 0,
+γ(t+1)
+j,r,j
+= γ(t)
+j,r,j − I(r ∈ Sj
+signal)η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, µyi
+��
+∥µyi∥2
+2 I(yi = j),
+γ(t+1)
+j,r,k = γ(t)
+j,r,k − I((mj,r)k = 1)η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, µyi
+��
+∥µyi∥2
+2 I(yi = k), ∀j ̸= k,
+ζ(t+1)
+j,r,i
+= ζ(t)
+j,r,i − η
+n · ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, ξi
+�� ����ξj,r,i
+���
+2
+2 I(j = yi),
+ω(t+1)
+j,r,i
+= ω(t)
+j,r,i − η
+n · ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, ξi
+�� ����ξj,r,i
+���
+2
+2 I(j ̸= yi),
+where {γ(t)
+j,r,j}T
+t=1, {ζ(t)
+j,r,i}T
+t=1 are increasing sequences and {γ(t)
+j,r,k}T
+t=1, {ω(t)
+j,r,i}T
+t=1 are decreasing se-
+quences, because −ℓ′(t)
+j,i ≥ 0 when j = yi, and −ℓ′(t)
+j,i ≤ 0 when j ̸= yi. By Lemma D.4, we have
+pd > n+K, and hence the set of vectors {µk}K
+k=1
+�{�ξi}n
+i=1 is linearly independent with probability
+measure 1 over the Gaussian distribution for each j ∈ [K], r ∈ [m]. Therefore the decomposition is
+unique.
+D
+Proof of Theorem 3.1
+We first formally restate Theorem 3.1.
+Theorem D.1 (Formal Restatement of Theorem 3.1). Under Condition 2.2, choose initialization
+variance σ0 = �Θ(m−4n−1µ−1) and learning rate η ≤ �O(1/µ2). For ϵ > 0, if p ≥ C1
+log d
+m
+for some
+sufficiently large constant C1, then with probability at least 1 − O(d−1) over the randomness in the
+data, network initialization and pruning, there exists T = �O(Kη−1σ2−q
+0
+µ−q + K2m4µ−2η−1ϵ−1)
+such that the following holds:
+1. The training loss is below ϵ: LS(�
+W(T)) ≤ ϵ.
+2. The weights of the CNN highly correlate with its corresponding class signal: maxr γ(T)
+j,r,j ≥
+Ω(m−1/q) for all j ∈ [K].
+3. The weights of the CNN doesn’t have high correlation with the signal from different classes:
+maxj̸=k,r∈[m] |γ(T)
+j,r,k| ≤ �O(σ0µ).
+4. None of the weights is highly correlated with the noise: maxj,r,i ζ(T)
+j,r,i = �O(σ0σn
+√pd), maxj,r,i |ω(T)
+j,r,i| =
+�O(σ0σn
+√pd).
+19
+
+Moreover, the testing loss is upper-bounded by
+LD(�
+W(T)) ≤ O(Kϵ) + exp(−n2/p).
+The proof of Theorem 3.1 consists of the analysis of the pruning on the signal and noise for
+three stages of gradient descent: initialization, feature growing phase, and converging phase, and the
+establishment of the generalization property. We present these analysis in detail in the following
+subsections.
+A special note is that the constant C showing up in the following proof of each
+subsequent Lemmas is defined locally instead of globally, which means the constant C within each
+Lemma is the same but may be different across different Lemma.
+D.1
+Initialization
+We analyze the effect of pruning on weight-signal correlation and weight-noise correlation at the
+initialization. We first present a few supporting lemmas, and finally provide our main result of
+Lemma D.7, which shows that if the pruning is mild, then it will not hurt the max weight-signal
+correlation much at the initialization.
+On the other hand, the max weight-noise correlation is
+reduced by a factor of √p.
+Lemma D.2. Assume n = Ω(K2 log Kd). Then, with probability at least 1 − 1/d,
+|{i ��� [n] : yi = j}| = Θ(n/K)
+∀j ∈ [K].
+Proof. By Hoeffding’s inequality, with probability at least 1 − δ/2K, for a fixed j ∈ [K], we have
+�����
+1
+n
+n
+�
+i=1
+I(yi = j) − 1
+K
+����� ≤
+�
+log(4K/δ)
+2n
+.
+Therefore, as long as n ≥ 2K2 log(4K/δ), we have
+�����
+1
+n
+n
+�
+i=1
+I(yi = j) − 1
+K
+����� ≤
+1
+2K .
+Taking a union bound over j ∈ [K] and making δ = 1/d yield the result.
+Lemma D.3. Assume pm = Ω(log d) and m = poly log d. Then, with probability 1 − 1/d, for all
+j ∈ [K], k ∈ [K], we have �m
+r=1(mj,r)k = Θ(pm), which implies that |Sj
+signal| = Θ(pm) for all
+j ∈ [K].
+Proof. When pm = Ω(log d), by multiplicative Chernoff’s bound, for a given k ∈ [K], we have
+P
+������
+m
+�
+r=1
+(mj,r)k − pm
+����� ≥ 0.5pm
+�
+≤ 2 exp {−Ω (pm)} .
+Take a union bound over j ∈ [K], k ∈ [K], we have
+P
+������
+m
+�
+r=1
+(mj,r)k − pm
+����� ≥ 0.5pm, ∀j ∈ [K], k ∈ [K]
+�
+≤ 2K2 exp {−Ω (pm)} ≤ 1/d.
+20
+
+Lemma D.4. Assume p = 1/ poly log d. Then with probability at least 1 − 1/d, for all j ∈ [K],
+r ∈ [m], �d
+i=1(mj,r)i = Θ(pd).
+Proof. By multiplicative Chernoff’s bound, we have for a given j, r
+P
+������
+d
+�
+i=1
+(mj,r)i − pd
+����� ≥ 0.5pd
+�
+≤ 2 exp{−Ω(pd)}.
+Take a union bound over j, r, we have
+P
+������
+d
+�
+i=1
+(mj,r)i − pd
+����� ≥ 0.5pd, ∀j ∈ [K], r ∈ [m]
+�
+≤ 2Km exp{−Ω(pd)} ≤ 1/d,
+where the last inequality follows from our choices of p, K, m, d.
+Lemma D.5. Suppose p = Ω(1/ poly log d), and m, n = poly log d. With probability at least 1−1/d,
+we have
+����ξj,r,i
+���
+2
+2 = Θ(σ2
+npd),
+���
+�
+�ξj,r,i, ξi′
+���� ≤ O(σ2
+n
+�
+pd log d),
+���
+�
+µk, �ξj,r,i
+���� ≤ | ⟨µ, ξi⟩ | ≤ O(σnµ
+�
+log d),
+for all j ∈ {−1, 1}, r ∈ [m], i, i′ ∈ [n] and i ̸= i′.
+Proof. From Lemma D.4, we have with probability at least 1 − 1/d,
+d
+�
+k=1
+(mj,r)k = Θ(pd),
+∀j ∈ [K], r ∈ [m].
+For a set of Gaussian random variable g1, . . . , gN ∼ N(0, σ2), by Bernstein’s inequality, with prob-
+ability at least 1 − δ, we have
+�����
+N
+�
+i=1
+g2
+i − σ2N
+����� ≲ σ2
+�
+N log 1
+δ .
+Thus, by a union bound over j, r, i, with probability at least 1 − 1/d, we have
+����ξj,r,i
+���
+2
+2 = Θ(σ2
+npd).
+For i ̸= i′, again by Bernstein’s bound, we have with probability at least 1 − δ,
+���
+�
+�ξj,r,i, ξi′
+���� ≤ O
+�
+σ2
+n
+�
+pd log Kmn
+δ
+�
+,
+for all j, r, i. Plugging in δ = 1/d gives the result. The proof for | ⟨µ, ξi⟩ | is similar.
+21
+
+Lemma D.6. Suppose we have m independent Gaussian random variables g1, g2, . . . , gm ∼ N(0, σ2).
+Then with probability 1 − δ,
+max
+i
+gi ≥ σ
+�
+log
+m
+log 1/δ.
+Proof. By the standard tail bound of Gaussian random variable, we have for every x > 0,
+�σ
+x − σ3
+x3
+� e−x2/2σ2
+√
+2π
+≤ P [g > x] ≤ σ
+x
+e−x2/2σ2
+√
+2π
+.
+We want to pick a x⋆ such that
+P
+�
+max
+i
+gi ≤ x⋆
+�
+= (P [gi ≤ x⋆])m = (1 − P [gi ≥ x⋆])m ≤ e−m P[gi≥x⋆] ≤ δ
+⇒ P[gi ≥ x⋆] = Θ
+�log(1/δ)
+m
+�
+⇒ x⋆ = Θ(σ
+�
+log(m/(log(1/δ) log m))).
+Lemma D.7 (Formal Restatement of Lemma 3.2). With probability at least 1−2/d, for all i ∈ [n],
+σ0σn
+�
+pd ≤ max
+r
+�
+�w(0)
+j,r , ξi
+�
+≤
+�
+2 log(Kmd)σ0σn
+�
+pd.
+Further, suppose pm ≥ Ω(log(Kd)). Then with probability 1 − 2/d, for all j ∈ [K],
+σ0 ∥µj∥2 ≤
+max
+r∈Sj
+signal
+�
+�w(0)
+j,r , µj
+�
+≤
+�
+2 log(8pmKd)σ0 ∥µj∥2 .
+Proof. We first give a proof for the second inequality. From Lemma D.3, we know that |Sj
+signal| =
+Θ(pm). The upper bound can be obtained by taking a union bound over r ∈ Sj
+signal, j ∈ [K]. To
+prove the lower bound, applying Lemma D.6, with probability at least 1−δ/K, we have for a given
+j ∈ [K]
+max
+r∈Sj
+signal
+�
+�w(0)
+j,r , µj
+�
+≥ σ0 ∥µj∥2
+�
+log
+pm
+log K/δ.
+Now, notice that we can control the constant in pm (by controlling the constant in the lower bound
+of p) such that pm/ log(Kd) ≥ e. Thus, taking a union bound over j ∈ [K] and setting δ = 1/d
+yield the result.
+The proof of the first inequality is similar.
+D.2
+Supporting Properties for Entire Training Process
+This subsection establishes a few properties (summarized in Proposition D.10) that will be used in
+the analysis of feature growing phase and converging phase of gradient descent presented in the next
+two subsections. Define T ⋆ = η−1 poly(1/ϵ, µ, d−1, σ−2
+n , σ−1
+0 n, m, d). Denote α = Θ(log1/q(T ⋆)), β =
+22
+
+2 maxi,j,r,k
+����
+�
+�w(0)
+j,r , µk
+���� ,
+���
+�
+�w(0)
+j,r , ξi
+����
+�
+. We need the following bound holds for our subsequent
+analysis.
+4m1/q max
+j,r,i
+��
+�w(0)
+j,r , µyi
+�
+, Cnαµ√log d
+σnpd ,
+�
+�w(0)
+j,r , ξi
+�
+, 3Cnα
+�
+log d
+pd
+�
+≤ 1
+(D.1)
+Remark D.8. To see why Equation (D.1) can hold under Condition 2.2, we convert everything in
+terms of d. First recall from Condition 2.2 that m, n = poly(log d) and µ = Θ(σn
+√
+d log d) = Θ(1).
+In both mild pruning and over pruning we require p ≥ Ω(1/poly log d). Since α = Θ(log1/q(T ⋆)), if
+we assume T ⋆ ≤ O(poly(d)) for a moment (which we are going to justify in the next paragraph),
+then α = O(log1/q(d)). Then if we set d to be large enough, we have 4m1/qCnα µ√log d
+σnpd
+≤ poly log d
+√
+d
+≤
+1. Finally for the quantity 4m1/q maxj,r,i{⟨�w(0)
+j,r , µyi⟩, ⟨�w(0)
+j,r , ξi⟩}, by Lemma 3.2, our assumption
+of K = O(log d) in Condition 2.2 and our choice of σ0 = �Θ(m−4n−1µ−1) in Theorem 3.1 (or
+Theorem D.1), we can easily see that this quantity can also be made smaller than 1.
+Now, to justify that T ⋆ ≤ O(poly(d)), we only need to justify that all the quantities T ⋆ depend on
+is polynomial in d. First of all, based on Condition 2.2, n, m = poly log(d) and µ = Θ(σn
+√
+d log d) =
+Θ(1) further implies σ−2
+n
+= Θ(d log2 d). Since Theorem 3.1 only requires σ0 = �Θ(m−4n−1µ−1), this
+implies σ−1
+0
+≤ O(poly log d). Hence σ−1
+0 n = O(poly log d). Together with our assumption that
+ϵ, η ≥ Ω(1/ poly(d)) (which implies 1/ϵ, 1/η ≤ O(poly(d))), we have justified that all terms involved
+in T ⋆ are at most of order poly(d). Hence T ⋆ = poly(d).
+Remark D.9. Here we make remark on our assumption on ϵ and η in Condition 2.2.
+For our assumption on ϵ, since the cross-entropy loss is (1) not strongly-convex and (2) achieves
+its infimum at infinity. In practice, the cross-entropy loss is minimized to a constant level, say
+0.001. We make this assumption to avoid the pathological case where ϵ is exponentially small in
+d (say ϵ = 2−d) which is unrealistic. Thus, for realistic setting, we assume ϵ ≥ Ω(1/ poly(d)) or
+1/ϵ ≤ O(poly(d)).
+To deal with η, the only restriction we have is η = O(1/µ2) in Theorem 3.1 and Theorem 4.1.
+However, in practice, we don’t use a learning rate that is exponentially small, say η = 2−d. Thus,
+like dealing with ϵ, we assume η ≥ Ω(1/ poly(d)) or 1/η ≤ O(poly d).
+We make the above assumption to simplify analysis when analyzing the magnitude of Fj(X)
+for j ̸= y given sample (X, y).
+Proposition D.10. Under Condition 2.2, during the training time t < T ⋆, we have
+1. γ(t)
+j,r,j, ζ(t)
+j,r,i ≤ α,
+2. ω(t)
+j,r,i ≥ −β − 6Cnα
+�
+log d
+pd .
+3. γ(t)
+j,r,k ≥ −β − 2Cnα µ√log d
+σnpd .
+Notice that the lower bound has absolute value smaller than the upper bound.
+Proof of Proposition D.10. We use induction to prove Proposition D.10.
+23
+
+Induction Hypothesis:
+Suppose Proposition D.10 holds for all t < T ≤ T ⋆.
+We next show that this also holds for t = T via the following a few lemmas.
+Lemma D.11. Under Condition 2.2, for t < T, there exists a constant C such that
+�
+�w(t)
+j,r − �w(0)
+j,r , µk
+�
+=
+�
+γ(t)
+j,r,k ± Cnαµ√log d
+σnpd
+�
+I((mj,r)k = 1),
+�
+�w(t)
+j,r − �w(0)
+j,r , ξi
+�
+= ζ(t)
+j,r,i ± 3Cnα
+�
+log d
+pd ,
+�
+�w(t)
+j,r − �w(0)
+j,r , ξi
+�
+= ω(t)
+j,r,i ± 3Cnα
+�
+log d
+pd .
+Proof. From Lemma D.5, there exists a constant C such that with probability at least 1 − 1/d,
+���
+�
+�ξj,r,i, ξi′
+����
+����ξj,r,i
+���
+2
+2
+≤ C
+�
+log d
+pd ,
+���
+�
+�ξj,r,i, µk
+����
+����ξj,r,i
+���
+2
+2
+≤ C µ√log d
+σnpd ,
+| ⟨µk, ξi⟩ |
+∥µk∥2
+2
+≤ C σn
+√log d
+µ
+.
+Using the signal-noise decomposition and assuming (mj,r)k = 1, we have
+���
+�
+�w(t)
+j,r − �w(0)
+j,r , µk
+�
+− γ(t)
+j,r,k
+��� =
+�����
+n
+�
+i=1
+ζ(t)
+j,r,i ·
+����ξj,r,i
+���
+−2
+2
+·
+�
+�ξj,r,i, µk
+�
++
+n
+�
+i=1
+ω(t)
+j,r,i
+����ξj,r,i
+���
+−2
+2
+·
+�
+�ξj,r,i, µk
+������
+≤ C µ√log d
+σnpd
+n
+�
+i=1
+���ζ(t)
+j,r,i
+��� + C µ√log d
+σnpd
+n
+�
+i=1
+���ω(t)
+j,r,i
+���
+≤ 2C µ√log d
+σnpd nα.
+where the second last inequality is by Lemma D.5 and the last inequality is by induction hypothesis.
+To prove the second equality, for j = yi,
+���
+�
+�w(t)
+j,r − �w(0)
+j,r , ξi
+�
+− ζ(t)
+j,r,i
+��� =
+�������
+K
+�
+k=1
+γ(t)
+j,r,k · ⟨µk, ξi⟩
+∥µk∥2
+2
++
+�
+i′̸=i
+ζ(t)
+j,r,i′ ·
+�
+�ξj,r,i′, ξi
+�
+����ξj,r,i′
+���
+2
+2
++
+n
+�
+i′=1
+ω(t)
+j,r,i′
+�
+�ξj,r,i′, ξi
+�
+����ξj,r,i′
+���
+2
+2
+�������
+≤ C σn
+√log d
+µ
+K
+�
+k=1
+|γ(t)
+j,r,k| + C
+�
+log d
+pd
+�
+i′̸=i
+|ζ(t)
+j,r,i′| + C
+�
+log d
+pd
+n
+�
+i′=1
+|ω(t)
+j,r,i′|
+= C σn
+√log d
+µ
+Kα + 2Cnα
+�
+log d
+pd
+24
+
+≤ 3Cnα
+�
+log d
+pd .
+where the last inequality is by n ≫ K and µ = Θ(σn
+√
+d log d). The proof for the case of j ̸= yi is
+similar.
+Lemma D.12 (Off-diagonal Correlation Upper Bound). Under Condition 2.2, for t < T, j ̸= yi,
+we have that
+�
+�w(t)
+j,r, µyi
+�
+≤
+�
+�w(0)
+j,r , µyi
+�
++ Cnαµ√log d
+σnpd ,
+�
+�w(t)
+j,r, ξi
+�
+≤
+�
+�w(0)
+j,r , ξi
+�
++ 3Cnα
+�
+log d
+pd ,
+Fj(�
+W(t)
+j , xi) ≤ 1.
+Proof. If j ̸= yi, then γ(t)
+j,r,k ≤ 0 and we have that
+�
+�w(t)
+j,r, µyi
+�
+≤
+�
+�w(0)
+j,r , µyi
+�
++
+�
+γ(t)
+j,r,yi + Cnαµ√log d
+σnpd
+�
+I((mj,r)yi = 1)
+≤
+�
+�w(0)
+j,r , µyi
+�
++ Cnαµ√log d
+σnpd .
+Further, we can obtain
+�
+�w(t)
+j,r, ξi
+�
+≤
+�
+�w(0)
+j,r , ξi
+�
++ ω(t)
+j,r,i + 3Cnα
+�
+log d
+pd
+≤
+�
+�w(0)
+j,r , ξi
+�
++ 3Cnα
+�
+log d
+pd .
+Then, we have the following bound:
+Fj(�
+W(t)
+j , xi) =
+m
+�
+r=1
+[σ(⟨�wj,r, µyi⟩) + σ(⟨�wj,r, ξi⟩)]
+≤ m2q+1 max
+j,r,i
+��
+�w(0)
+j,r , µyi
+�
+, Cnαµ√log d
+σnpd ,
+�
+�w(0)
+j,r , ξi
+�
+, 3Cnα
+�
+log d
+pd
+�q
+≤ 1.
+where the first inequality is by Equation (D.1).
+Lemma D.13 (Diagonal Correlation Upper Bound). Under Condition 2.2, for t < T, j = yi, we
+have
+�
+�w(t)
+j,r, µj
+�
+≤
+�
+�w(0)
+j,r , µj
+�
++ γ(t)
+j,r,j + Cnαµ√log d
+σnpd ,
+25
+
+�
+�w(t)
+j,r, ξi
+�
+≤
+�
+�w(0)
+j,r , ξi
+�
++ ζ(t)
+j,r,i + 3Cnα
+�
+log d
+pd .
+If max{γ(t)
+j,r,j, ζ(t)
+j,r,i} ≤ m−1/q, we further have that Fj(�
+W(t)
+j , xi) ≤ O(1).
+Proof. The two inequalities are immediate consequences of Lemma D.11. If max{γ(t)
+j,r,j, ζ(t)
+j,r,i} ≤
+m−1/q, we have
+Fj(�
+W(t)
+j , xi) =
+m
+�
+r=1
+[σ(⟨�wj,r, µj⟩) + σ(⟨�wj,r, ξi⟩)]
+≤ 2 · 3qm max
+j,r,i
+�
+γ(t)
+j,r, ζ(t)
+j,r,i,
+���
+�
+�w(0)
+j,r , µj
+���� ,
+���
+�
+�w(0)
+j,r , ξi
+���� , Cnαµ√log d
+σnpd , 3Cnα
+�
+log d
+pd
+�q
+≤ O(1).
+Lemma D.14. Under Condition 2.2, for t ≤ T, we have that
+1. ω(t)
+j,r,i ≥ −β − 6Cnα
+�
+log d
+pd ;
+2. γ(t)
+j,r,k ≥ −β − 2Cnα µ√log d
+σnpd .
+Proof. When j = yi, we have ω(t)
+j,r,i = 0. We only need to consider the case of j ̸= yi. When
+ω(T−1)
+j,r,i
+≤ −0.5β − 3Cnα
+�
+log d
+pd , by Lemma D.11 we have
+�
+�w(T−1)
+j,r
+, ξi
+�
+≤
+�
+�w(0)
+j,r , ξi
+�
++ ω(T−1)
+j,r,i
++ 3Cnα
+�
+log d
+pd
+≤ 0.
+Thus,
+ω(T)
+j,r,i = ω(T−1)
+j,r,i
+− η
+n · ℓ′(T−1)
+j,i
+· σ′ ��
+�w(T−1)
+j,r
+, ξi
+�� ����ξj,r,i
+���
+2
+2 I(j ̸= yi)
+= ω(T−1)
+j,r,i
+≥ −β − 6Cnα
+�
+log d
+pd .
+When ω(T−1)
+j,r,i
+≥ −0.5β − 3Cnα
+�
+log d
+pd , we have
+ω(T)
+j,r,i = ω(T−1)
+j,r,i
+− η
+n · ℓ′(T−1)
+j,i
+· σ′ ��
+�w(T−1)
+j,r
+, ξi
+�� ����ξj,r,i
+���
+2
+2 I(j ̸= yi)
+≥ −0.5β − 3Cnα
+�
+log d
+pd
+− η
+nσ′
+�
+0.5β + 3Cnα
+�
+log d
+pd
+� ����ξj,r,i
+���
+2
+2
+≥ −β − 6Cnα
+�
+log d
+pd ,
+26
+
+where the last inequality is by setting η ≤ nq−1 �
+0.5β + 3Cnα
+�
+log d
+pd
+�2−q
+(C2σ2
+nd)−1 and C2 is the
+constant such that
+����ξj,r,i
+���
+2
+2 ≤ C2σ2
+npd for all j, r, i in Lemma D.5.
+For γ(t)
+j,r,k, the proof is similar. Consider I((mj,r)k) = 1. When γ(t)
+j,r,k ≤ −0.5β − Cnα µ√log d
+σnpd , by
+Lemma D.11, we have
+�
+�w(t)
+j,r, µk
+�
+≤
+�
+�w(0)
+j,r , µk
+�
++ γ(t)
+j,r,k + Cnαµ√log d
+σnpd
+≤ 0.
+Hence,
+γ(T)
+j,r,k = γ(T−1)
+j,r,k
+− η
+n
+n
+�
+i=1
+ℓ′(T−1)
+j,i
+σ′ ��
+�w(T−1)
+j,r
+, µk
+��
+µ2I(yi = k)
+= γ(T−1)
+j,r,k
+≥ −β − 2Cnαµ√log d
+σnpd .
+When γ(t)
+j,r,k ≥ −0.5β − Cnα µ√log d
+σnpd , we have
+γ(T)
+j,r,k = γ(T−1)
+j,r,k
+− η
+n
+n
+�
+i=1
+ℓ′(T−1)
+j,i
+σ′ ��
+�w(T−1)
+j,r
+, µk
+��
+µ2I(yi = k)
+≥ −0.5β − Cnαµ√log d
+σnpd
+− C2
+η
+K σ′
+�
+0.5β + Cnαµ√log d
+σnpd
+�
+µ2
+≥ −β − 2Cnαµ√log d
+σnpd ,
+where the first inequality follows from the fact that there are Θ( n
+K ) samples such that I(yi = k),
+and the last inequality follows from picking η ≤ K(0.5β + Cnα µ√log d
+σnpd )2−qµ−2q−1C−1
+2 .
+Lemma D.15. Under Condition 2.2, for t ≤ T, we have γ(t)
+j,r,j, ζ(t)
+j,r,i ≤ α.
+Proof. For yi ̸= j or r /∈ Sj
+signal, γ(t)
+j,r,j, ζ(t)
+j,r,i = 0 ≤ α.
+If yi = j, then by Lemma D.12 we have
+���ℓ′(t)
+j,i
+��� = 1 − logitj(F; X) =
+�
+i̸=j eFi(X)
+�K
+i=1 eFi(X) ≤
+Ke
+eFj(X) .
+(D.2)
+Recall that
+γ(t+1)
+j,r,j
+= γ(t)
+j,r,j − I(r ∈ Sj
+signal)η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, µyi
+��
+∥µyi∥2
+2 I(yi = j),
+ζ(t+1)
+j,r,i
+= ζ(t)
+j,r,i − η
+n · ℓ′(t)
+j,i · σ′ ��
+�w(t)
+j,r, ξi
+�� ����ξj,r,i
+���
+2
+2 I(j = yi).
+27
+
+We first bound ζ(T)
+j,r,i. Let Tj,r,i be the last time t < T that ζ(t)
+j,r,i ≤ 0.5α. Then we have
+ζ(T)
+j,r,i = ζ(Tj,r,i)
+j,r,i
+− η
+nℓ′(Tj,r,i)
+i
+· σ′ ��
+�w(Tj,r,i)
+j,r
+, ξi
+��
+I(yi = j)
+����ξj,r,i
+���
+2
+2
+�
+��
+�
+I1
+−
+�
+Tj,r,i<t<T
+η
+nℓ′(Tj,r,i)
+j,i
+σ′ ��
+�w(t)
+j,r, ξi
+��
+I(yi = j)
+����ξj,r,i
+���
+2
+2
+�
+��
+�
+I2
+.
+We bound I1, I2 separately. We first bound I1 as follows.
+|I1| ≤ q η
+n
+�
+ζ(Tj,r,i)
+j,r,i
++ 0.5β + 3Cnα
+�
+log d
+pd
+�q−1
+C2σ2
+npd ≤ q2qn−1ηαq−1C2σ2
+npd ≤ 0.25α,
+where the first inequality follows from Lemma D.13, the second inequality follows because β ≤ 0.1α
+and 3Cnα
+�
+log d
+pd ≤ 0.1α, and the last inequality follows because η ≤ n/(q2q+2αq−2σ2
+nd).
+For Tj,r,i < t < T, by Lemma D.11, we have that
+�
+�w(t)
+j,r, ξi
+�
+≥ 0.5α−0.5β−3Cnα
+�
+log d
+pd ≥ 0.25α
+and
+�
+�w(t)
+j,r, ξi
+�
+≤ α + 0.5β + 3Cnα
+�
+log d
+pd ≤ 2α.
+Now we bound I2 as follows
+|I2| ≤
+�
+Tj,r,i<t<T
+η
+nKe exp {−Fj(X)} σ′ ��
+�w(t)
+j,r, ξi
+��
+I(yi = j)
+����ξj,r,i
+���
+2
+2
+≤
+�
+Tj,r,i<t<T
+η
+nKe exp
+�
+−σ
+��
+�w(t)
+j,r, ξi
+���
+σ′ ��
+�w(t)
+j,r, ξi
+��
+I(yi = j)
+����ξj,r,i
+���
+2
+2
+≤ qKeη2q−1T ⋆
+n
+exp(−αq/4q)αq−1σ2
+npd
+≤ 0.25T ⋆ exp(−αq/4q)αq−2α
+≤ 0.25α,
+where the first inequality follows from Equation (D.2), the second inequality follows because
+Fj(X) ≥ σ
+��
+�w(t)
+j,r, ξi
+��
+, the fourth inequality follows by choosing η ≤ n/(qKe2q+1σ2
+nd), and the
+last inequality follows by choosing α = Θ(log1/q(T ⋆)).
+Plugging the bounds on I1, I2 finishes the proof for ζ(T)
+j,r,i.
+To prove γ(t)
+j,r,j ≤ α, we pick η ≤ 1/(qe2q+2µ2) and the rest of the proof is similar.
+Lemma D.14 and Lemma D.15 imply Proposition D.10 holds for all t ≤ T.
+Induction Ends
+D.3
+Feature Growing Phase
+In this subsection, we first present a supporting lemma, and then provide our main result of
+Lemma D.17, which shows that the signal strength is relatively unaffected by pruning while the
+28
+
+noise level is reduced by a factor of √p.
+During the feature growing phase of training, the output of Fj(X) = O(1) for all j ∈ [K].
+Therefore, logiti(F, X) = O( 1
+K ) and 1 − logiti(F, X) = Θ(1) until
+�
+�w(t)
+j,r, µj
+�
+reaches m−1/q.
+Lemma D.16. Under the same assumption as Theorem D.1, for T =
+nη−1C4σ2−q
+0
+(σn
+√pd)−q
+C3(2C1)q−1[log d](q−1)/2 , the
+following results hold:
+• |ζ(t)
+j,r,i| = O(σ0σn
+√pd) for all j ∈ [K], r ∈ [m], i ∈ [n] and t ≤ T.
+• |ω(t)
+j,r,i| = O(σ0σn
+√pd) for all j ∈ [K], r ∈ [m], i ∈ [n] and t ≤ T.
+Proof. Define Ψ(t) = maxj,r,i{ζ(t)
+j,r,i, |ω(t)
+j,r,i|}. Then we have
+Ψ(t+1)
+≤ Ψ(t) + max
+j,r,i
+�
+�
+�
+�
+�
+η
+n|ℓ′(t)
+j,i | · σ′
+�
+�
+�
+�
+�w(0)
+j,r , ξi
+�
++
+K
+�
+k=1
+γ(t)
+j,r,k
+⟨µk, ξi⟩
+∥µk∥2
+2
++
+n
+�
+i′=1
+Ψ(t)
+�
+�ξj,r,i′, ξi
+�
+����ξj,r,i′
+���
+2
+2
+�
+�
+�
+����ξj,r,i
+���
+2
+2
+�
+�
+�
+�
+�
+≤ Ψ(t) + η
+nq
+�
+O(
+�
+log dσ0σn
+�
+pd) + K log1/q T ⋆ µσn
+√log d
+µ2
++ O(σ2
+npd) + nO(σ2
+n
+√pd log d)
+Θ(σ2npd)
+Ψ(t)
+�q−1
+O(σ2
+npd)
+≤ Ψ(t) + η
+n
+�
+O(
+�
+log dσ0σn
+�
+pd) + O(Ψ(t))
+�q−1
+O(σ2
+npd),
+where the second inequality follows by |ℓ′(t)
+j,i | and applying the bounds from Lemma D.5, and the last
+inequality follows by choosing 2K log T ⋆
+σnd√p
+= �O(1/
+√
+d) ≪ σ0. Let C1, C2, C3 be the constants for the up-
+per bound to hold in the big O notation. For any T = nη−1C4σ2−q
+0
+(σn
+√pd)−q
+C3(2C1)q−1[log d](q−1)/2 = Θ( nη−1σ2−q
+0
+(σn
+√pd)−q
+[log d](q−1)/2
+),
+we use induction to show that
+Ψ(t) ≤ C4σ0σn
+�
+pd, ∀t ∈ [T].
+(D.3)
+Suppose that Equation (D.3) holds for t ∈ [T ′] for T ′ ≤ T − 1. Then
+Ψ(T ′+1) ≤ Ψ(T ′) + η
+n
+�
+C1
+�
+log dσ0σn
+�
+pd + C2C4σ0σn
+�
+pd
+�q−1
+C3σ2
+npd
+≤ Ψ(T ′) + η
+n
+�
+2C1
+�
+log dσ0σn
+�
+pd
+�q−1
+C3σ2
+npd
+≤ (T ′ + 1)η
+n
+�
+2C1
+�
+log dσ0σn
+�
+pd
+�q−1
+C3σ2
+npd
+≤ T η
+n
+�
+2C1
+�
+log dσ0σn
+�
+pd
+�q−1
+C3σ2
+npd
+≤ C4σ0σn
+�
+pd,
+where the last inequality follows by picking T =
+nη−1C4σ2−q
+0
+(σn
+√pd)−q
+C3(2C1)q−1[log d](q−1)/2 = Θ( nη−1σ2−q
+0
+(σn
+√pd)−q
+[log d](q−1)/2
+).
+Therefore, by induction, we have Ψ(t) ≤ C4σ0σn
+√pd for all t ∈ [T].
+Lemma D.17 (Formal Restatement of Lemma 3.3). Under the same assumption as Theorem D.1,
+there exists time T1 =
+log 2m−1/q
+log(1+Θ( η
+K )µqσq−2
+0
+) = O(Kη−1σ2−q
+0
+µ−q log 2m−1/q) such that
+29
+
+1. maxr γ(T1)
+j,r,j ≥ m−1/q for j ∈ [K].
+2. |ζ(t)
+j,r,i|, |ω(t)
+j,r,i| ≤ O(σ0σn
+√pd) for all j ∈ [K], r ∈ [m], i ∈ [n] and t ≤ T1.
+3. |γ(t)
+j,r,k| ≤ O(σ0µ poly log d) for all j, k ∈ [K], j ̸= k, r ∈ [m] and t ≤ T1.
+Proof. Consider a fixed class j ∈ [K]. Denote T1 to be the last time for t ∈
+�
+0, nη−1C4σ2−q
+0
+(σn
+√pd)−q
+C3(2C1)q−1[log d](q−1)/2
+�
+satisfying maxr γ(t)
+j,r ≤ m−1/q. Then for t ≤ T1, maxj,r,i ζ(t)
+j,r,i, |ω(t)
+j,r,i| ≤ O(σ0σp
+√pd) ≤ O(m−1/q)
+and maxj,r γ(t)
+j,r,j. Thus, by Lemma D.13, we obtain that Fj(�
+W(t), xi) ≤ O(1), ∀yi = j. Thus,
+ℓ′(t)
+j,i = Θ(1). For j ∈ Sj
+signal, we have
+γ(t+1)
+j,r,j
+= γ(t)
+j,r,j − η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i · σ′
+�
+�
+�
+�
+�w(0)
+j,r , µj
+�
++ γ(t)
+j,r,j +
+n
+�
+i′=1
+ζ(t)
+j,r,i
+�
+�ξj,r,i, µj
+�
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i′=1
+ω(t)
+j,r,i
+�
+�ξj,r,i, µj
+�
+����ξj,r,i
+���
+2
+2
+�
+�
+� ∥µj∥2
+2 I(yi = j)
+≥ γ(t)
+j,r,j − η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i σ′
+��
+�w(0)
+j,r , µj
+�
++ γ(t)
+j,r,j − O(nσ0σnpdσnµ√log d
+σ2npd
+)
+�
+I(yi = j).
+Let �γ(t)
+j,r,j = γ(t)
+j,r,j +
+�
+�w(0)
+j,r , µj
+�
+− O(nσ0σn
+√pd σnµ√log d
+σ2npd
+) and A(t) = maxr �γ(t)
+j,r,j.
+Note that by
+our choice of µ, we have
+nµ√log d
+σnpd
+= o(1).
+Since maxr
+�
+�w(0)
+j,r , µj
+�
+≥ Ω(σ0µ) by Lemma D.7,
+maxr
+�
+�w(0)
+j,r , µj
+�
+≥ Ω(σ0µ) − O(nσ0σnpd σnµ√log d
+σ2npd
+) = Ω(σ0µ). Then we have
+A(t+1) ≥ A(t) − η
+n
+n
+�
+i=1
+ℓ′(t)
+j,i σ′(A(t))µ2I(yi = j)
+≥ A(t) + Θ( η
+K )µ2[A(t)]q−1
+≥ (1 + Θ( η
+K µ2[A(t)]q−2))A(t)
+≥ (1 + Θ( η
+K µqσq−2
+0
+))A(t).
+Therefore, the sequence A(t) will exponentially grow and will reach 2m−1/q within
+log 2m−1/q
+log(1+Θ( η
+K )µqσq−2
+0
+) =
+O(Kη−1σ2−q
+0
+µ−q log 2m−1/q) ≤ Θ( nη−1σ2−q
+0
+(σn
+√pd)−q
+[log d](q−1)/2
+). Thus, maxr γ(t)
+j,r ≥ A(t)−maxj,r |
+�
+�w(0)
+j,r , µj
+�
+| ≥
+2m−1/q − O(σ0µ) ≥ 2 − m−1/q = m−1/q.
+Now we prove that under the same assumption as Theorem D.1, for T = O(Kη−1σ2−q
+0
+µ−q), we
+have |γ(t)
+j,r,k| ≤ O(σ0µ poly log d) for all r ∈ [m], j, k ∈ [K], j ̸= k and t ≤ T.
+We show that there exists a time T ′ ≥ T such that for all t ≤ T ′, maxj,r,k |γ(t)
+j,r,k| ≤ O(σ0µ poly log d).
+Let T ′ = O(K2η−1σ2−q
+0
+µ−q log d).
+Define Φ(t) = maxr∈[m], j,k∈[K], j̸=k{|γ(t)
+j,r,k|}. Since we assume T ≤ Θ( nη−1σ2−q
+0
+(σn
+√pd)−q
+[log d](q−1)/2
+), by
+30
+
+Lemma D.16, we have ζ(t)
+j,r,i, |ω(t)
+j,r,i| ≤ O(σ0σn
+√pd).
+Φ(t+1)
+≤ Φ(t) + max
+j,r,k,i
+�
+�
+�
+�
+�
+η
+n
+n
+�
+i=1
+I(yi = k)|ℓ′(t)
+j,i |σ′
+�
+�
+�
+�
+�w(0)
+j,r , µk
+�
++
+n
+�
+i′=1
+ζ(t)
+j,r,i′
+�
+�ξj,r,i′, µk
+�
+����ξj,r,i′
+���
+2
+2
++
+n
+�
+i′=1
+ω(t)
+j,r,i′
+�
+�ξj,r,i′, µk
+�
+����ξj,r,i′
+���
+2
+2
+�
+�
+� µ2
+�
+�
+�
+�
+�
+≤ Φ(t) + η
+K
+1
+K q
+�
+O(σ0µ
+�
+log d) + nO(σ0σn
+�
+pd)σnµ√log d
+σ2npd
+)
+�q−1
+µ2
+≤ Φ(t) + qη
+K2
+�
+O(σ0µ
+�
+log d)
+�q−1
+µ2,
+where the first inequality follows because γ(t)
+j,r,k < 0, the second inequality follows because there are
+Θ(n/K) samples from a given class k and |ℓ′(t)
+j,i | = Θ( 1
+K ), and the last inequality follows because
+µ = σn
+√
+d log d. Now, let C be the constant such that the above holds with big O. Then, we use
+induction to show that Φ(t) ≤ C2σ0µ for all t ≤ T. We proceed as follows.
+Φ(t+1) ≤ Φ(t) + qη
+K2
+�
+Cσ0µ
+�
+log d
+�q−1
+µ2
+≤ T qη
+K2
+�
+Cσ0µ
+�
+log d
+�q−1
+µ2
+≤ C2σ0µpoly log d,
+where the last inequality follows by picking T = C2K2η−1σ2−q
+0
+µ−q√log d
+Cq−1
+= O(K2η−1σ2−q
+0
+µ−q log d).
+D.4
+Converging Phase
+In this subsection, we show that gradient descent can drive the training loss toward zero while
+the signal in the feature is still large.
+An important intermediate step in our argument is the
+development of the following gradient upper bound for multi-class cross-entropy loss.
+In this phase, we are going to show that
+• maxr γ(t)
+j,r,j ≥ m1/q for all j ∈ [K].
+• maxj̸=k,r∈[m] |γ(t)
+j,r,k| ≤ β1 where β1 = �O(σ0µ).
+• maxj,r,i{ζ(t)
+j,r,i, |ω(t)
+j,r,i|} ≤ β2 where β2 = O(σ0σn
+√pd)
+Define W⋆ as follows:
+w⋆
+j,r = w(0)
+j,r + Θ(m log(1/ϵ))µj
+µ2 .
+Lemma D.18. Based on the result from the feature growing phase,
+����
+W(T1) − �
+W⋆���
+2
+F ≤ O(Km3 log2(1/ϵ)µ−2).
+Proof. We first compute
+����
+W(T1) − �
+W(0)���
+2
+F
+31
+
+=
+K
+�
+j=1
+m
+�
+r=1
+�������
+γ(T1)
+j,r,j
+µj ⊙ mj,r
+µ2
++
+�
+k̸=j
+γ(T1)
+j,r,k
+µk ⊙ mj,r
+µ2
++
+�
+i
+ζ(T1)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
++
+�
+i
+ω(T1)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+�������
+2
+2
+≤
+�
+j
+�
+r
+�
+�γ(T1)
+j,r,j
+1
+µ +
+�
+k̸=j
+γ(T1)
+j,r,k
+1
+µ +
+�
+i
+ζ(T1)
+j,r,i
+1
+����ξj,r,i
+���
+2
++
+�
+i
+ω(T1)
+j,r,i
+1
+����ξj,r,i
+���
+2
+�
+�
+2
+≤
+�
+j
+�
+r
+�
+�O( 1
+µ) + K �O(σ0) + n �O(σ0)
+�2
+≤
+�
+j
+�
+r
+�O( 1
+µ2 )
+= �O(Km 1
+µ2 ),
+where the first inequality follows from triangle inequality, the second inequality follows from Lemma
+D.17, and the last inequality follows from our choice of σ0. On the other hand,
+����
+W(0) − �
+W⋆���
+2
+F =
+�
+j,r
+m2 log2(1ϵ) 1
+µ2 = O(Km3 log2(1/ϵ) 1
+µ2 ).
+Thus, we obtain
+����
+W(T1) − �
+W⋆���
+2
+F ≤ 4
+����
+W(T1) − �
+W(0)���
+2
+F + 4
+����
+W(0) − �
+W⋆���
+2
+F ≤ O(Km3 log2(1/ϵ) 1
+µ2 ).
+Lemma D.19 (Gradient Upper Bound). Under Condition 2.2, for t ≤ T ⋆, there exists constant
+C = O(Km2/q max{µ2, σ2
+npd}) such that
+���∇LS(�
+W(t)) ⊙ M
+���
+2
+F ≤ CLS(�
+W(t)).
+Proof. We need to prove that |ℓ′(t)
+yi,i|
+���∇F(�
+W(t), xi) ⊙ M
+���
+2
+F ≤ C. Assume yi ̸= j. Then we obtain
+���∇Fj(�
+Wj, xi) ⊙ M
+���
+F ≤
+�
+r
+���σ′ ��
+�w(t)
+j,r, µyi
+��
+µyi + σ′ ��
+�w(t)
+j,r, ξi
+��
+�ξi
+���
+2
+≤
+�
+r
+σ′ ��
+�w(t)
+j,r, µyi
+��
+∥µyi∥2 + σ′ ��
+�w(t)
+j,r, ξi
+�� ����ξi
+���
+2
+≤ m1/q �
+Fj(�
+Wj, xi)
+�(q−1)/q
+max{µ, Cσn
+�
+pd}
+≤ m1/q max{µ, Cσn
+�
+pd},
+where the first and second inequality follow from triangle inequality, the third inequality follows
+from H¨older’s inequality, and the last inequality follows from Lemma D.12. Similarly, on the other
+32
+
+hand, if yi = j, then
+���∇Fyi(�
+W) ⊙ M
+���
+F ≤ m1/q �
+Fyi(�
+Wyi, xi)
+�(q−1)/q
+max{µ, Cσn
+�
+pd}.
+Therefore,
+�
+j̸=yi
+|ℓ′(t)
+j,i |
+���∇Fj(�
+Wj, xi) ⊙ Mj
+���
+2
+F ≤
+�
+j̸=yi
+|ℓ′(t)
+j,i |m2/qO(max{µ2, σ2
+npd})
+= |ℓ′(t)
+yi,i|m2/qO(max{µ2, σ2
+npd})
+≤ Ke exp{−Fyi(xi)}m2/qO(max{µ2, σ2
+npd}),
+and
+|ℓ′(t)
+yi,i|
+���∇Fyi(�
+Wyi, xi) ⊙ Myi
+���
+2
+F
+≤ Ke exp{−Fyi(xi)}m2/q �
+Fyi(�
+Wyi, xi)
+�2(q−1)/q
+O(max{µ2, σ2
+npd},
+where the inequality follows from Equation (D.2). Thus,
+K
+�
+j=1
+|ℓ′(t)
+j,i |2 ���∇Fj(�
+Wj, xi) ⊙ Mj
+���
+2
+F
+≤ |ℓ′(t)
+yi,i|
+K
+�
+j=1
+|ℓ′(t)
+j,i |
+���∇Fj(�
+Wj, xi) ⊙ Mj
+���
+2
+F
+≤ |ℓ′(t)
+yi,i|Ke exp{−Fyi(xi)}m2/qO(max{µ2, σ2
+npd}
+��
+Fyi(�
+Wyi, xi)
+�(q−1)/q
++ 1
+�
+≤ |ℓ′(t)
+yi,i|O(Km2/q max{µ2, σ2
+npd}),
+(D.4)
+where the first inequality follows because |ℓ′(t)
+j,i | ≤ |ℓ′(t)
+yi,i|, and the last inequality uses the fact that
+exp{−x}(1 + x(q−1)/q) = O(1) for all x ≥ 0.
+The gradient norm can be bounded by
+���∇LS(�
+W(t))
+���
+2
+F ≤
+�
+1
+n
+n
+�
+i=1
+���∇L(�
+W(t), xi)
+���
+F
+�2
+=
+�
+� 1
+n
+n
+�
+i=1
+�
+�
+�
+�
+K
+�
+j=1
+|ℓ′(t)
+j,i |2
+���∇Fj(�
+W(t)
+j , xi)
+���
+2
+F
+�
+�
+2
+≤
+�
+1
+n
+n
+�
+i=1
+|ℓ′(t)
+yi,i|
+���∇F(�
+W(t)
+j , xi)
+���
+F
+�2
+≤
+�
+1
+n
+n
+�
+i=1
+�
+|ℓ′(t)
+yi,i|O(Km2/q max{µ2, σ2nd})
+�2
+33
+
+≤ O(Km2/q max{µ2, σ2
+nd}) 1
+n
+n
+�
+i=1
+|ℓ′(t)
+yi,i|
+≤ O(Km2/q max{µ2, σ2
+nd})LS(�
+W(t)),
+where the first inequality uses triangle inequality, the second inequality follows because |ℓ′(t)
+j,i | ≤
+|ℓ′(t)
+yi,i|, the third inequality uses the bound (D.4), the fourth inequality uses Jensen’s inequality and
+the last inequality follows because |ℓ′(t)
+yi,i| ≤ ℓ(t)
+i .
+Lemma D.20. For T1 ≤ t ≤ T ⋆, we have for all j ̸= yi,
+�
+∇Fyi(�
+W(t)
+yi , xi), �
+W⋆
+yi
+�
+−
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+≥ q log 2qK
+ϵ
+.
+Proof of Lemma D.20. The proof of this lemma depends on the next two lemmas.
+Lemma D.21. For T1 ≤ t ≤ T ⋆ and j = yi, we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+≥ Θ(m1/q log(1/ϵ)).
+Proof. By Lemma D.17, we have
+max
+r
+��
+�w(t)
+j,r, µj
+��
+= max
+r
+�
+�
+�
+�
+�
+�
+�w(0)
+j,r , µj
+�
++ γ(t)
+j,r,j +
+n
+�
+i=1
+ζ(t)
+j,r,i
+�
+�ξj,r,i, µj
+�
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i=1
+ω(t)
+j,r,i
+�
+�ξj,r,i, µj
+�
+����ξj,r,i
+���
+2
+2
+�
+�
+�
+�
+�
+≥ m−1/q − O(σ0µ
+�
+log d) − O(nσ0σn
+√
+dµ√log d
+σnpd )
+≥ Θ(m−1/q),
+where the last inequality follows by picking σ0 ≤ O(m−1n−1µ−1(log d)−1/2). On the other hand,
+���
+�
+�w(t)
+j,r, ξi
+���� ≤
+���
+�
+�w(0)
+j,r , ξi
+���� + |ω(t)
+j,r,i| + |ζ(t)
+j,r,i| + O(n
+�
+log d
+pd α) + O(nµ√log d
+σnpd α) ≤ O(1),
+(D.5)
+where the first inequality follows from Lemma D.11 and the second inequality follows from Equa-
+tion (D.1) and Proposition D.10. Therefore,
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+=
+�
+r
+σ′ ��
+�w(t)
+j,r, µj
+�� �
+µj, �w⋆
+j,r
+�
++
+�
+r
+σ′ ��
+�w(t)
+j,r, ξi
+�� �
+ξi, �w⋆
+j,r
+�
+≥
+�
+r
+σ′ ��
+�w(t)
+j,r, µj
+��
+Θ(m log(1/ϵ)) −
+�
+r
+O(σ0σn
+�
+pd log d + σn
+√log d
+µ
+m log(1/ϵ))
+≥ Θ(m1/q log(1/ϵ)) − O(mσ0σn
+�
+pd log d + σn
+√log d
+µ
+m2 log(1/ϵ))
+≥ Θ(m1/q log(1/ϵ)),
+where the last inequality follows because mσ0σn
+√pd log d = o(1) and σn
+√log d
+µ
+m2 = o(1) by our
+choices of µ, σ0.
+34
+
+Lemma D.22. For T1 ≤ t ≤ T and j ̸= yi, we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+≤ O(1).
+Proof. First we have
+�
+�w(t)
+j,r, µyi
+�
+=
+�
+�w(0)
+j,r , µyi
+�
++ γ(t)
+j,r,yi +
+n
+�
+i=1
+ζ(t)
+j,r,i
+�
+�ξj,r,i, µj
+�
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i=1
+ω(t)
+j,r,i
+�
+�ξj,r,i, µj
+�
+����ξj,r,i
+���
+2
+2
+≤ O(σ0µ
+�
+log d + σ0µ poly log d + nσ0σn
+�
+pdσnµ√log d
+σ2npd
+)
+≤ O(1),
+(D.6)
+where the first inequality follows from Lemma D.7, Lemma D.5 and Lemma D.17, and the last
+inequality follows from our choices of σ0, µ. Then, we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+=
+�
+r
+σ′ ��
+�w(t)
+j,r, µyi
+�� �
+µyi, �w⋆
+j,r
+�
++
+�
+r
+σ′ ��
+�w(t)
+j,r, ξi
+�� �
+ξi, �w⋆
+j,r
+�
+≤ mO(σ0µ
+�
+log d) + mO(σ0σn
+�
+d log d + m log(1/ϵ)σn
+√log d
+µ
+)
+≤ O(1),
+where the second inequality follows from Equation (D.6) and Equation (D.5), and the last inequality
+follows from our choices of µ, σ0.
+Applying the lower bound and upper bound from Lemma D.21 and Lemma D.22, we have
+�
+∇Fyi(�
+W(t)
+yi , xi), �
+W⋆
+yi
+�
+−
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+≥ Θ(m1/q log(1/ϵ)) − O(1)
+≥ q log 2qK
+ϵ
+.
+Lemma D.23. Under the same assumption as Theorem D.1, we have
+����
+W(t) − �
+W⋆���
+2
+F −
+����
+W(t+1) − �
+W⋆���
+2
+F ≥ 5ηLS(�
+W(t)) − ηϵ.
+Proof. To simplify our notation, we define �F (t)
+j (xi) =
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+.
+We use the fact that the network is q-homogeneous.
+����
+W(t) − �
+W⋆���
+2
+F −
+����
+W(t+1) − �
+W⋆���
+2
+F
+= 2η
+�
+∇LS(�
+W(t)) ⊙ M, �
+W(t) − �
+W⋆�
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+= 2η
+n
+n
+�
+i=1
+K
+�
+j=1
+ℓ′(t)
+j,i
+�
+qFj(�
+W(t)
+j ; xi, yi) −
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+��
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+35
+
+≥ 2qη
+n
+n
+�
+i=1
+�
+�log(1 +
+K
+�
+j=1
+eFj−Fyi) − log(1 +
+K
+�
+j=1
+e( �Fj− �Fyi)/q)
+�
+� − η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ 2qη
+n
+n
+�
+i=1
+�
+ℓ(�
+W(t); xi, yi) − log(1 + Ke− log(2qK/ϵ))
+�
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ 2qη
+n
+n
+�
+i=1
+�
+ℓ(�
+W(t); xi, yi) − ϵ
+2q
+�
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ CηLS(�
+W(t)) − ηϵ,
+where the first inequality follows from the convexity of the cross-entropy loss with softmax, the
+second inequality follows from Lemma D.20, the third inequality follows because log(1 + x) ≤ x,
+and the last inequality follows from Lemma D.19 for some constant C.
+Lemma D.24 (Formal Restatement of Lemma 3.5). Under the same assumption as Theorem D.1,
+choose T2 = T1+ ∥�
+W(T1)−�
+W⋆∥
+2
+F
+2ηϵ
+= T1+ �O(Km3 log2(1/ϵ)µ−2). Then for any time t during this stage,
+we have maxr γ(t)
+j,r,j ≥ m1/q for all j ∈ [K], maxj,r,i{|ζ(t)
+j,r,i|, |ω(t)
+j,r,i|} ≤ 2β1, maxj̸=k,r∈[m]{|γ(t)
+j,r,k|} ≤
+2β2, and
+1
+t − T1
+t
+�
+s=T1
+LS(�
+W(s)) ≤
+����
+W(T1) − �
+W⋆���
+2
+F
+Cη(t − T1)
++ ϵ
+C .
+Proof. From Lemma D.17, we have maxrγ(T1)
+j,r,j ≥ m1/q and since γ(t) is an increasing sequence over
+t, we have maxrγ(t)
+j,r,j ≥ m1/q for all t ∈ [T1, T2]. We have
+����
+W(s) − �
+W⋆���
+2
+F −
+����
+W(s+1) − �
+W⋆���
+2
+F ≥ CηLS(�
+W(s)) − ηϵ.
+Taking a telescopic sum from T1 to t yields
+t
+�
+s=T1
+LS(�
+W(s)) ≤
+����
+W(T1) − �
+W⋆���
+2
+F + ηϵ(t − T1)
+Cη
+.
+Combining Lemma D.18, we have
+t
+�
+s=T1
+LS(�
+W(s)) ≤ O(η−1 ����
+W(T1) − �
+W⋆���
+2
+F ) = O(η−1Km3 log2(1/ϵ)µ−2).
+(D.7)
+Define Ψ(t) = maxj,r,i{ζ(t)
+j,r,i, |ω(t)
+j,r,i|} and Φ(t) = maxj̸=k,r∈[m] |γ(t)
+j,r,k| and β2 = �O(σ0µ). Now we use
+induction to prove Ψ(t) ≤ 2β1 and Φ(t) ≤ 2β2. Suppose the result holds for time t ≤ t′. Then
+Ψ(t+1) ≤ Ψ(t) + max
+j,r,i
+�
+�
+�
+�
+�
+η
+n|ℓ′(t)
+j,i | · σ′
+�
+�
+�
+�
+�w(0)
+j,r , ξi
+�
++
+K
+�
+k=1
+γ(t)
+j,r,k
+⟨µk, ξi⟩
+∥µk∥2
+2
++
+n
+�
+i′=1
+Ψ(t)
+�
+�ξj,r,i′, ξi
+�
+����ξj,r,i′
+���
+2
+2
+�
+�
+�
+����ξj,r,i
+���
+2
+2
+�
+�
+�
+�
+�
+36
+
+≤ Ψ(t) + η
+nq max
+i
+|ℓ′(t)
+yi,i|
+�
+O(
+�
+log dσ0σn
+�
+pd) + K log1/q T ⋆ µσn
+√log d
+µ2
++ O(σ2
+npd) + nO(σ2
+n
+√pd log d)
+Θ(σ2npd)
+Ψ(t)
+�q−1
+O(σ2
+npd)
+≤ Ψ(t) + η
+n
+n
+�
+i=1
+|ℓ′(t)
+yi,i|
+�
+O(
+�
+log dσ0σn
+�
+pd) + O(Ψ(t))
+�q−1
+O(σ2
+npd),
+where the second inequality follows by |ℓ′(t)
+j,i | ≤ |ℓ′(t)
+yi,i| and applying the bounds from Lemma D.5,
+and the last inequality follows by choosing K log1/q T ⋆
+√
+d
+= �O( 1
+√
+d) ≪ σ0σn
+√pd. Unrolling the recursion
+by taking a sum from T1 to t′ we have
+Ψ(t′+1) (i)
+≤ Ψ(T1) + η
+n
+t′
+�
+s=T1
+n
+�
+i=1
+|ℓ′(s)
+yi,i|O(σ2
+npd poly log d)βq−1
+1
+(ii)
+≤ Ψ(T1) + η
+nO(σ2
+npd poly log d)βq−1
+1
+t′
+�
+s=T1
+n
+�
+i=1
+ℓ(s)
+i
+= Ψ(T1) + η
+nO(σ2
+npd poly log d)βq−1
+1
+t′
+�
+s=T1
+LS(�
+W(s))
+(iii)
+≤ Ψ(T1) + 1
+nO(Km3µ−2σ2
+npd poly log d)βq−1
+1
+(iv)
+≤ β1 + �O(Km3)βq−1
+1
+(v)
+≤ 2β1,
+where (i) follows from induction hypothesis Ψ(t) ≤ 2β1, (ii) follows from the property of cross-
+entropy loss with softmax |ℓ′
+j,i| ≤ |ℓ′
+yi,i| ≤ ℓi, (iii) follows from Equation (D.7), (iv) follows from
+our choice of µ, n, K, and (v) follows because �O(Km3)β1q−2 ≤ �O(Km3σ0σn
+√pd) ≤ 1. Therefore,
+by induction Ψ(t) ≤ 2β1 holds for time t ≤ t′ + 1.
+On the other hand,
+Φ(t′+1)
+(i)
+≤ Φ(t) + max
+j,r,k,i
+�
+�
+�
+�
+�
+η
+n
+n
+�
+i=1
+I(yi = k)|ℓ′(t)
+j,i |σ′
+�
+�
+�
+�
+�w(0)
+j,r , µk
+�
++
+n
+�
+i′=1
+ζ(t)
+j,r,i′
+�
+�ξj,r,i′, µk
+�
+����ξj,r,i′
+���
+2
+2
++
+n
+�
+i′=1
+ω(t)
+j,r,i′
+�
+�ξj,r,i′, µk
+�
+����ξj,r,i′
+���
+2
+2
+�
+�
+� µ2
+�
+�
+�
+�
+�
+(ii)
+≤ Φ(t) + Θ( η
+K ) max
+j,i |ℓ′(t)
+j,i |
+�
+O(σ0µ
+�
+log d) + nO(σ0σn
+�
+pd)σnµ√log d
+σ2npd
+)
+�q−1
+µ2
+(iii)
+≤ Φ(T1) + Θ( η
+K )µ2
+t
+�
+s=T1
+n
+�
+i=1
+ℓ(s)
+i
+�
+O(σ0µ
+�
+log d)
+�q−1
+(iv)
+≤ β2 + O(m3)βq−1
+2
+37
+
+(v)
+≤ 2β2,
+where (i) follows because γ(t)
+j,r,k ≤ 0, (ii) follows from Lemma D.7 and Lemma D.5, (iii) follows
+because maxj,i |ℓ′(t)
+j,i | ≤ maxi |ℓ′(t)
+yi,i| ≤ maxi ℓ(t)
+i
+≤ �
+i ℓ(t)
+i , (iv) follows from Equation (D.7), and (v)
+follows because O(m3)βq−2
+2
+≤ �O(m3σ0µ) ≤ 1.
+D.5
+Generalization Analysis
+In this subsection, we show that pruning can purify the feature by reducing the variance of the
+noise by a factor of p when a new sample is given.
+Now the network has parameter
+�w⋆
+j,r = �w(0)
+j,r +
+K
+�
+k=1
+γ⋆
+j,r,k
+µk ⊙ mj,r
+µ2
++
+n
+�
+i=1
+ζ⋆
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i=1
+ω⋆
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+.
+We have
+����w⋆
+j,r
+���
+2 = O(σ0
+√pd + µ−1 log1/q(T ⋆) + Kσ0 poly log d + nσ0σn
+√pd
+1
+σn
+√pd) = O(σ0
+√pd).
+Lemma D.25 (Formal Restatement of Lemma 3.6). With probability at least 1−2Km exp
+�
+− (2m)−4/q
+O(σ2
+0σ2npd)
+�
+,
+max
+j,r
+����w⋆
+j,r, ξ
+��� ≤ (2m)−2/q.
+Proof. Since
+�
+�w⋆
+j,r, ξ
+�
+follows a Gaussian distribution with variance O(σ2
+0σ2
+npd), we have
+P
+�����w⋆
+j,r, ξ
+��� ≥ (2m)−2/q�
+≤ 2 exp
+�
+− (2m)−4/q
+O(σ2
+0σ2npd)
+�
+.
+Applying a union bound over j ∈ [K], r ∈ [m] gives the result.
+Theorem D.26 (Formal Restatement of Generalization Part of Theorem 3.1). Under the same
+assumptions as Theorem D.1, within �O(Kη−1σ2−q
+0
+µ−q + K2m4µ−2η−1ϵ−1) iterations, we can find
+�
+W⋆ such that
+• LS(�
+W⋆) ≤ ϵ.
+• LD ≤ O(Kϵ) + exp(−n2/p).
+Proof. Let E be the event that Lemma D.25 holds. Then, we can divide LD(�
+W⋆) into two parts:
+E[ℓ(F(�
+W⋆, x))] = E[I(E)ℓ(F(�
+W⋆, x))]
+�
+��
+�
+I1
++ E[I(Ec)ℓ(F(�
+W⋆, x))]
+�
+��
+�
+I2
+.
+Since LS(�
+W⋆) ≤ ϵ, for each class j ∈ [K] there must exist one training sample (xi, yi) ∈ S with yi =
+j such that ℓ(F(�
+W⋆, xi)) ≤ Kϵ ≤ 1 by pigeonhole principle. This implies that �
+j′̸=j exp(Fj′(xi) −
+Fj(xi)) ≤ 2Kϵ. Conditioning on the event E, by Lemma D.25, we have
+|Fj(�
+W⋆, x) − Fj(�
+W⋆, xi)| ≤
+�
+r
+σ(
+��w⋆
+j,r, ξi
+�
+) +
+�
+r
+σ(
+��w⋆
+j,r, ξ
+�
+)
+38
+
+≤
+�
+r
+(2m)−1 +
+�
+r
+(2m)−1
+≤ 1.
+Thus, we have exp(Fj′(x) − Fj(x)) ≤ 2Kϵe2 = O(Kϵ). Next we bound the term I2.
+ℓ(F(�
+W⋆, x)) = log
+�
+�1 +
+�
+j′̸=y
+exp(Fj′(x) − Fy(x))
+�
+�
+≤ log
+�
+�1 +
+�
+j′̸=y
+exp(Fj′(x))
+�
+�
+≤
+�
+j′̸=y
+log(1 + exp(Fj′(x)))
+≤ K +
+�
+j′̸=y
+Fj′(x)
+= K +
+�
+j′̸=y
+σ(
+��w⋆
+j′,r, µy
+�
+) + σ(
+��w⋆
+j′,r, ξ
+�
+)
+≤ K + Km(O(σ0µ
+�
+log d))q + �O(m(σ0σn
+√
+d)q) ∥ξ/σn∥q
+2
+≤ 2K + ∥ξ/σn∥q
+2 ,
+(D.8)
+where the first inequality follows because Fy(x) ≥ 0, the second and third inequalities follow from
+the property of log function, and the last inequality follows from our choice of σ0 ≤ �O(m−4n−1σ−1
+n d−1/2).
+We further have
+I2 ≤
+�
+E[I(E)]
+�
+E[ℓ(F(�
+W⋆, x))2]
+≤
+�
+P(Ec)
+�
+4K2 + E ∥ξ/σn∥2q
+2
+≤ exp(−Cm−2/qσ−2
+0 σ−2
+n p−1d−1 + log(d))
+≤ exp(−n2/p),
+where the first inequality follows from Cauchy-Schwarz inequality, the second inequality follows
+from Equation (D.8), the third inequality follows from Lemma D.25, and the last inequality follows
+because σ0 ≤ �O(m−4n−1σ−1
+n d−1/2).
+E
+Proof of Theorem 4.1
+In this section, we show that there exists a relatively large pruning fraction (i.e., small p) such that
+while gradient descent is still able to drive the training error toward zero, the learned model yields
+poor generalization. We first provide a formal restatement of Theorem 4.1.
+Theorem E.1 (Formal Restatement of Theorem 4.1). Under Condition 2.2, choose initialization
+variance σ0 = �Θ(m−4n−1µ−1) and learning rate η ≤ �O(1/µ2). For ϵ > 0, if p = Θ(
+1
+Km log d), then
+with probability at least 1−1/ log(d), there exists T = O(η−1nσq−2
+0
+σ−q
+n (pd)−q/2+η−1ϵ−1m4nσ−2
+n (pd)−1)
+39
+
+such that the following holds:
+1. The training loss is below ϵ: LS(�
+W(T)) ≤ ϵ.
+2. The model weight doesn’t learn any of its corresponding signal at all: γ(t)
+j,r,j = 0 for all j ∈
+[K], r ∈ [m].
+3. The model weights is highly correlated with the noise: maxr∈[m] ζ(T)
+j,r,i ≥ Ω(m−1/q) if yi = j.
+Moreover, the testing loss is large:
+LD(�
+W(T)) ≥ Ω(log K).
+The proof of Theorem 4.1 consists of the analysis of the over-pruning for three stages of gradient
+descent: initialization, feature growing phase, and converging phase, and the establishment of the
+generalization property. We present these analysis in detail in the following subsections.
+E.1
+Initialization
+Lemma E.2. When m = poly log d and p = Θ(
+1
+Km log d), with probability 1 − O(1/ log d), for all
+class j ∈ [K] we have |Sj
+signal| = 0.
+Proof. First, the probability that a given class j receives no signal is (1−p)m. We use the inequality
+that
+1 + t ≥ exp {O(t)}
+∀t ∈ (−1/4, 1/4).
+Then the probability that |Sj
+signal| = 0, ∀j ∈ [K] is given by
+(1 − p)Km ≥ exp {−O (pKm)} ≥ 1 − O
+�
+1
+log d
+�
+.
+E.2
+Feature Growing Phase
+Lemma E.3 (Formal Restatement of Lemma 4.3). Under the same assumption as Theorem E.1,
+there exists T1 < T ⋆ such that T1 = O(η−1nσq−2
+0
+σ−q
+n (pd)−q/2) and we have
+• maxr ζyi,r,i ≥ m−1/q for all i ∈ [n].
+• maxj,r,i |ω(t)
+j,r,i| = �O(σ0σn
+√pd).
+• maxj,r,k |γ(t)
+j,r,k| ≤ �O(σ0µ).
+Proof. First of all, recall that from Definition C.1 we have for j = yi
+�
+�w(t)
+j,r, ξi
+�
+=
+�
+�w(0)
+j,r , ξi
+�
++ ζ(t)
+j,r,i +
+�
+k̸=j
+γ(t)
+j,r,k
+�
+µk, �ξj,r,i
+�
+µ2
++
+�
+i′̸=i
+ζ(t)
+j,r,i
+�
+�ξj,r,i′, ξi
+�
+����ξj,r,i′
+���
+2
+2
++
+n
+�
+i′=1
+ω(t)
+j,r,i
+�
+�ξj,r,i′, ξi
+�
+����ξj,r,i′
+���
+2
+2
+.
+40
+
+Let
+B(t)
+i
+= max
+j=yi,r
+�
+ζ(t)
+j,r,i +
+�
+�w(0)
+j,r , ξi
+�
+− O(n log1/q T ⋆
+�
+log d
+pd ) − O(nσ0σn
+�
+pd
+�
+log d
+pd )
+�
+.
+Since maxj=yi,r
+�
+�w(0)
+j,r , ξi
+�
+≥ Ω(σ0σn
+√pd), we have
+B(0)
+i
+≥ Ω(σ0σn
+�
+pd) − O(n log1/q T ⋆
+�
+log d
+pd ) − O(nσ0σn
+�
+pd
+�
+log d
+pd ) ≥ Ω(σ0σn
+�
+pd).
+Let Ti to be the last time that ζ(t)
+j,r,i ≤ m−1/q. We can compute the growth of B(t)
+i
+as
+B(t+1)
+i
+≥ B(t)
+i
++ Θ(ησ2
+npd
+n
+)[B(t)
+i ]q−1
+≥ B(t)
+i
++ Θ(ησ2
+npd
+n
+)[B(0)
+i
+]q−2B(t)
+i
+≥
+�
+1 + Θ
+�
+ησq−2
+0
+σq
+npq/2dq/2
+n
+��
+B(t)
+i .
+Therefore, B(t)
+i
+will reach 2m−1/q within �O(η−1nσq−2
+0
+σ−q
+n (pd)−q/2) iterations.
+On the other hand, by Proposition D.10, we have |ω(t)
+j,r,i| ≤ β+6Cnα
+�
+log d
+pd = O(σ0σn
+√pd log d).
+E.3
+Converging Phase
+From the first stage we know that
+�w(T1)
+j,r
+= �w(0)
+j,r + +
+�
+k̸=j
+γ(t)
+j,r,k
+µk ⊙ mj,r
+µ2
++
+n
+�
+i=1
+ζ(T1)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i=1
+ω(T1)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+.
+Now we define �
+W⋆ as follows:
+�w⋆
+j,r = �w(0)
+j,r + Θ(m log(1/ϵ))
+�
+��
+n
+�
+i=1
+I(j = yi)
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+�
+�� .
+Lemma E.4. Based on the result from feature growing phase,
+����
+W(T1) − �
+W⋆���
+F ≤ O(m2n1/2 log(1/ϵ)σ−1
+n (pd)−1/2).
+Proof. We derive the following bound:
+����
+W(T1) − �
+W⋆���
+F
+≤
+����
+W(T1) − �
+W(0)���
+F +
+����
+W(0) − �
+W⋆���
+F
+41
+
+≤
+�
+j,r
+�
+�
+�
+������
+�
+k̸=j
+γ(t)
+j,r,k
+µk
+µ2
+������
+2
++
+�������
+n
+�
+i=1
+ζ(T1)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+�������
+2
++
+�������
+n
+�
+i=1
+ω(T1)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+�������
+2
+�
+�
+� + Θ(m2n1/2 log(1/ϵ)σ−1
+n (pd)−1/2)
+≤ Km(O(
+√
+Kσ0) + O(n1/2σ−1
+n (pd)−1/2 log1/q T ⋆)) + �O(m2n1/2 log(1/ϵ)σ−1
+n (pd)−1/2)
+≤ �O(m2n1/2 log(1/ϵ)σ−1
+n (pd)−1/2),
+where the first inequality follows from triangle inequality, the second inequality follows from the
+expression of W(T1), W⋆, and the third inequality follows from Lemma D.5 and the fact that
+ζ(t)
+j,r,i > 0 if and only if j = yi.
+Lemma E.5. For T1 ≤ t ≤ T ⋆, we have
+�
+∇Fyi(�
+Wyi, xi), �
+W⋆
+yi
+�
+−
+�
+∇Fj(�
+Wj, xi), �
+W⋆
+j
+�
+≥ q log 2qK
+ϵ
+.
+Lemma E.6. For T1 ≤ t ≤ T ⋆ and j = yi, we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+≥ Θ(m1/q log(1/ϵ)).
+Proof. By Lemma D.5, we have
+�
+�ξj,r,i, �w⋆
+j,r
+�
+= Θ(m log(1/ϵ)) and by Lemma E.3 for j = yi,
+maxr
+�
+�w(t)
+j,r, ξi
+�
+≥ maxr ζj,r,i − maxr
+�
+�w(0)
+j,r , ξi
+�
+− O(n
+�
+log d
+d α) ≥ Θ(m−1/q). Then we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+=
+m
+�
+r=1
+σ′ ��
+�w(t)
+j,r, ξi
+�� �
+�ξj,r,i, �w⋆
+j,r
+�
+≥ Θ(m1/q log(1/ϵ)).
+Lemma E.7. For T1 ≤ t ≤ T ⋆ and j ̸= yi, we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+≤ O(1).
+Proof. We first compute
+�
+�w⋆
+j,r, ξi
+�
+=
+�
+�w(0)
+j,r , ξi
+�
++Θ(m log(1/ϵ)) �n
+i=1 I(j = yi)⟨�ξj,r,i,ξi⟩
+∥�ξj,r,i∥
+2
+2
+= O(σ0σn
+√pd log d).
+Further,
+�
+�w(t)
+j,r, ξi
+�
+=
+�
+�w(0)
+j,r , ξi
+�
++
+�
+k̸=j
+γ(t)
+j,r,k
+�
+µk, �ξj,r,i
+�
+µ2
++
+n
+�
+i=1
+ζ(t)
+j,r,i
+�
+�ξj,r,i, ξi
+�
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i=1
+ω(t)
+j,r,i
+�
+�ξj,r,i, ξi
+�
+����ξj,r,i
+���
+2
+2
+≤ O(σ0σn
+�
+pd log d),
+42
+
+where the inequality follows from Lemma D.5 and Lemma D.15. Thus, we have
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+=
+m
+�
+r=1
+σ′ ��
+�w(t)
+j,r, ξi
+�� �
+�ξj,r,i, �w⋆
+j,r
+�
+≤ mO
+�
+σ0σn
+�
+pd log d
+�q
+≤ O(1),
+where the last inequality follows from our choice of σ0 ≤ �O(m−1/qµ−1).
+Lemma E.8. Under the same assumption as Theorem E.1, we have
+���W(t) − W⋆���
+2
+F −
+���W(t+1) − W⋆���
+2
+F ≥ CηLS(�
+W(t)) − ηϵ.
+Proof. To simplify our notation, we define �F (t)
+j (xi) =
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+�
+. The proof is exactly
+the same as the proof of Lemma D.23.
+����
+W(t) − �
+W⋆���
+2
+F −
+����
+W(t+1) − �
+W⋆���
+2
+F
+= 2η
+�
+∇LS(�
+W(t)) ⊙ M, �
+W(t) − �
+W⋆�
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+= 2η
+n
+n
+�
+i=1
+K
+�
+j=1
+ℓ′(t)
+j,i
+�
+qFj(�
+W(t)
+j ; xi, yi) −
+�
+∇Fj(�
+W(t)
+j , xi), �
+W⋆
+j
+��
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ 2qη
+n
+n
+�
+i=1
+�
+�log(1 +
+K
+�
+j=1
+eFj−Fyi) − log(1 +
+K
+�
+j=1
+e( �Fj− �Fyi)/q)
+�
+� − η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ 2qη
+n
+n
+�
+i=1
+�
+ℓ(�
+W(t); xi, yi) − log(1 + Ke− log(2qK/ϵ))
+�
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ 2qη
+n
+n
+�
+i=1
+�
+ℓ(�
+W(t); xi, yi) − ϵ
+2q
+�
+− η2 ���∇LS(�
+W(t)) ⊙ M
+���
+2
+F
+≥ CηLS(�
+W(t)) − ηϵ,
+where the first inequality follows from the convexity of the cross-entropy loss with softmax, the
+second inequality follows from Lemma D.20, the third inequality follows because log(1 + x) ≤ x,
+and the last inequality follows from Lemma D.19 for some constant C > 0.
+Lemma E.9 (Formal Restatement of Lemma 4.4). Under the same assumption as Theorem E.1,
+choose T2 = T1 +
+�
+∥�
+W(T1)−�
+W⋆∥
+2
+F
+2ηϵ
+�
+= T1 + �O(η−1ϵ−1m4nσ−2
+n (pd)−1). Then for any time t during
+this stage we have maxj,r |ω(t)
+j,r,i| = O(σ0
+√pd) and
+1
+t − T1
+t
+�
+s=T1
+LS(�
+W(s)) ≤
+����
+W(T1) − �
+W⋆���
+2
+F
+Cη(t − T1)
++ ϵ
+C .
+43
+
+Proof. We have
+����
+W(s) − ���
+W⋆���
+2
+F −
+����
+W(s+1) − �
+W⋆���
+2
+F ≥ CηLS(�
+W(s)) − ηϵ.
+Taking a telescopic sum from T1 to t yields
+t
+�
+s=T1
+LS(�
+W(s)) ≤
+����
+W(T1) − �
+W⋆���
+2
+F + ηϵ(t − T1)
+Cη
+.
+Combining Lemma E.4, we have
+t
+�
+s=T1
+LS(�
+W(s)) ≤ O(η−1 ����
+W(T1) − �
+W⋆���
+2
+F ) = �O(η−1m4nσ−2
+n (pd)−1).
+E.4
+Generalization Analysis
+Theorem E.10 (Formal Restatement of the Generalization Part of Theorem 4.1). Under the same
+assumption as Theorem E.1, within O(η−1nσq−2
+0
+σ−q
+n (pd)−q/2+η−1ϵ−1m4nσ−2
+n (pd)−1) iterations, we
+can find �
+W(T) such that LS(�
+W(T)) ≤ ϵ, and LD(�
+W(t)) ≥ Ω(log K).
+Proof. First of all, from Lemma E.9 we know there exists t ∈ [T1, T2] such that LS(�
+W(T)) ≤ ϵ.
+Then, we can bound
+����w(t)
+j,r
+���
+2 =
+�������
+�w(0)
+j,r +
+�
+k̸=j
+γ(t)
+j,r,k
+µk
+µ2 +
+n
+�
+i=1
+ζ(t)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
++
+n
+�
+i=1
+ω(t)
+j,r,i
+�ξj,r,i
+����ξj,r,i
+���
+2
+2
+�������
+2
+≤
+����w(0)
+j,r
+���
+2 +
+�
+k̸=j
+|γ(t)
+j,r,k| 1
+µ +
+n
+�
+i=1
+ζ(t)
+j,r,i
+1
+����ξj,r,i
+���
+2
++
+n
+�
+i=1
+|ω(t)
+j,r,i|
+1
+����ξj,r,i
+���
+2
+≤ O(σ0
+√
+d) + �O(nσ−1
+n (pd)−1/2).
+Consider a new example (x, y). Taking a union bound over r, with probability at least 1 − d−1, we
+have
+���
+�
+w(t)
+y,r, ξ
+���� = �O(σ0σn
+√
+d + n(pd)−1/2),
+for all r ∈ [m]. Then,
+Fy(x) =
+m
+�
+r=1
+σ
+��
+�w(t)
+j,r, µy
+��
++ σ
+��
+�w(t)
+j,r, ξ
+��
+≤ m max
+r
+���
+�
+w(t)
+y,r, ξ
+����
+q
+≤ m �O(σq
+0σq
+ndq/2 + nq(pd)−q/2)
+≤ 1,
+44
+
+where the last inequality follows because σ0 ≤ �O(m−1/qµ−1) and d ≥ �Ω(m2/qn2).
+Thus, with
+probability at least 1 − 1/d,
+ℓ(F(�
+W(t); x)) ≥ log(1 + (K − 1)e−1).
+45
+