On the Generalization of Neural Networks Trained with SGD: Information-Theoretical Bounds and Implications

Understanding the generalization behaviour of deep neural networks is an important1 theme of modern research in machine learning . In this paper , we follow up on2 a recent work of Neu [ 49 ] and present new information-theoretic upper bounds3 for the generalization error of neural networks trained with SGD . Our bounds and4 experimental study provide new insights on the SGD training of neural networks.5 They also point to a new and simple regularization scheme which we show performs6 comparably to the current state of the art.7 1 Introduction8 The outstanding performance of deep learning has brought to the surface some intriguing properties9 of deep neural networks , one of which is the observation that despite their high capacity , deep neural10 networks tend to generalize well [ 80 ] . This contradicts classical wisdom in statistical learning theory11 ( e.g. , [ 71 ] ) and has stimulated intense research interest in understanding the generalization behaviour12 of modern neural networks.13 One theme of research focuses on the study of over-parameterized neural networks , where generaliza-14 tion bounds are obtained [ 22 , 1 , 6 , 50 , 52 , 53 , 2 , 3 ] and a curious “ double descent ” phenomenon is15 observed and analyzed [ 9 , 46 , 78 ] . New bounding techniques for analyzing generalization have also16 been developed , utilizing information theoretic quantities [ 64 , 65 , 77 , 5 , 13 , 69 , 23 , 7 , 4 , 30 , 84 ] . The17 bounds provided by these techniques have the advantages of accounting for both the data structure18 and the learning algorithm.19 The generalization ability of neural networks trained with mini-batched stochastic gradient descent20 [ 61 ] , simply referred to SGD in this paper , has also been widely studied . Specifically , built on a21 connection between stability and generalization [ 12 ] , a stability-based bound is first presented in [ 32 ] ,22 followed by a surge of research effort exploiting similar approaches [ 44 , 17 , 26 , 43 , 8 ] . Information-23 theoretic bounding techniques have also demonstrated great power in analyzing SGD-like algorithms.24 For example , [ 55 ] is the first to utilize information-theoretical bound in analyzing the generalization25 ability of SGLD [ 28 , 74 ] . The bound was subsquently improved by [ 47 , 31 , 62 , 72 ] . Inspired by the26 work of [ 55 ] , [ 49 ] presents an information- theoretic analysis of the models trained with SGD . The27 analysis of [ 49 ] constructs an auxiliary weight process parallel to SGD training and upper-bounds the28 generalization error through this auxiliary process.29 Another line of research connects the generalization of neural networks with the flatness of loss30 minima [ 35 ] found by SGD or its variant [ 40 , 20 , 24 , 51 , 16 , 37 , 38 , 83 , 27 ] . This understanding31 has led to the discovery of new SGD-based training algorithms for improved generalization . For32 example , in a concurrent development by [ 83 ] and [ 27 ] , a local “ max-pooling ” operation is applied to33 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . the loss landscape prior to the SGD updates . This approach , referred to as AMP [ 83 ] or SAM [ 27 ] , is34 shown to make SGD favor flatter minima and achieve the state-of-the-art performance among various35 competitive regularization schemes [ 83 ] .36 In this paper , we focus on investigating the generalization of neural networks trained with SGD.37 We build upon the work of [ 49 ] . Following the same construction of the auxiliary weight process38 in [ 49 ] , we present upper bounds of generalization error that improve upon [ 49 ] in two ways . The39 first improvement is via removing an unnecessary term in the bounds of [ 49 ] by invoking the HWI40 inequality [ 60 ] . The second improvement is via replacing a sample-level mutual information term in41 [ 49 ] with an instance-level mutual information term , exploiting a recent result of [ 13 ] . The bounds we42 obtain decompose into two terms , one measuring the impact of training trajectories ( “ the trajectory43 term ” ) and the other measuring the impact of the flatness of the found solution ( “ the flatness term ” ) .44 We empirically validate the derived bounds . Various insights are also obtained experimentally45 concerning the generalization of neural networks under SGD training . For example , the batch size of46 SGD appears to impact the trajectory term and the flatness term in opposite ways , which complicates47 the overall dependency of generalization error on batch sizes . A particular interesting observation48 from our experiments is that a key quantity arising in the trajectory term of the bounds , which we49 refer to as gradient dispersion1 , reveals a double descent phenomenon with respect to training epochs.50 Most intriguingly , the valley in the double descent curve appears to mark the great divide between the51 “ generalization regime ” and the “ memorization regime ” of training . Furthering from this observation,52 we also show that it is possible to reduce the memorization effect by dynamically clipping the gradient53 and reducing its dispersion.54 Our bounds also inspire a natural and simple solution to alleviate generalization error . Specifically,55 we propose a new training scheme , referred to as Gaussian model perturbation ( GMP ) , aiming at56 reducing the flatness term of the bounds . This scheme effectively applies a local “ average pooling ” to57 the empirical risk surface prior to SGD , greatly resembling the “ max-pooling ” approach adopted in58 AMP [ 83 ] . We demonstrate experimentally that GMP achieves a competitive performance with the59 current art of regularization schemes.60 Length constraints precludes elaboration at places . The reader is referred to supplementary materials61 for proofs and additional information.62 Other Related Literature Gradient dispersion is mostly studied from optimization perspectives [ 11,63 63 , 39 , 75 , 25 ] . Prior to this work , only a few works relate gradient dispersion with the generalization64 behaviour of the networks . In [ 49 , 72 ] , gradient dispersion also appears in the generalization bounds.65 In [ 38 ] , gradient dispersion is argued to capture a notion of “ flatness ” of the local minima of the loss66 landscape , thereby correlating with generalization.67 Injecting noise in the training process has been proposed in various regularization schemes , for68 example , [ 10 , 14 , 15 , 68 , 73 ] . But unlike the Gaussian model perturbation scheme derived in this69 paper , where noise is injected to the model parameters , noise in those schemes is injected either to70 the training data or to the network activation.71 Gradient clipping is a common technique for preventing gradient exploding ( see , e.g. , [ 45 , 56 ] ) .72 This technique is also used in [ 82 ] to accelerate training . In this paper , gradient clipping is used to73 investigate and control the impact of gradient dispersion on generalization error.74 2 Preliminaries75 Population Risk , Empirical Risk and Generalization Error Unless otherwise noted , a random76 variable will be denoted by a capitalized letter ( e.g. , Z ) , and its realization denoted by the correspond-77 ing lower-case letter ( e.g . z ) . Let Z be the instance space of interest and µ be an unknown distribution78 on Z , specifying random variable Z . Let W ⊆ Rd be the space of hypotheses . Suppose that a79 training sample S = ( Z1 , Z2 , . . . , Zn ) is drawn i.i.d . from µ and that a stochastic learning algorithm80 A takes S as its input and outputs a hypothesis W ∈ W according to some conditional distribution81 PW |S mapping Zn toW . Let ` : W × Z → R+ be a loss function , where ` ( w , z ) measures the82 “ unfitness ” or “ error ” of any z ∈ Z with respect to a hypothesis w ∈ W . The population risk , for any83 1The quantity is often referred to as gradient variance in the literature [ 49 , 72 ] , but we prefer “ dispersion ” to “ variance ” so as to better comply with the mathematical conventions and avoid possible confusion . w ∈ W , is defined as84 Lµ ( w ) , EZ∼µ [ ` ( w , Z ) ] . The goal of learning is to find a hypothesis w that minimizes the population risk . But since µ is only85 partially accessible via the sample S , in practice , we instead turn to the empirical risk , defined as86 LS ( w ) , 1 n n∑ i=1 ` ( w , Zi ) . The expected generalization error of the learning algorithm A is then defined as87 gen ( µ , PW |S ) , EW , S [ Lµ ( W ) − LS ( W ) ] , where the expectation is taken over the joint distribution of ( S , W ) ( i.e. , µn ⊗ PW |S ) .88 Throughout this paper , we take ` as a continuous function ( adopting the usual notion “ surrogate loss ” 89 [ 66 ] . ) . Additionally , we assume that ` is differentiable almost everywhere with respect to both w and90 z . Furthermore we assume that ` ( w , Z ) is R-subgaussian2 for any w ∈ W . Note that a bounded loss91 is guaranteed to be subgaussian for all µ and all w ∈ W . Let I ( X ; Y ) denote the mutual information92 [ 18 ] between any pair of random variables ( X , Y ) . The following results are known.93 Lemma 1 ( [ 77 , Theorem 1. ] ) . The expected generalization error of algorithm A is bounded by94 |gen ( µ , PW |S ) | ≤ √ 2R2 n I ( W ; S ) , 95 Lemma 2 ( [ 13 , Proposition 1. ] ) . The expected generalization error of algorithm A is bounded by96 |gen ( µ , PW |S ) | ≤ 1 n n∑ i=1 √ 2R2I ( W ; Zi ) , 97 Stochastic Gradient Descent We now restrict the learning algorithm A to be the mini-batched98 stochastic gradient descent ( SGD ) algorithm for empirical risk minimization . For each training epoch,99 the dataset S is randomly split into m disjoint mini-batches , each having size b , namely , n = mb.100 Based on each batch , one parameter update is performed . Specifically , let Bt denote the batch used101 for the tth update . Define102 g ( w , Bt ) , 1 b ∑ z∈Bt ∇w ` ( w , z ) , namely , g ( w , Bt ) is the average gradient computed for the batch Bt with respect to parameter w. The103 rule for the tth parameter update is then104 Wt , Wt−1 − λtg ( Wt−1 , Bt ) , where λt is the learning rate at the step t. The initial parameter setting W0 is assumed to be drawn105 from the zero-mean spherical Gaussian N ( 0 , σ20Id ) with variance σ20 in each dimension . We will106 assume that the SGD algorithm stops after T updates and outputs WT as the learned model parameter.107 Given the training sample S , let ξ govern the randomness in the sequence ( B1 , B2 , . . . , BT ) of108 batches . For the simplicity of notion , we will fix the configuration of ξ . That is , we will assume a109 fixed “ batching trajectory ” , or a fixed way to shuffle the example indices { 1 , . . . , n } and divide them110 into m batches in each epoch . The presented generalization bounds of this paper can be extended111 to the case where the batching trajectory is uniformly random ( as we set up above ) . This merely112 involves averaging over all batching trajectories or taking expectation over ξ.113 Auxiliary Weight Process We now associate with the SGD algorithm an auxiliary weight process114 { W̃t } . Let σ2 be given , and let σ1 , σ2 , . . . , σT be a sequence of positive real numbers . Define115 W̃0 , W0 , and W̃t , W̃t−1 − λtg ( Wt−1 , Bt ) +Nt , for t > 0 , where Nt ∼ N ( 0 , σ2t Id ) is a Gaussian noise . The relationship between this auxiliary weight process116 { W̃t } and the weight process { Wt } in SGD is shown in the Bayesian network below.117 2Recall that a random variable X is R-subgaussian [ 60 ] if for any ρ , logE exp ( ρ ( X − EX ) ) ≤ ρ2R2/2 . N1 N2 · · · NT−1 NT ↓ ↓ ↓ ↓ W̃0 → W̃1 → W̃2 → · · · → W̃T−1 → W̃T ‖ ↗ ↗ ↗ ↗ W0 → W1 → W2 → · · · → WT−1 → WT 118 Let ∆t = ∑t τ=1Nτ . Noting that the weight updates in { W̃t } uses the same gradient signal as that119 used in { Wt } ( which depends onWt−1 not W̃t−1 ) , it is immediate that W̃t = Wt+∆t . Note that this120 auxiliary process follows the same construction as [ 49 ] , which we will use to study the generalization121 error of SGD.122 To that end , define gradient dispersion at parameter w by123 V ( w ) , E [ ||∇w ` ( w , Z ) − E [ ∇w ` ( w , Z ) ] ||22 ] , where the expectation is taken over Z ∼ µ.124 For a given sample s ∈ Zn , define125 γ ( w , s ) , E [ Ls ( w + ∆T ) − Ls ( w ) ] , where the expectation is taken over ∆T and Ls ( w ) is the empirical risk of s at parameter w.126 In the remainder of the paper , let S′ denote another sample drawn from µn , independent of all other127 random variables . The main generalization bound in [ 49 ] is re-stated below.128 Lemma 3 ( [ 49 , Theorem 1. ] ) . The generalization error of SGD is upper bounded by129 |gen ( µ , PWT |S ) | ≤ √√√√2R2 n T∑ t=1 λ2t σ2t E [ Ψ ( Wt−1 ) + V ( Wt−1 ) b ] + |E [ γ ( WT , S ) − γ ( WT , S′ ) ] | , where Ψ ( wt−1 ) , E [ ||∇w ` ( wt−1 , Z ) −∇w ` ( wt−1 + ζ , Z ) ||22 ] and ζ ∼ N ( 0 , 2 ∑t−1 i=1 σ 2 i Id ) .130 The term Ψ ( wt−1 ) in the bound is referred to as “ local gradient sensitivity ” in [ 49 ] .131 3 New Generalization Bounds for SGD132 We first prove that the generalization bound in Lemma 3 can be tightened by removing the local133 gradient sensitivity term Ψ ( wt−1 ) . The key observation is that an independence condition used for134 establishing Lemma 3 in [ 49 ] is unnecessary ( see Lemma 4 in [ 49 ] ) . This requires invoking a vector135 version of the HWI inequality [ 60 , Lemma 3.4.2 ] , which we prove in this paper.136 Lemma 4 . Let X and Y be two random vectors in Rd , and let N ∼ N ( 0 , Id ) be independent of137 ( X , Y ) . Then , for every t > 0 , DKL ( PX+√tN ||PY+√tN ) ≤ 1 2tE [ ||X − Y ||2 ] .138 Here DKL is the KL divergence . Note that the bound in Lemma 3 relies on a similar result which139 however requires the independence of X and Y . Using Lemma 4 , we obtain the following theorem.140 Theorem 1 . The generalization error of SGD is upper bounded by141 |gen ( µ , PWT |S ) | ≤ √√√√2R2 nb T∑ t=1 λ2t σ2t E [ V ( Wt−1 ) ] + |E [ γ ( WT , S ) − γ ( WT , S′ ) ] | . 142 The proof of this theorem , as the bounds in [ 49 ] , relies on Lemma 1 and the sample-level mutual143 information bound therein . This theorem can be further tightened by exploiting the fact that the144 instance-level mutual information bound in Lemma 2 is in fact tighter than the sample-level mutual145 information bound in Lemma 1 , as shown in [ 13 ] . The main ingredient to proceed in this direction is146 the following lemma.147 Lemma 5 . LetGt = −λtg ( Wt−1 , Bt ) . If Zi ∈ Bt , then I ( Gt+Nt ; Zi|W̃t−1 ) ≤ λ 2 t σ2t b 2E [ V ( Wt−1 ) ] .148 149 In this lemma , the mutual information I ( Gt +Nt ; Zi|W̃t−1 ) roughly indicates the degree by which150 the SGD ’ s updating signal Gt ( smoothed with noise ) depends on an individual training instance151 Zi , when Zi is used for computing the gradient . When this dependency is strong ( giving rise to a152 high value of the mutual information ) , the model conceivably tends to overfit the individual training153 instances . This lemma suggests that the strength of this dependency can be upper-bounded by the154 expected gradient dispersion at the current weight configuration . In our experiments , we will estimate155 the expected gradient dispersion and validate this intuition.156 It is remarkable that the noise { Nt } plays an important role for the bound to hold . To see this,157 consider b = 1 and Z is countable and large . Then I ( Gt ; Zt|Wt−1 ) is merely the conditional entropy158 H ( Zi|Wt−1 ) , which would grow with sample size n at least as log n. Upper-bounding it with a159 quantity independent of n would be impossible – This justifies the construction of the auxiliary160 weight process.161 We now state our main theorem . Unlike Theorem 1 , which considers a random batching trajectory,162 this theorem considers a fixed batching trajectory to keep the expression less cluttered . For that163 batching trajectory , we will use Ti to denote the set of indices of batches Bt containing instance Zi.164 Theorem 2 . The expected generalization error of SGD is bounded by165 |gen ( µ , PWT |S ) | ≤ R nb n∑ i=1 √√√√∑ t∈Ti 2λ2t σ2t E [ V ( Wt−1 ) ] + |E [ γ ( WT , S ) − γ ( WT , S′ ) ] | . 166 With an additional assumption , the second term in the bound can be re-expressed , as shown in the167 following corollary.168 Corollary 1 . Assume Lµ ( wT ) ≤ E∆ [ Lµ ( wT + ∆T ) ] , then the following holds,169 gen ( µ , PWT |S ) ≤ R nb n∑ i=1 √√√√∑ t∈Ti 2λ2t σ2t E [ V ( Wt−1 ) ] + ∑T t=1 σ 2 t 2 Tr ( E [ HWT ( Z ) ] ) , where HWT is the Hessian matrix of the loss with respect to WT and Tr ( · ) denotes trace.170 Corollary 1 follows directly from the second order Taylor expansion of the second term in the bound171 of Theorem 2 . The condition Lµ ( wT ) ≤ E∆ [ Lµ ( wT + ∆T ) ] indicates that the perturbation does172 not decrease the population risk . This is also assumed in [ 27 ] in the derivation of a PAC-Bayesian173 generalization bound.174 Notably , in the bound of Theorem 2 , the first term captures the impact of the training trajectory175 ( “ trajectory term ” ) , and the second term captures the impact of the final solution . As seen in Corollary176 1 , this term in fact measures the flatness for the loss landscape at the found solution ( “ flatness term ” ) .177 The previous bound of [ 49 ] ( Lemma 3 ) and its tightened version in Theorem 1 also similarly contain a178 trajectory term and a flatness term . Despite that the flatness term there are identical to that in Theorem179 2 , we now show the trajectory term in Theorem 2 does improve on its counter-part in Theorem 1.180 Lemma 6 . Assume the instances are sampled without replacement in every epoch . Then the trajectory181 term in Theorem 2 is upper-bounded by182 min Rn T∑ t=1 √ 2λ2t σ2t E [ V ( Wt−1 ) ] , √√√√2R2 nb T∑ t=1 λ2t σ2t E [ V ( Wt−1 ) ] . 183 The condition in Lemma 6 is usually satisfied in practice . This lemma then immediately implies that184 the trajectory term in Theorem 2 is no worse than that in Theorem 1 . Incorporating this result , if we185 restrict the smoothness of the loss function ` , we may obtain another version of the generalization186 bound ( although the flatness term therein is expected to be looser than that in Corollary 1 ) .187 Corollary 2 . If the loss function is differentiable and β-smooth with respect to w , then under the188 condition of Lemma 6,189 |gen ( µ , PWT |S ) | ≤min Rn T∑ t=1 √ 2λ2t σ2t E [ V ( Wt−1 ) ] , √√√√2R2 nb T∑ t=1 λ2t σ2t E [ V ( Wt−1 ) ] + βd T∑ t=1 σ2t . 190 To conclude , we remark that these bounds suggest that in order for the model to generalize well , both191 the trajectory term and the flatness term need to be small — the former involves the interaction of the192 learning rate and batch size with the gradient dispersion along the training trajectory , whereas the193 latter depends on the flatness of the empirical risk surface at the found solution.194 4 Experimental Study195 Bound Verification We first verify our bound in Corollary 1 by training an MLP ( with one hidden196 layer ) and an AlexNet [ 42 ] on MNIST and CIFAR10 [ 41 ] , respectively . To simplify estimation , we197 fix the weight initialization and set σt and λt to be constants σ and λ , respectively . To compute198 ∑n i=1 √∑ t∈Ti E [ V ( Wt−1 ) ] , we compute the gradient dispersion as its empirical estimate from a199 batch , utilizing a PyTorch [ 54 ] library BackPack [ 19 ] . To compute Tr ( E [ HWT ( Z ) ] ) , we randomly200 sample 10 % of the training data and use the PyHessian library [ 79 ] to compute the Hessian . Since201 every choice of σ gives a valid generalization bound in Corollary 1 , we need to find the optimal202 σ , which gives the tightest bound . This can be done by simply utilizing the fact A/σ + σ2B ≥203 3 ( A/2 ) 2/3B1/3 for any positive A and B , where the equality is achieved by the optimal σ . We set204 the sub-gaussian parameter R = 0.1 . The implementation in this paper is on PyTorch , and all the205 experiments are carried out on NVIDIA Tesla V100 GPUs ( 32 GB ) .206 We perform experiments with varying network width and varying levels of label noise . Specifically,207 label noise level refers to the setting where we replace the labels of fraction of the training and208 testing instances with random labels . The estimated bound is compared against the true generalization209 gap , namely , the difference between the training loss and testing loss , and is shown in Figure 1.210 In Figure 1 , we see that in all cases the estimated bound follows closely the trend of the true211 generalization gap . The fact that the bound curve consistently tracks the gap curve under various212 label noise levels indicates that our bound very well captures the changes of the data distribution.213 Note that in Figure 1 ( a ) and ( b ) , our bound decays with the increase of the model size , showing a214 trend as opposite to the bounds obtained in classical learning theory . But such a trend clearly better215 explains the generalization behaviour of modern neural networks.216 Learning Rate and Batch Size . The learning rate and batch size in SGD have explicitly appeared217 in the trajectory term of the bound in Theorem 2 . From the way they appear in the bound , one may be218 tempted to assert that a small learning rate or large batch size will improve generalization . This would219 then contradict some previous observations [ 37 , 76 , 33 ] , in which increasing the ratio of learning rate220 to batch size will benefit generalization . We now investigate this by performing experiments with221 varying learning rates and batch sizes . In our experiments , the model is continuously updated until222 the average training loss drops below 0.0001 . We separate trajectory and flatness terms of the bound223 and plot them in Figure 2.224 A key observation in Figure 2 is that the learning rate impacts the trajectory term and the flatness term225 in opposite ways , as seen , for example , in ( a ) and ( b ) , where the two set of curves swap their orders in226 the two figures . On the other hand , the batch size also impacts the two terms in opposite ways , as seen227 in ( a ) and ( b ) where curves decrease in ( a ) but increase in ( b ) . This makes the generalization bound,228 i.e. , the sum of the two terms , have a rather complex relationship with the settings of learning rate229 and batch size . This relationship is further complicated by the fact that a small learning rate requires230 a longer training time , or a larger number T of training iterations , which increases the number that231 are summed over in the trajectory term . Nonetheless , we do observe that a smaller batch size gives a232 lower value of the flatness term ( ( b ) and ( d ) ) , confirming the previous wisdom that small batch sizes233 enable the neural network to find a flat minima [ 40 ] .234 Double Descent of Gradient Dispersion We experimentally investigate the impact of gradient235 dispersion on the training of the neural networks by fixing the learning rate , batch size and weight236 initialization for the each model ( MLP for MNIST , AlexNet for CIFAR10 ) . For each model and237 various label noise levels , we plot in Figure 3 the evolution of the ( empirical ) gradient dispersion238 V̂ ( wt ) , training accuracy and testing accuracy across training epochs.239 An intriguing epoch-wise “ double descent ” phenomenon is observed , particularly when the labels are240 noisy . According to the double descent curve , the training may be split into three phases ( e.g. , Figure241 3 ( h ) ) . In the first phase , the gradient dispersion rapidly descends and maintains a very low level . In242 this phase , both training and test accuracies increase while maintaining a very small generalization243 gap . This suggests that the network in this phase is extracting useful patterns and generalizes well.244 In the second phase , the gradient dispersion starts increasing until it reaches a peak value . In this245 phase , the training and testing accuracies gradually diverge , marking the model entering an overfitting246 or “ memorization ” regime – when the data contains the noisy labels , the network mostly tries to247 memorize the labels in the training set . In the third phase , the gradient dispersion descends again,248 reaching a low value . In this phase , the model continuously overfits the training data , until the training249 and testing curves reach their respective maximum and minimum . It appears that the timing of the250 three phases depends on the dataset and the label noise level . For simpler data ( e.g . MNIST ) and251 cleaner datasets ( e.g . CIFAR10 with low label noise ) , the first phase may be shorter . This is arguably252 because in these datasets , extracting useful patterns is relatively easier . Nonetheless , the valley in the253 double-descent curve appears to mark a “ great divide ” between generalization and memorization.254 Dynamic Gradient Clipping Inspired by our generalization bounds and above observations , one255 way to reduce the generalization error is to control the trajectory term of the bounds by reducing the256 gradient dispersion in each training step . Here we investigate a simple scheme that dynamically clips257 the gradient norm so as to reduce the gradient dispersion . Specifically , whenever the current gradient258 norm is larger than the gradient norm K steps earlier , or ||g ( Wt , Bt ) ||2 > ||g ( Wt−K , Bt−K ) ||2 ( i.e.,259 the model is expected to have entered the “ memorization ” regime ) , we reduce the norm of the current260 gradient g ( Wt , Bt ) to α fraction of ||g ( Wt−K , Bt−K ) ||2 , for some prescribed value α < 1 . The261 effectiveness of this scheme is best demonstrated when the labels contain noise . As shown in Figure262 4 , dynamic gradient clipping significantly closes the gap between the training accuracy and the testing263 accuracy . The models trained with this scheme maintain a near-optimal testing accuracy ( e.g. , about264 80 % when the label noise level of MNIST is 0.2 ) , without suffering from the severe memorization265 effect as seen in models trained without this scheme . Further understanding of the double-descent266 phenomenon of the gradient dispersion may enable more delicate design of such a dynamic clipping267 scheme and potentially lead to novel and powerful regularization techniques.268 5 A Practical Implication : Gaussian Model Perturbation269 The appearance of the flatness term in our generalization bounds suggests that for an empirical270 risk minimizer w∗ to generalize well , it is necessary that the empirical risk surface at w∗ is flat,271 or insensitive to a small perturbation of w∗ . This naturally motivates a training scheme using the272 following regularized loss:273 min w Ls ( w ) + ρ E ∆∼N ( 0 , σ2Id ) [ Ls ( w + ∆ ) − Ls ( w ) ] , where ρ is a hyper-parameter . Replacing the expectation above with its stochastic approximation274 using k realizations of ∆ gives rise to the following optimization problem.275 min w 1 b ∑ z∈B ( ( 1− ρ ) ` ( w , z ) + ρ1 k k∑ i=1 ( ` ( w + δi , z ) ) ) . We refer to the SGD training scheme using this loss as Gaussian model perturbation or GMP . Notably,276 GMP requires k + 1 forward passes for every parameter update . Empirical evidence shows that a277 small k , for example , k = 3 , already gives competitive performance . Implementing the k+ 1 forward278 passes on parallel processors further reduces the computation load.279 We experimentally compare GMP with several major regularization schemes in the current art , includ-280 ing Dropout [ 68 ] , label smoothing [ 70 ] , Flooding [ 36 ] , MixUp [ 81 ] , adversarial training [ 29 ] , and281 AMP [ 83 ] . The compared schemes are evaluated on three popular benchmark image classification282 datasets SVHN [ 48 ] , CIFAR-10 and CIFAR-100 [ 41 ] . Two representative deep architectures PreAc-283 tResNet18 [ 34 ] and VGG16 [ 67 ] are taken as the underlying model . We train the models for 200284 epochs by SGD . The learning rate is initialized as 0.1 and divided by 10 after 100 and 150 epochs.285 For all compared models , the batch size is set to 50 and weight decay is set to 10−4 . For GMP , we286 choose ρ = 0.5 and set the standard deviation of the Gaussian noise ∆ to 0.03 . The value of k is287 chosen as 3 and 10 respectively ( referred to as GMP3 and GMP10 ) .288 The performances of all compared schemes are given in Table 1 . For the compared regularization289 schemes except GMP , we directly report their performances as given in [ 83 ] . Performances of vanilla290 ERM without regularization are also included as a reference.291 Table 1 demonstrates the effectiveness of GMP . Overall GMP performs comparably to the current292 art of regularization schemes , although appearing slighly inferior to the most recent record given by293 AMP [ 83 ] . Noting that the key ingredient of AMP , “ max-pooling ” in the parameter space , greatly294 resembles regularization term in GMP , which may be seen as “ average-pooling ” in the same space.295 6 Conclusion and Outlook296 This paper presents new generalization bounds for neural networks trained with SGD , improving297 upon the results of [ 49 ] . Our bounds naturally point to new and effective regularization schemes . At298 the same time , they reveal interesting phenomena in the SGD training of neural networks . While299 these phenomena deserve further investigation in their own right , we here suggest another direction300 for improving the bounds , namely , via the use of strong data-processing inequalities ( DPI ) [ 57–59 ] 301 ( noting that the standard DPI is in fact needed for establishing Theorem 2 ) .302 For any Markov chain U → X → Y , we will denote by U , X , and Y the spaces in which U ,303 X , Y take values , respectively . For any distribution P on X , we will use PY |X ◦ P to denote the304 distribution on Y induced by the push-forward of the distribution P by PY |X , namely , for any y ∈ Y ,305 ( PY |X ◦ P ) ( y ) , ∫ PY |X ( y|x ) P ( x ) dx . Let S ( U ) be the support of PU and H ( U , PX|U ) be the306 convex hull of { PX|U=u : u ∈ S ( U ) } . Define307 η ( U → X → Y ) , sup P , Q∈H ( U , PX|U ) DKL ( PY |X ◦ P ||PY |X ◦Q ) DKL ( P ||Q ) Lemma 7 . For any Markov chain U → X → Y , I ( U ; Y ) ≤ η ( U → X → Y ) I ( U ; X ) .308 Here η ( U → X → Y ) serves as the “ contraction coefficient ” for the stochastic kernel PY |X ,309 characterizing the greatest extent by which the kernel may bring closer any two distributions on S ( U ) 310 in its output space . It is easy to see that η ( U → X → Y ) ≤ 1 , giving rise to a stronger DPI.311 Denote Vt , W̃t−1 + Gt . It can be verified that Zi → Vt → W̃t form a Markov chain . Denote312 ηi , t , η ( Zi → Vt → W̃t ) , and Γti = { t+ 1 , t+ 2 , . . . , T } \ Ti . 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Summary: This paper studies the generalization of neural networks trained using stochastic gradient descent (SGD) using information-theoretic bounds. It builds upon existing work that bounds the generalization error using the average square root of the mutual information of the weights with each training datum and related work by Neu 2021 [49] that refines this bound using quantities that correspond to the local gradient sensitivity, the variance/dispersion of the mini-batch gradients, and a stability term that captures the loss at the terminal weights on independent test data. The main contributions of this manuscript are as follows.  1. The paper introduces a coupled auxiliary weight process which refers to the weights updated using additive noise in the gradient descent direction calculated by SGD using the original weights. This allows the authors to study the mutual information of the descent direction with a particular datum in Lemma 5. 2. Theorem 1 establishes that the gradient sensitivity term can be eliminated using an application of Otto and Villani’s HWI inequality that relates the entropy of the weight posterior with the Wassserstein distance and the Fisher information. Theorem 2 and Corollary 1 expand on this result. The main conclusion is that the generalization error can be broken down into two terms, one that depends upon the ratio of the learning rate and batch-size and another that depends upon the stability term referred to above (bounded in Corollary 1 using the trace of the Hessian). 3. The bound in Theorem 2 is verified using a fully-connected network on MNIST and a convolutional network on CIFAR-10.  4. The paper also develops a loss function that training neural networks that averages the loss of the network across different perturbations of the weights by Gaussian noise. This new loss function is evaluated on MNIST, CIFAR-10 and CIFAR-100.