Title: Sharpness Minimization Algorithms Do Not Only Minimize Sharpness To Achieve Better Generalization
Venue: NeurIPS 2023
Accepted: yes
Keywords: Sharpness

Flatness

Generalization

Generalization Bound

SAM
Abstract: Despite extensive studies, the underlying reason as to why overparameterized
neural networks can generalize remains elusive. Existing theory shows that common stochastic optimizers prefer flatter minimizers of the training loss, and thus
a natural potential explanation is that flatness implies generalization. This work
critically examines this explanation. Through theoretical and empirical investigation, we identify the following three scenarios for two-layer ReLU networks: (1)
flatness provably implies generalization; (2) there exist non-generalizing flattest
models and sharpness minimization algorithms fail to generalize poorly, and (3)
perhaps most strikingly, there exist non-generalizing flattest models, but sharpness
minimization algorithms still generalize. Our results suggest that the relationship
between sharpness and generalization subtly depends on the data distributions
and the model architectures and sharpness minimization algorithms do not only
minimize sharpness to achieve better generalization. This calls for the search for
other explanations for the generalization of over-parameterized neural networks

Sharpness Minimization Algorithms Do Not Only Minimize Sharpness To Achieve Better Generalization

Abstract

Despite extensive studies, the underlying reason as to why overparameterized neural networks can generalize remains elusive. Existing theory shows that common stochastic optimizers prefer flatter minimizers of the training loss, and thus a natural potential explanation is that flatness implies generalization. This work critically examines this explanation. Through theoretical and empirical investigation, we identify the following three scenarios for two-layer ReLU networks: (1) flatness provably implies generalization; (2) there exist non-generalizing flattest models and sharpness minimization algorithms fail to generalize poorly, and (3) perhaps most strikingly, there exist non-generalizing flattest models, but sharpness minimization algorithms still generalize. Our results suggest that the relationship between sharpness and generalization subtly depends on the data distributions and the model architectures and sharpness minimization algorithms do not only minimize sharpness to achieve better generalization. This calls for the search for other explanations for the generalization of over-parameterized neural networks.

1 Introduction

It remains mysterious why stochastic optimization methods such as stochastic gradient descent (SGD) can find generalizable models even when the architectures are overparameterized (Zhang et al., 2016; Gunasekar et al., 2017; Li et al., 2017; Soudry et al., 2018; Woodworth et al., 2020). Many empirical and theoretical studies suggest that generalization is correlated with or guaranteed by the flatness of the loss landscape at the learned model (Hochreiter & Schmidhuber, 1997; Keskar et al., 2016; Dziugaite & Roy, 2017; Jastrzebski et al., 2017; Neyshabur et al., 2017; Wu et al., 2018; Jiang et al., 2019; Blanc et al., 2019; Wei & Ma, 2019a,b; HaoChen et al., 2020; Foret et al., 2021; Damian et al., 2021; Li et al., 2021; Ma & Ying, 2021; Ding et al., 2022; Nacson et al., 2022; Wei et al., 2022; Lyu et al., 2022; Norton & Royset, 2021; Wu & Su, 2023). Thus, a natural theoretical question is

Question 0. Does the flatness of the minimizers always correlate with the generalization capability?

The answer to the question turns out to be false. First, Dinh et al. (2017) theoretically construct very sharp networks with good generalization. Second, recent empirical results (Andriushchenko et al., 2023b) find that sharpness may not have a strong correlation with test accuracy for a collection of modern architectures and settings, partly due to the same reason-there exist sharp models with good generalization. We note that, technically speaking, Question 0 is ill-defined without specifying the collection of models on which the correlation is evaluated. However, those sharp but generalizable models appear to be the main cause for the non-correlation.

Table 1: Overview of Our Results. Each row in the table corresponds to one architecture. The second column indicates whether all flattest minimizers of training loss generalize well. ✓ indicates that all (near) flattest minimizers of training loss provably generalize well and ✗ indicates that there provably exists flattest minimizers that generalize poorly. The third column indicates whether the sharpness minimization algorithms generalize well in our experiments. Results in row 2 and 4 deny Question 1 and Question 2 respectively.

Observing the existing theoretical and empirical evidence, it is natural to ask the one-side version of Question 0, where we are only interested in whether sharpness implies generalization but not vice versa.

Question 1. Do all the flattest neural network minimizers generalize well?

Though there are some theoretical works that answer Question 1 affirmatively for simplified linear models (Li et al., 2021; Ding et al., 2022; Nacson et al., 2022; Gatmiry et al., 2023), the answer to Question 1 for standard neural networks remains unclear. Those theoretical results linking generalization to sharpness for more general architectures typically also involve other terms in generalization bounds, such as parameter dimension or norm (Neyshabur et al., 2017; Foret et al., 2021; Wei & Ma, 2019a,b; Norton & Royset, 2021), thus do not answer Question 1 directly.

Our first contribution is a theoretical analysis showing that the answer to Question 1 can be false , even for simple architectures like 2-layer ReLU networks. Intriguingly, we also find that the answer to Question 1 subtly depends on the architectures of neural networks. For example, simply removing the bias in the first layer turns the aforementioned negative result into a positive result, as also shown in the Theorem 4.3 of Wu & Su (2023) (that the authors only came to be aware of after putting this work online).

More concretely, we show that for the 2 parity xor problem with mean square loss and with data sampled from hypercube {-1 , 1 } d , all flattest 2-layer ReLU neural networks without bias provably generalize. However, when bias is added, for the same data distribution and loss function, there exists a flattest minimizer that fails to generalize for every unseen data. Since adding bias in the first layer can be interpreted as appending a constant input feature, this result suggests that the generalization of the flattest minimizer is sensitive to both network architectures and data distributions.

Recent theoretical studies (Wu et al., 2018; Blanc et al., 2019; Damian et al., 2021; Li et al., 2021; Arora et al., 2022; Wen et al., 2022; Nacson et al., 2022; Lyu et al., 2022; Bartlett et al., 2022; Li et al., 2022) also show that optimizers including SGD with large learning rates or label noise and Sharpness-Aware Minimization (SAM, Foret et al. (2021)) may implicitly regularize the sharpness of the training loss landscape. These optimizers are referred to as sharpness minimization algorithms in this paper. Because Question 1 is not always true, it is then natural to hypothesize that sharpnessminimization algorithms will fail for architectures and data distributions where Question 1 is not true.

Question 2. Will sharpness minimization algorithm fail to generalize when there exist nongeneralizing flattest minimizers?

A priori, the authors were expecting that the answer to Question 2 is affirmative, which means that a possible explanation is that the sharpness minimization algorithm works if and only if for certain architecture and data distribution, Question 1 is true. However, surprisingly, we also answer this question negatively for some architectures and data distributions. In other words, we found that sharpness-minimization algorithms can still generalize well even when the answer to Question 1 is false. The result is consistent with our theoretical discovery that for many architectures, there exist both non-generalizing and generalizing flattest minimizers of the training loss. We show empirically that sharpness-minimization algorithms can find different types of minimizers for different architectures.

Our results are summarized in Table 1. We show through theoretical and empirical analysis that the relationship between sharpness and generalization can fall into three different regimes depending on the architectures and distributions. The three regimes include:

Scenario 1. Flattest minimizers of training loss provably generalize and sharpness minimization algorithms find generalizable models. This regime (Theorems 3.1 and 3.2) includes 2-layer ReLU MLP without bias and 2-layer ReLU MLP with a simplified BatchNorm (without mean subtraction and bias). We answer both the Question 1 and Question 2 affirmatively in this scenario. 1

Scenario 2. There exists a flattest minimizer that has the worst generalization over all minimizers. Also, sharpness minimization algorithms fail to find generalizable models. This regime includes 2 layer ReLU MLP with bias. We deny Question 1 while affirm Question 2 in this scenario.

Scenario 3. There exist flattest minimizers that do not generalize but the sharpness minimization algorithm still finds the generalizable flattest model empirically. This regime includes 2-layer ReLU MLP with a simplified LayerNorm (without mean subtraction and bias). In this scenario, the sharpness minimization algorithm relies other unknown mechanisms beyond minimizing sharpness to find a generalizable model. We deny both Question 1 and Question 2 in this scenario.

2 Setup

Rademacher Complexity. Given n data S = { x i } n i =1 , the empirical Rademacher complexity of function class F is defined as R S ( F ) = 1 n E ϵ ∼{± 1 } n sup f ∈F ∑ n i =1 ϵ i f ( x i ) . Architectures. As summarized in Table 1, we will consider multiple network architectures and discuss how architecture influences the relationship between sharpness and generalization. For each model f θ parameterized by θ , we will use d to denote the input dimension and m to denote the network width. We will now describe the architectures in detail.

2-MLP-No-Bias . f nobias θ ( x ) = W 2 relu ( W 1 x ) with θ = ( W 1 , W 2 ) .

2-MLP-Bias . f bias θ ( x ) = W 2 relu ( W 1 x + b 1 ) with θ = ( W 1 , b 1 , W 2 ) . We additionally define MLP-Bias as f bias , D θ ( x ) = W D relu · · · W 2 relu ( W 1 x + b 1 ) ,

2-MLP-Sim-BN . f sbn θ ( x, { x i } i ∈ [ n ] ) = W 2 SBN γ ( relu ( W 1 x + b 1 ) , { relu ( W 1 x i + b 1 ) } ) , where the simplified BatchNorm SBN is defined as ∀ m,n ∈ N, ∀ i ∈ [ n ] , x, x i ∈ R m , j ∈ [ m ] , SBN γ ( x, { x i } i ∈ [ n ] )[ j ] = γx [ j ] / (∑ n i =1 ( x i [ j ]) 2 /n ) 1 / 2 and θ = ( W 1 , b 1 , γ, W 2 ) .

2-MLP-Sim-LN . f sln θ ( x ) = W 2 relu ( W 1 x + b 1 ) max {∥ relu ( W 1 x + b 1 ) ∥ 2 ,ϵ } where ϵ is a sufficiently small positive constant.

Surprisingly, our results show that the relationships between sharpness and generalization are strikingly different among these simple yet similar architectures.

Data Distribution. We will consider a simple data distribution as our testbed. Data distribution P xor is a joint distribution over data point x and label y . The data point is sampled uniformly from the hypercube {-1 , 1 } d and the label satisfies y = x [1] x [2] . Many of our results, including our generalization bound in Section 3 and experimental observations can be generalized to broader family of distributions (Appendix B).

Loss. We will use mean squared error ℓ mse for training and denote the training loss as L . In Appendix B, we will show that all our theoretical results and empirical observations hold for logistic loss with label smoothing probability p > 0 . We will also consider zero one loss Pr( yf θ ( x ) > 0) for evaluating the model. We will use interpolating model to denote the model with parameter θ that minimizes L .

Definition 2.1 (Interpolating Model) . A model f θ interpolates the dataset { ( x i , y i ) } n i =1 if and only if ∀ i, f θ ( x i ) = y i .

Sharpness. Our theoretical analysis focuses on understanding the sharpness of the trained models. Precisely, for a model f θ parameterized by θ , a dataset { ( x i , y i ) } n i =1 and loss function ℓ , we will use the trace of Hessian of loss function, Tr( ∇ 2 L ( θ )) to measure how sharp the loss is at θ , which is a proxy for the sharpness along a random direction (Wen et al., 2022), or equivalently, the expected increment of loss under a random gaussian perturbation (Foret et al., 2021; Orvieto et al., 2022) .

Tr( ∇ 2 L ( θ )) is not the only choice for defining sharpness, but theoretically many sharpness minimization algorithms have been shown to minimize this term over interpolating models. In particular,

(a) Baseline

(b) 1-SAM

175

Sharpness

1.0

250

Sharpness

1.0

Valid acc

Valid acc

narpness

S

150

Train acc

narpness

200

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0.9

0.8

0.8

125

Acc

150

AcC

0.7

h

100

100

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0.6

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75

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101

103

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101

103

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Epoch

under the assumptions that the minimizer of the training loss form a smooth manifold Cooper (2018); Fehrman et al. (2020), Sharpness-Aware Minimization (SAM) (Foret et al., 2021) with batch size 1 and sufficiently small learning rate η and perturbation radius ρ (Wen et al., 2022; Bartlett et al., 2022), or Label Noise SGD with sufficiently small learning rate η (Blanc et al., 2019; Damian et al., 2021; Li et al., 2021), prefers interpolating models with small trace of Hessian of the loss. Hence, we choose to analyze trace of Hessian of the loss and will use SAM with batch size 1 (we denote it by 1-SAM) as our sharpness minimization algorithm in our experiments.

Notations. We use Tr to denote the trace of a matrix and x [ i ] to denote the value of the i -th coordinate of vector x . We will use ⊙ to represent element-wise product. We use 1 as the (coordinate-wise) indicator function, for example, 1 [ x > 0] is a vector of the same length as x whose j -th entry is 1 if x [ j ] > 0 and 0 otherwise. We will use ˜ O ( x ) to hide logarithmic multiplicative factors.

3 Scenario I: All Flattest Models Generalize

3.1 Flattest models provably generalize

When the architecture is 2-MLP-No-Bias , we will show that the flattest models can provably generalize, hence answering Question 1 affirmatively for this architecture and data distribution P xor .

Theorem 3.1. For any δ ∈ (0 , 1) and input dimension d , for n = Ω ( d log ( d δ )) , with probability at least 1 -δ over the random draw of training set { ( x i , y i ) } n i =1 from P n xor , let L ( θ ) ≜ 1 n ∑ n i =1 ℓ mse ( f nobias θ ( x i ) , y i ) be the training loss for 2-MLP-No-Bias , it holds that for all θ ∗ ∈ arg min L ( θ )=0 Tr ( ∇ 2 L ( θ ) ) , we have that

Theorem 3.1 shows that for P xor , flat models can generalize under almost linear sample complexity with respect to the input dimension. We note that Theorem 3.1 implies that Pr x,y ∼P xor [ f nobias θ ∗ ( x ) y > 0 ] ≤ ˜ O ( d/n ) . because if f nobias θ ∗ ( x ) y ≤ 0 , it holds that ℓ mse ( f nobias θ ∗ ( x ) , y ) ≥ 1 . This shows that the model can classify the input with high accuracy. The major proof step is relating sharpness to the norm of the weight itself.

Lemma 3.1. Define Θ C ≜ { θ = ( W 1 , W 2 ) | ∑ m j =1 ∥ W 1 ,j ∥ 2 | W 2 ,j | ≤ C } . Under the setting of Theorem 3.1, there exists a absolute constant C independent of d and δ , such that with probability at least 1 -δ , arg min L ( θ )=0 Tr ( ∇ 2 L ( θ ) ) ⊆ Θ C and R S ( { f nobias θ | θ ∈ Θ C } ) ≤ ˜ O ( √ d/n ) .

We would like to note that similar results of Theorem 3.1 and lemma 3.1 have also been shown in a prior work Wu & Su (2023) (that the authors were not aware of before the first version of this work was online).

The almost linear complexity in Theorem 3.1 is not trivial. For example, Wei et al. (2019) shows that learning the distribution will require Ω( d 2 ) samples for Neural Tangent Kernel (NTK) (Jacot et al.,

(a) 2-layer MLP with simplified BN

W

1

,i

[1]

W

1

,i

[2]

∥

W

1

,i

[3 :

d

]

∥

2

18.581

-18.582

0.02

-14.363

-14.363

0.03

13.768

13.771

0.03

-12.601

12.601

0.01

(b) Weights of the four neurons with the largest norm in the first Layer

1.0

25

Sharpness

Valid acc

0.9

Sharpness

S

20

Trainacc

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15

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0

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2018). In contrast, our result shows that learning the distribution only requires ˜ O ( d ) samples as long as the flatness of the model is controlled.

Beyond reducing model complexity, flatness may also encourage the model to find a more interpretable solution. We prove that under a stronger than i.i.d condition over the training set, the near flattest interpolating model with architecture 2-MLP-Sim-BN will provably generalize and the weight of the first layer will be centered on the first two coordinates of the input, i.e., ∥ W 1 ,i [3 : d ] ∥ 2 ≤ ϵ ∥ W 1 ,i ∥ 2 .

Condition 1 (Complete Training Set Condition) . There exists set S ⊂ {-1 , 1 } d -2 , such that the linear space spanned by S -S = { s 1 -s 2 | s 1 , s 2 ∈ S } has rank d -2 and the training set is { ( x, y ) | x ∈ R d , x [3 : d ] ∈ S, x [1] , x [2] ∈ {-1 , 1 } , y = x [1] × x [2] } .

Theorem 3.2. Given any training set { ( x i , y i ) } n i =1 satisfying Condition 1, for any width m and any ϵ > 0 , there exists constant κ > 0 , such that for any widthm 2-MLP-Sim-BN , f sbn , satisfying f sbn θ interpolates the training set and Tr ( ∇ 2 L ( θ ) ) ≤ κ +inf L ( θ ′ )=0 Tr ( ∇ 2 L ( θ ′ ) ) , it holds that ∀ x ∈ {-1 , 1 } d , ∣ ∣ x [1] x [2] -f θ ( x ) ∣ ∣ ≤ ϵ and that ∀ i ∈ [ m ] , ∥ W 1 ,i [3 : d ] ∥ 2 ≤ ϵ ∥ W 1 ,i ∥ 2 .

One may notice that in Theorem 3.2 we only consider the approximate minimizer of sharpness. This is because the gradient of output with respect to W 1 , b 1 , despite never being zero, will converge to zero as the norm of W 1 , b 1 converges to ∞ .

Condition 1 may seem stringent. In practice (Figure 2b), we find it not necessary for 1-SAM to find a generalizable solution. We hypothesize that this condition is mainly technical. Theorem 3.2 shows that sharpness minimization may guide the model to find an interpretable and low-rank representation. Similar implicit bias of SAM has also been discussed in Andriushchenko et al. (2023a) The proof is deferred to Appendix B.1

3.2 SAM empirically finds the flattest model that generalizes

Weuse 1-SAM to train 2-MLP-No-Bias on data distribution P xor to verify our Theorem 3.1 (Figure 1). As expected, the model interpolates the training set and reaches a flat minimum that generalizes perfectly to the test set.

We then verify our Theorem 3.2 by training a 2-layer MLP with simplified BN on data distribution P xor (Figure 2a). Here we do not enforce the strong theoretical Condition 1. However, we still observe that SAM finds a flat minimum that generalizes well. We then perform a detailed analysis of the model and find that the model is indeed interpretable. For example, the four largest neurons in the first layer approximately extract features { relu ( c 1 x [1] + c 2 x [2]) | c 1 , c 2 ∈ {-1 , 1 }} (Figure 2b). Also, the first 2 columns of the weight matrix of the first layer, corresponding to the useful features { relu ( c 1 x [1] + c 2 x [2]) | c 1 , c 2 ∈ {-1 , 1 }} , have norms 42 . 47 and 42 . 48 , while the largest column norm of the rest of the weight matrix is only 5 . 65 .

4 Scenario II: Both Flattest Generalizing and Non-generalizing Models Exist, and SAM Finds the Former

4.1 Both generalizing and non-generalizing solutions can be flattest

In previous section, we show through Theorems 3.1 and 3.2 that sharpness benefits generalization under some assumptions. It is natural to ask whether it is possible to extend this bound to general architectures. However, in this section, we will show that the generalization benefit depends on model architectures. In fact, simply adding bias to the first layer of 2-MLP-No-Bias makes nonvacuous generalization bound impossible for distribution P xor . This then leads to a negative answer to Question 1.

Definition 4.1 (Set of extreme points) . A finite set S ⊂ R d is a set of extreme points if and only if for any x ∈ S , x is a vertex of the convex hull of S .

Definition 4.2 (Memorizing Solutions) . A D -layer network is a memorizing solution for a training dataset if (1) the network interpolates the training dataset, and (2) for any depth k ∈ [ D -1] , there is an injection from the input data to the neurons on depth k , such that the activations in layer k for each input data is a one-hot vector with the non-zero entry being the corresponding neuron.

Theorem 4.1. For any D ≥ 2 , if the input data points { x i } of the training set form a set of extreme points (Definition 4.1), then there exists a width n layer D MLP-Bias that is a memorizing solution (Definition 4.2) for the training dataset and has minimal sharpness over all the interpolating solutions.

As one may suspect, these memorizing solutions can have poor generalization performance.

Proposition 4.1. For data distribution P xor , for any number of samples n , there exists a widthn 2MLP-Bias that memorizes the training set as in Theorem 4.1, reaches minimal sharpness over all the interpolating models and has generalization error max { 1 -n/ 2 d , 0 } measured by zero one error.

This corollary shows that a flat model can generalize poorly. Comparing Theorems 3.1 and 4.1, one may observe the perhaps surprising difference caused by slightly modifying the architectures (adding bias or removing the BatchNorm). To further show the complex relationship between sharpness and generalization, the following theorem suggests, despite the existence of memorizing solutions, there also exists a flattest model that can generalize well.

Proposition 4.2. For data distribution P xor , for any number of samples n , there exists a widthn 2MLP-Bias that interpolates the training dataset, reaches minimal sharpness over all the interpolating models, and has zero generalization error measured by zero one error.

The flat solution constructed is highly simple. It contains four activated neurons, each corresponding to one feature in ± x [1] ± x [2] (Equation (5)).

̸

Proof sketch. For simplicity, we will consider 2-MLP-Bias here. The construction of the memorizing solution in Theorem 4.1 is as follows (visualized in Figure 3). As the input data points form a set of extreme points (Definition 4.1), for each input data point x i , there exists a vector ∥ w i ∥ = 1 , w i ∈ R d , such that ∀ j = i, w ⊤ i x i > w ⊤ i x j . We can then choose

Here r i = ( ∥ x i ∥ 2 + 1) 1 / 2 and ϵ is a sufficiently small positive number. Then it holds that relu ( W 1 x i + b 1 ) = √ r i | y i | e i , where e i is the i -th coordinate vector. This shows there is a one-to-one correspondence between the input data and the neurons. It is easy to verify that the model interpolates the training dataset. Furthermore, for P xor and sufficiently small ϵ , for any input x ̸∈ { x i } i ∈ [ n ] , it holds that relu ( W 1 x + b 1 ) = 0 . Hence the model will output the same label 0 for all the data points outside the training set. This indicates Proposition 4.1.

To show the memorization solution has minimal sharpness, we need the following lemma that relates the sharpness and the Jacobian of the model.

Lemma 4.1. For mean squared error loss l mse , if model f θ is differentiable and interpolates dataset { ( x i , y i ) } i ∈ [ n ] , then Tr ( ∇ 2 L ( θ ) ) = 2 n ∑ n i =1 ∥∇ θ f θ ( x i ) ∥ 2 .

x

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x

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Proof of Lemma 4.1. By standard calculus, it holds that,

The first equation in Equation (1) use ∀ i, f θ ( x i ) = y i . The proof is then complete.

After establishing Lemma 4.1, one can then explicitly calculate the lower bound of ∥∇ θ f θ ( x i ) ∥ 2 condition on f θ ( x i ) = y i . For simplicity of writing, we will view the bias term as a part of the weight matrix by appending a 1 to the input data point. Precisely, we will use notation x ′ i ∈ R d +1 to denote transformed input satisfying ∀ j ∈ [ d ] , x ′ i [ j ] = x i [ j ] , x ′ i [ d +1] = 1 and W ′ 1 = [ W 1 , b 1 ] ∈ R m × ( d +1) to denote the transformed weight matrix.

By the chain rule, we have,

In Equation (3), we use condition f θ ( x i ) = y i . Finally, notice that the lower bound is reached when

Condition Equation (4) is clearly reached for the memorization construction we constructed, where both sides of the equation are equal to √ | y i | / ∥ x ′ i ∥ e i . This completes the proof of Theorem 4.1.

However, the memorization network is not the only parameter that can reach the lower bound. For example, for distribution P xor , if parameter θ satisfies,

(a) Baseline

(b) 1-SAM

180

Sharpness

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Valid acc

Valid acc

less

160

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0.9

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AcC

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with r = ( d 2 +1) 1 / 4 . then for any x ∈ {-1 , 1 } d , it holds that relu ( W 1 x + b 1 ) = re 5 / 2 -x [1] -x [2] / 2 and f θ ( x ) = x [1] × x [2] . Hence it is possible for Equation (5) to hold while the model has perfect generalization performance.

4.2 SAM empirically finds the non-generalizing solutions

In this section, we will show that in multiple settings, SAM can find solutions that have low sharpness but fail to generalize compared to the baseline full batch gradient descent method with weight decay. This proves that flat minimization can hurt generalization performance. However, one should note that Question 2 is not denied for the current architectures.

Converged models found by SAM fail to generalize. We perform experiments on data distribution P xor in Figure 4. We apply small learning rate gradient descent with weight decay as our baseline and observe that the converged model found by SAM has a much lower sharpness than the baseline. However, the generalization performance of SAM is much worse than the baseline. Moreover, the generalization performance even starts to degenerate after 4000 epochs. We conclude that in this scenario, sharpness minimization can empirically hurt generalization performance.

1-SAM may fail to generalize with other activation functions. A natural question is whether the phenomenon that 1-SAM fails to generalize is limited to ReLU activation. In Figure 5, we show empirically that 1-SAM fails to generalize for 2-layer networks with softplus activation trained on the same dataset, although there is no known guarantee for the existence of memorizing solutions.

(a) Standard Training

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5 Scenario III: Both Flattest Generalizing and Non-generalizing Models Exist, and SAM Finds the Latter

5.1 Both generalizing and non-generalizing solutions can be flattest

Despite the surprising contrary between Theorems 3.1 and 4.1, experiments show that Question 2 consistently hold. However, we will provide a counterexample in this section. Specifically, we will consider data distribution P xor and 2-layer ReLU MLP with simplified LayerNorm. One can first show both generalizing and non-generalizing solutions exist similar to Theorem 4.1 and propositions 4.1 and 4.2.

Theorem 5.1. If the input data points { x i } of the training set form a set of extreme points (Definition 4.1), for sufficiently small ϵ , then there exists a widthn 2-MLP-Sim-LN with hyperparameter ϵ that is a memorizing solution (Definition 4.2) for the training dataset and has minimal sharpness over all the interpolating solutions.

Proposition 5.1. For data distribution P xor , for sufficiently small ϵ , for any number of samples n , there exists a widthn 2-MLP-Sim-LN with hyperparameter ϵ that memorizes the training set as in Theorem 4.1, reaches minimal sharpness over all the interpolating models and has generalization error max { 1 -n/ 2 d , 0 } measured by zero one error.

Proposition 5.2. For data distribution P xor , for sufficiently small ϵ , for any number of samples n , there exists a widthn 2-MLP-Sim-LN with hyperparameter ϵ that interpolates the training dataset, reaches minimal sharpness over all the interpolating models, and has zero generalization error measured by zero one error.

The construction and intuition behind Theorem 5.1 and propositions 5.1 and 5.2 are similar to that of Theorem 4.1 and propositions 4.1 and 4.2. The proof is deferred to Appendix B.

5.2 SAM empirically finds generalizing models

Notice in Section 5.1 our theory makes the same prediction as in Section 4. However, strikingly, the experimental observation is reversed (Figure 6). Now running SAM can greatly improve the generalization performance till the model perfectly generalizes. This directly denies Question 2 as now we have a scenario in which sharpness minimization algorithms can improve generalization till perfect generalization while there exists a flattest minimizer that will generalize poorly.

6 Discussion and Conclusion

We present theoretical and empirical evidence for (1) whether sharpness minimization implies generalization subtly depends on the choice of architectures and data distributions, and (2) sharpness minimization algorithms including SAM may still improve generalization even when there exist flattest models that generalize poorly. Our results suggest that low sharpness may not be the only cause of the generalization benefit of sharpness minimization algorithms.

References

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper sheds light on the relationship between three concepts: (a) generalization (b) flatness and (c) explicitly inducing flatness via SAM. The paper identifies that the relationship is nuanced and is based on certain architectural properties of the model.\n\n Specifically, depending on the architecture, there are three possible regimes:\n1. A regime where _every flat minimizer generalizes well_ and SAM finds these minimizers.\n2. A regime where _there exists flat minimizers that generalize poorly_ but SAM does _not_ find these minimizers.\n3. A regime where _there exists flat minimizers that generalize poorly_ and SAM _does_ find these minimizers.\n\nThe paper shows that \n\n- Regime (1) happens for 2-layer RELU networks _without_ bias\n- Regime (2) happens for 2-layer RELU networks _with_ bias\n- Regime (3) happens for 2-layer RELU networks _with_ bias, _with_ a simplified Layer Norm.\n\nNote that in all cases, the existence of a flat solution that generalizes/memorizes is proven theoretically; while the effect of SAM is demonstrated empirically.\n\nSTRENGTHS:\n1. The questions formalized in the paper are novel, and well-formulated. \n2. I gained a much clearer understanding of some generally unclear and confusing concepts: (a) flatness, (b) generalization and (c) explicitly inducing flatness. All these ideas are dealt with care and rigor. \n3. The introduction is well-written and the chain of arguments are explained clearly. Despite being a theoretical and nuanced paper, I found it extremely easy to follow the arguments in each section. I thoroughly enjoyed reading the paper!\n4. Rather than unnecessarily complicate the theory, the authors come up with a reasonably minimal settings of a 2-layer _non-linear_ model and support their arguments with either a formal result or with a rightly-designed empirical result. \n5. That there are situations where the flattest solution does not generalize well is surprising from a theoretical point of view.\n\nWEAKNESSES:\nI currently do not have any major concerns about this paper. Most of my comments are follow-up questions, which I enumerate in the next section. My most important question --- which may or may not hint at a weakness --- is below:\n\n### Major question\n\nQ1a: Could you be more explicit about any parameter count dependence in the Rademacher complexity term of Thm 3.1 and Lemma 3.1? I am unsure if this is orthogonal to the main point of the result, so I'd also appreciate some clarity on that.  \n\nQ1b: On that note, I'd also like to know why this bound would break down upon the introduction of bias into the architecture. Would one of the terms within the Rademacher complexity end up scaling with the number of datapoints?\n\nQUESTIONS:\n1. Def 4.1 for the memorizing solutions appears to be incomplete and can be more formal. First, by _input data_, please clarify that you are referring to the training data. Furthermore, you also require the additional criterion that the activations of the model on _test_ data are all zero --- is this right? If not, the definition appears to be incomplete as even the raw inputs must more or less satisfy the orthogonality assumption with high probability? i.e., when we sample from a high-dimensional hypercube, two samples must more or less be orthogonal. So this alone shouldn't characterize a memorization solution? \n\n2. Could you provide intuition for what the \"flattest solution\" in Eq 4 satisfies? It seems to require that _the activated top-layer weights have correspondence to the activations themselves_. I don't think Fig 3 captures this specific structure?\n\n3. Do you empirically find that SAM learns a solution whose parameters resemble Eq 4 in any way?\n\n4. The motivation behind the choice of loss function can be made more transparent. In particular, the main loss function used is squared error loss. In the appendix, the authors also use logistic loss _but with label smoothing_. If I am not wrong these choices ensure that the flattest minimizer exists and is unique up to some permutations. However, with pure logistic loss, the flattest minimizer would become ill-defined since there would be many different flat minimizers \"at $\\infty$\"? I believe this is the reason behind a line of work in PAC-Bayes literature that tries to \"normalize\" flatness in a well-defined way. In other words, this work \"sidesteps\" this annoying issue; this is okay, as it still helps outline an elegant insight. But this needs to be clearly stated so that readers can put this result in context. \n\n### Minor suggestions\n\n1. I'd suggest making the specific notion of trace-of-hessian-based flatness clear in the abstract and in the beginning of the introduction. Since there are many notions of flatness, it would be good to be upfront about which notion is being discussed here.\n\n2. In the introduction, I'd suggest clarifying that the SAM-related results are purely empirical.\n\n3.  Section 4 title seems incorrect. In this regime, SAM finds the _latter_ non-generalizing solution not the former one.\n\n4. Line 167 makes an abrupt reference to a theorem that's not discussed until then. \n\n5. It's not clear until we get to the discussion of SAM as to whether the proof is empirical or not. \n\n6. In Sec 5.1, instead of simply referring to Thm 3.1 and 4.1, I would add a verbal description so that it's easier to follow.\n\n\n### Summary\n\nOverall, I found this paper to be insightful, rigorous and clear in its message. The community would find this valuable given the confusing nature of literature in flatness. \n\n### Post-rebuttal update\n\nI have increased my score since the authors addressed my response satisfactorily. \n**I'd also like to respectfully express my strong disagreement with concerns raised in the other reviews.**\n- Regarding \"Two-layer network is too simplistic\": When a certain truth has not been proven in any setting to begin with, and when someone makes headway by providing the first proof in a simple setting, it is unfair and counter-productive to hold it against them. For without that headway, there is little hope for more general analyses. In fact, IMO proofs in more general settings (like a multi-layered network) get so unwieldy as to become uninsightful. Furthermore, in this scenario, the setting is a 2-layer network which does not reduce to a linear function of its parameters. IMO, the statements in this paper are already highly non-trivial to identify, let alone prove. \n- Regarding considering other alternate versions of SAM, this is an interesting reformulation of the problem. Thanks to this work, we can be better posed to answer more important questions. Hence, again I don't want to fault this paper for not proving its elegant theory in various sophisticated versions of SAM. I would share this review's concern had the considered setting been far removed from practice, and promised no tangible way of extending to other situations --- this doesn't seem to be the case here, in my humble opinion.\n\nLIMITATIONS:\nThe paper lists reasonable limitations in the conclusion.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-06-15"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the connections between sharpness of the training loss and generalization performance under a simplified MLP architecture. In particular, authors show scenarios of when flat solution do/do not generalize, as well as cases when Sharpness-Aware Minimization (SAM) does/does not generalize.\n\nSTRENGTHS:\nThe theoretical insights of the paper are interesting and hopefully can help better understand the relationship between flatness and generalization.\n\nWEAKNESSES:\nI think the paper has several shortcomings, outlined below.\n\n- Model setup: I think the model setup is too simplistic. The input space is binary, which I think is fine. However, no noise is considered in the model and the output label $y=x[1]x[2]$ satisfies a very specific structure. Therefore, it is not clear to me why results discussed in the paper would generalize to broader model setups, and the observations discussed here are not just an artifact of the simplistic model. As shown in Lemma 3.1, the sharpness under the paper's setup is closely related to the weight norm, which may or may not be the case in a more realistic setup. \n\n- Experiments: I don't find the experiments on SAM convincing. How is SAM implemented? How are the algorithm hyper-parameters chosen? How is SAM initialized? Maybe changing the SAM parameters and initialization would change the outcome of the experiments.  Can the observation of SAM converging to non-generalizing flat solution be replicated in a more realistic setup? Is weight decay used for SAM, as specially, in this case sharpness seems to be closely related to the weight norm.\n\n- The authors miss [1]. As this paper discusses the connections between generalization and sharpness/SAM, the authors should discuss and compare their results to [1], specially as [1] discusses noisy setups.\n\n\n[1] Behdin, K., & Mazumder, R. (2023). Sharpness-aware minimization: An implicit regularization perspective. arXiv preprint arXiv:2302.11836.\n\nQUESTIONS:\n- Maybe some visualization might help address some of my concerns in the previous part? For example visualizing the SAM/GD trajectories?\n\n- Additional experiments either on noisy data settings or more realistic deep networks might help fill some of the gaps in the paper.\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work aims to examine different cases for which flatness implies generalization. The conclusion heavily depends on the underlying model, while there are three different scenarios that contradict with each other. Additionally, the theoretical guarantees study the generalization error at the solution, however the empirical guarantees consider the generalization error at the last iteration of the SAM algorithm. Despite the convergence of the SAM algorithm, the training horizon may also affect generalization. With alternative sampling schedules of the data-set, it can be possible to minimize the training horizon (for instance see Repeated Random Sampling for Minimizing the Time-to-Accuracy of Learning by Okanovic et al.). I think it would be useful to examine if the SAM algorithm generalizes better when considering alternative sampling schedules. In such a case, the batch-selection policy could be more significant than the sharpness minimization. As the author also mention flattest solutions are not directly associated with efficient generalization in general.\n\nFurther, there is a deep literature for the edge of stability phenomenon (Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability by Cohen et al.). In this case, the sharpness increases at termination (Understanding Gradient Descent on the Edge of Stability in Deep Learning by Arora et al., Section 2. Related Works), however good generalization performance have been observed in some cases. These cases might be considerd by the authors to provide a more complete picture for the empirical part of the paper, and additional conclusion about sharpness minimization and generalization.\n\nSTRENGTHS:\nThe results are original and provide a partial evidence between sharpness and generalization, however the conclusion is not clear. Theorem 3.1 is an interesting result, however it is limited to a simple architecture but it is valuable.\n\nWEAKNESSES:\nSome parts of the paper are not clear enough. For instance, there exist statements like \"All flattest models generalize\" while the statement holds only for certain cases. It appears that the theoretical guarantees study generalization at the solution, however in practice the generalization of the algorithm might be different (see also the Summary). The main question of the paper has been considered in prior works as well.\n\nQUESTIONS:\nIn theorem 3.1 the variable $\\theta^*$ should be a random variable that depends on the training data-set. Is this true? In such a case the authors may consider to use the notation of the conditional expectation in the statement of the Theorem to make this fact clear.\n\nLIMITATIONS:\nThe authors have addressed the limitations sufficiently.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper explores the relationship between sharpness and generalization in overparameterized neural networks. The authors challenge the popular belief that flatness of the loss function implies better generalization and show that sharpness minimization algorithms do not always lead to better generalization. Through theoretical and empirical investigation, the authors identify three different scenarios for two-layer ReLU networks, showing that the relationship between sharpness and generalization is subtle and depends on the data distributions and the model architectures. The paper's contributions include a theoretical analysis of the relationship between sharpness and generalization.\n\nSTRENGTHS:\nThe authors identify three scenarios for two-layer ReLU networks and provide theoretical and empirical evidence to support their findings. I believe the subtle details in architecture and distribution located in this paper are remarkable. Overall, this paper makes significant contributions to the field of generalization theory and challenges the popular belief that the flatness of the loss function implies better generalization.\n\nWEAKNESSES:\n1.\t One potential weakness of the paper is that it only examines the relationship between sharpness and generalization for two-layer ReLU networks (personally, I think this is acceptable since it is very hard to analyze more complex scenarios). Additionally, the authors only examine the CIFAR-10 and CIFAR-100 datasets, and it would be valuable to examine their findings on other datasets to determine if their results generalize to other domains. \n2.\tThe expression of the article is not very friendly and a little difficult to understand.\n3.\tThere are some grammatical and formatting problems, some of which are shown as follows.\n1)\tPictures should be resized to make them look more coherent. In addition, the text in the image should be consistent with the font within the paper.\n2)\tThere are some spelling errors, such as “subtlely”.\n3)\tReferences are poorly formatted. Please use a uniform format for references. \n4)\tFormulas and variables should be consistent with the font within the paper.\n5)\tOperators should be carefully defined.\n6)\tThere are some formatting issues. For example, lowercase letters as the first letter of a sentence.\n\nQUESTIONS:\n1. In abstract, what does this sentence “there exist non-generalizing flattest models and sharpness minimization algorithms fail to generalize poorly” mean?\n\nLIMITATIONS:\nThe authors have adequately addressed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}