Why is SAM Robust to Label Noise?

Thank you for the feedback! We appreciate your recognition of the strengths of our paper. We address your concerns below.

1. Analysis limited to 2-layer linear networks, unclear if insights extend to deep nonlinear networks.

Great question! To reiterate, our nonlinear section makes the following connection between the 2-layer linear network analysis and deep networks:

• In 2-layer linear networks, SAM regularizes the norm of the intermediate features and last layer weights (Equation 4.6).
• If we apply a feature regularization and weight decay in a similar fashion to the last two layers of a deep model (ResNet18), we can close the gap between SAM and SGD halfway.

We’ve also added the following new experiment to further connect the two.

• When training with SAM, the norm of the last-layer activations and weights are smaller than SGD. In Figure 7 (Page 19), we plot the norm of the last-layer activations and weights during the training trajectory of SAM and it is indeed noticeably smaller.

  This suggests that SAM does implicit feature/weight norm regularization in deep models, similar to what we identified in the 2-layer linear network analysis, and this regularization is important for label noise.

• Under no label noise, we find that our regularization method (with hyperparameter search) only leads to a 1% boost in accuracy and the gain disappears at convergence. See Figure 5 (Page 18) for the test accuracy plots for SAM, SGD, and SGD + our weight/feature penalty.
  
  While under label noise, the last layer regularization closes half the SAM-SGD performance gap, the regularizer only closes an eighth of the performance gap with no label noise. This finding suggests that last-layer feature regularization, which is clearly only just a small aspect of SAM, is especially important under label noise.

2. Lacks comparison to other label noise robust methods. Could you compare the proposed simplified method to existing techniques

Thanks for the suggestion! We’d like to first emphasize that this paper is not proposing any state-of-the-art label noise robustness technique. Instead, we would like to understand why SAM, an optimizer designed to converge to flat minima, has surprisingly good label noise robustness close to state-of-the-art.

Previous works do a thorough comparison of SAM to other state-of-the-art methods:

• Table 4 of [1] compares SAM against methods including Co-teaching, MentorNet, Mixup, MentorMix, and Bootstrap. They observe SAM is better than these methods by 0-5% on CIFAR10.
• Table 9 of [2] compares SAM against logit clipping, and shows SAM can benefit from additional logit clipping by 3%.

In our paper, we restrict comparing the performance of our regularizer to SAM since our goal is not to achieve state-of-the-art but understand what makes SAM robust. We see that the penalty does halve the gap between SAM and SGD’s performance suggesting that feature regularization at the last layer can be a key component for explaining why SAM is robust to label noise.

We hope these adjustments address your main concerns. Please let us know if you have any further feedback!

References

[1] Foret et al (2020) Sharpness-Aware Minimization for Efficiently Improving Generalization [https://arxiv.org/abs/2010.01412]

[2] Wei et al (2022) Mitigating Memorization of Noisy Labels by Clipping the Model Prediction [https://arxiv.org/abs/2212.04055]

Thank you for the feedback! We appreciate your recognition of the strengths of our paper. We address your concerns below.

1. Equation 4.5 includes a stop gradient operator in a minimization target, which, to the reviewer's knowledge, is a non-standard way of writing. The reviewer would recommend rephrasing into an update rule.

Thank you for the suggestion, we’ve modified the paper accordingly.

2. How would the regularization method perform when there is no label noise present?

Under no label noise, we find that our regularization method (with hyperparameter search) only leads to a 1% boost in accuracy and the gain disappears at convergence. See Figure 5 (Page 18) for the test accuracy plots for SAM, SGD, and SGD with our regularization.

While under label noise, the last layer regularization closes half the SAM-SGD performance gap, the regularizer only closes an eighth of the performance gap with no label noise. This finding suggests that last-layer feature regularization, which is clearly only just a small aspect of SAM, is especially important under label noise.

3. Is the performance gain brought by SAM additive to current robust training algorithms or will this performance gain diminish when a more sophisticated training algorithm than SGD is used?

Great question! The analysis we provide for SAM in the linear section shows that in SAM has properties similar to other label noise robustness methods that also leverage the speed at which examples are learned i.e. clean points losses tend to go down faster than noisy point losses (see our Related Works). In direct relation are methods that prevent gradient starvation of low loss points either by logit clipping [1] or a more balanced loss function than cross-entropy [2].

Getting back to your original question, when applying SAM on top of label noise robustness methods, it may be possible to observe further improvements. See Table 4 in [3] where SAM observes further gains with bootstrapping and Table 9 of [1] where SAM is paired with logit clipping.

We hope these adjustments address your main concerns. Please let us know if you have any further feedback!

References

[1] Wei et al. (2022) Mitigating Memorization of Noisy Labels by Clipping the Model Prediction [https://arxiv.org/abs/2212.04055]

[2] Ghosh et al. (2017) Robust Loss Functions under Label Noise for Deep Neural Networks [https://arxiv.org/abs/1712.09482]

[3] Foret et al. (2020) Sharpness-Aware Minimization for Efficiently Improving Generalization [https://arxiv.org/abs/2010.01412]

Thank you for the constructive feedback! We appreciate your recognition of the strengths of our paper. We address your concerns below.

1. No analysis/empirical demonstrations on tasks other than classification are provided (e.g., regression tasks)

This is correct, we focus on understanding SAM’s regularization for the cross-entropy loss (classification), which may have different qualitative behaviors than SAM for the squared loss (regression). We hope this is clear from our abstract, and we believe our findings for classification alone can still be quite interesting to the research community.

2. Missing some references in Related Work, e.g.: https://arxiv.org/abs/1609.04836, https://arxiv.org/abs/1705.10694

Thank you for the pointers to these works! Keskar was already cited in Subsection 5.2, and we added Rolnick to Subsection 5.3 of our related works section.

3. How does robustness of SAM depend on the network width/number of parameters (d) and number of training samples (n)? Are there additional benefits (or otherwise) that SAM provide in the overparameterized regime (or some non-trivial regimes in terms of n and d)?

Great questions! To answer, we added the following ablation studies on Page 19 to observe the effect of width (or overparameterization) on the performance difference between SAM and SGD.

• Figure 9 (Page 19), SAM vs SGD in linear models trained on different numbers of examples.
• Figure 11 (Page 20) SAM versus SGD trained on different data percentages of CIFAR10 on ResNet18.
• Figure 12 (Page 20) SAM versus SGD on CIFAR10 for different number of convolution filters (to adjust for width) in ResNet18.

We observe that in both linear and nonlinear models, for the same SAM perturbation rho, the gap between SAM and SGD actually increases with the number of training examples. In particular, looking at the effect in linear models in Figure 9, one can observe that the difference between SAM and SGD is especially significant when the model is under-parametrized ($n > d$).

This may seem contrary to your expectations. However, note that we are not analyzing the performance of models at convergence, but the best test performance along its training trajectory. As a result, even when the loss is strictly convex where SAM and SGD may converge to the same solution, depending on the training trajectory, the early stopping accuracy can be very different between the two optimizers.

On the other hand, we see in Figure 12 that while decreasing the number of data (keeping the model size fixed) makes the model more “overparameterized”, but reduces the gap between SAM and SGD, overparameterization by increasing model width (and fixing data size) increases the gap between SAM and SGD. We suspect that the relationship between label noise and overparameterization/width is complicated, and do not draw any strong conclusions from these results.

4. How does robustness of SAM in the stage of SGD training depends on the learning rate? Does the learning rate need to be small enough to better isolate the benefits of SAM?

Great question! See Figure 10 (Page 20) and 13 (Page 21), where we conduct an ablation study over the learning rate for SGD and SAM in linear and nonlinear models respectively. We do a grid search over SAM’s hyperparameter rho for each learning rate. Indeed as the learning rate scales up, SGD’s early stopping test accuracy also increases. However, SAM’s performance also improves and SAM’s gains do not disappear at higher learning rates.

5. Typos in the formula 2.6 Thanks! We fixed this.

We hope these adjustments address your main concerns. Please let us know if you have any further feedback!

Thank you very much for the thorough feedback! We appreciate your recognition of the strengths of our paper. We address your concerns below

1. [De-emphasizing early stopping] I ask the authors to either provide more evidence on the special role of early stopping or edit the text in the abstract, introduction, and sections 5 and 6 to deemphasize the importance of early stopping.

Thank you for pointing out, we apologize for the confusion. We would first like to clarify that it is not that the difference in performance between SAM and SGD is aggravated with early stopping and that is not the point of our work. Empirically, the difference is aggravated with random label noise (reported in Table 4 of [1]; see our Figure 5 (Page 18) where with no label noise, SAM only observes an 8% boost in accuracy).

The reason we had emphasized early stopping is the following. To characterize SAM’s label noise robustness, we want to compare the best performance of SAM with the best performance of SGD. Under heavy label noise, the best test performance often occurs with early stopping for gradient based optimization methods [2]. Like SGD, SAM’s peak test accuracy also occurs before convergence as shown in Figures 1, 2, and 3.

To reiterate, we aren’t claiming that early stopping has a special role in observing improvements with SAM. The point is we want to study SAM under label noise, and the best performance often occurs before convergence when mislabeled training points have not been fit.

We’ve modified our submission to de-emphasize early stopping, and we hope this fix clarifies our motivation. Please let us know if the modifications are sufficient. We’re happy to make further improvements.

2. [Better depiction of faster learning of clean training examples] In addition, the presentation of the middle plot in Figure 1 is problematic: The caption says "SAM fits much more clean data however before beginning to overfit to mislabeled data" but the evidence is hard to infer from the plot. The revision should present this result more clearly.

We apologize for the confusion. To clarify this point, we’ve added another subfigure in Figure 1 (also in Figure 6 on Page 18) plotting the ratio of the accuracy of clean training examples over that of noisy training examples. As shown in the figure, the ratio of clean training accuracy over noisy training accuracy peaks to a noticeably higher value with SAM, and the ratio is highly correlated with the test accuracy. Does this help present the result more clearly? We will incorporate it into the main body for the final draft.

3. [Connecting the regularization to label noise] The discussion does not properly connect this regularization effect to the overall narrative about robustness to label noise…

Thank you for the feedback! First, we reiterate the connections we do make for context. Through deriving SAM’s updates for a two layer deep linear network, we identified that SAM regularizes the norm of the intermediate features and last layer weights (Equation 4.6). Applying a similar treatment to the last two layers in deep networks, we empirically observed that the difference in performance between SAM and SGD under label noise is more than halved, suggesting that this implicit regularization of SAM does provide robustness to label noise. .

Admittedly, we do not provide precise theoretical analysis of the robustness effects of SAM’s network Jacobian term. Yet we find our empirical results alone quite surprising and we hope it provides interesting insights about the behavior of SAM – a large proportion of SAM’s improvements above SGD can be retrieved by a simple regularization to the last layer weights and features of the network.

We contribute the following additional experimental results to strengthen the connection between the conclusion derived in 2-layer linear networks and label noise.

• In Figure 8 (Page 19), we plot the norm of the last-layer intermediate features and weights with SAM and it is indeed noticeably smaller than SGD. Thus, a similar regularization seems to be occurring at the final two layers with SAM in deep networks.
• Under no label noise, we find that our regularization method (with hyperparameter search) only leads to a 1% boost in accuracy (out of the 8% total performance gap between SAM and SGD) and the gain disappears at convergence. See Figure 6 (Page 18).
  
  While under label noise, the last layer regularization closes half the SAM-SGD performance gap, the regularizer only closes an eighth of the performance gap with no label noise. This finding suggests that last-layer feature regularization, which is clearly just one aspect of SAM, is especially important under label noise.

We think it would be interesting future work to make this connection theoretically precise.