Benign Overfitting in Single-Head Attention

We thank the reviewers for their thoughtful comments and valuable feedback.

Comparison with related work

First, we note that Jiang et al. [1] posted their paper on arxiv a few days before the ICLR submission deadline, so it should be considered concurrent work to ours. Moreover, to the best of our knowledge, our work is the first to establish benign overfitting in a softmax attention model with noisy data, in contrast to [1], which considers clean data. Note that our analysis shows that the learning patterns for clean and noisy samples differ significantly, which is a crucial component of our analysis.

Multi-token settings

While the setting of [1] involves multiple tokens, they rely on additional assumptions about the data distribution. In their Condition 4.1.(10), they assume that the second noise token is sufficiently larger than that of other noises, hence it essentially plays a similar role to the noise token in our setting.

We conducted some new experiments which show that our results also hold for the multi-token setting (see Fig. 6 in Appendix A.4 of the updated version). In this scenario, the key idea presented in our paper remains consistent: the attention layer focuses on optimal tokens to achieve benign overfitting. The main difference lies in the optimal token selection mechanism. As shown in Figure 6, the optimal tokens for clean samples are still the signal tokens. However, for noisy samples, the optimal tokens may be a combination of noise tokens. This figure also illustrates that, in the multi-token setting without additional assumptions, the interactions between noise tokens become more complex, making the analysis of their attention probabilities more challenging. Nevertheless, the key concept of "optimal token selection" continues to hold in more complex settings.

For our GD analysis, we believe that we know how to extend the result to multiple tokens, but we still need to verify a few technical details. The proof strategy is very similar and is based on analyzing the coefficients of the tokens in the attention head $v$, showing that the softmax weight $p$ assigns a high softmax probability to the signal token for clean samples, and a low softmax probability to the signal token for noisy samples (instead of showing a high softmax probability for a specific noisy token, as in the case of two tokens).

Upper Bound on SNR

Regarding the SNR upper bound, a full answer would require many technical details. The bottom line is that proving benign overfitting with a larger SNR requires sharper bounds on the softmax probabilities of optimal tokens, i.e., stronger than 1-c, for some constant c>0, as we have now. If needed, we can provide more technical details about that.

Different Step-size

Regarding the step size, with a smaller step size, more than 2 iterations will be required to fit the noisy samples and achieve benign overfitting. Roughly speaking, the loss of noisy samples will be smaller, therefore more iterations will be required to learn the noisy tokens of the noisy samples. We also demonstrate this in the experiments (see figure 2, line 495). With a larger step size, we suspect that the model will not converge, since clean and noisy samples have the same signal token, but with opposite directions, the model might alternate between fitting the noisy samples and fitting the clean samples.

Zero and Gaussian Initialization

We prove the result using zero initialization for simplicity, but the same results hold for Gaussian initialization with sufficiently small variance. This is because the model is smooth: the gradient with respect to the softmax weight p is Lipschitz (as shown in Lemma 6 of [1]), and the same holds also for the gradient with respect to the attention head v (f(X;p,v) is linear in v). Consequently, the loss and gradient for any sample under Gaussian initialization will closely resemble the loss and gradient under zero initialization. Using this reasoning, we conclude that the model also exhibits benign overfitting (after two iterations) with Gaussian initialization. To support this claim, we have included additional experiments using Gaussian initialization (See Figure 8 in the additional experiments section in the Appendix). If required, we can formally prove this result in the final version of the paper.

Heatmap for SNR Threshold

Thanks for the suggestion! See Figure 7 in the additional experiments section in the Appendix, which confirms the SNR threshold in our theory.

[1] Jiang, Jiarui, Wei Huang, Miao Zhang, Taiji Suzuki, and Liqiang Nie. "Unveil Benign Overfitting for Transformer in Vision: Training Dynamics, Convergence, and Generalization." arXiv preprint arXiv:2409.19345 (2024).

[2] Davoud Ataee Tarzanagh, Yingcong Li, Xuechen Zhang, and Samet Oymak. Max-margin token selection in attention mechanism

Thank you for your detailed response.

Benign overfitting in a softmax attention model with noisy data

Including the analysis with label flipping is interesting. However, I am still concerned about whether this constitutes a significant contribution. I reviewed your response to Reviewer 7cCk and remain curious about the role of attention in the presence of label noise. An experimental simulation/theoretical explanation addressing this question would be appreciated.

Upper Bound on SNR

Thank you for your explanation; I now understand the difficulties. However, could you please list some potential methods that might address this issue?

Heatmap for SNR Threshold

Thank you. I am glad to see this new result.

Verification of Theorem 6, 8, 10

To verify Theorems under the max-margin learning rule, we trained using GD with weight-decay (as GD with weight decay corresponds to norm-minimization/margin-maximization) and got similar plots to the ones of GD without weight decay (see Figure 10, page 69 in the updated version).

We also refer to Figure 7 (page 68 in the updated version) which presents a heatmap of the test accuracy, plotted across varying SNR and sample sizes n, which also demonstrates the tight bound on the SNR. These empirical results support Theorem 6,8,10.

Experiments on more complex model

We appreciate the question regarding the extension to more general settings and we have conducted additional experiments on the self-attention model, multi-token data model and multi-layer transformer, demonstrating that our analysis can extend to more complex settings.

First, for self-attention model $v^{\top} X^{\top} S(X W x_1)$ where $x_1$ is the first token, the results in Fig.5 (Appendix A.4 in the updated version) confirm the presence of benign overfitting. The softmax probabilities assigned to the signal and noise tokens align with the optimal token selection mechanism outlined in our paper.

Second, for multi-token data, we consider T=5 tokens, where all noise tokens are generated in the same approach. The results in Fig.6 show that the model continues to exhibit benign overfitting. For clean samples, most of the softmax attention is focused on the signal token, which corresponds to the optimal token in our setting. In contrast, for noisy samples, the softmax probabilities are more evenly distributed across all tokens, with significantly lower attention on the signal token.

Finally, for multi-layer attention model, we consider a 4-layer single-head attention and the data model remains the same. The results in Fig.9 indicate that benign overfitting phenomenon still appears and the attention weights of the first layer have the same optimal token selection mechanism.

Justification of Model Simplification

Our setting corresponds to self-attention where q is a token concatenated to the raw input X to get $[q, X^{\top}]^{\top}$, and thus our output corresponding to position q is $v^{\top} X^{\top} S(XWX^{\top})$. In practice, this token serves as a learnable parameter that enables the model to adapt flexibly to new tasks. This setup is widely used in applications, such as the [CLS] token in BERT and Vision Transformers (ViT) for classification tasks. Also, the prompt-tuning technique in [2] and [3] leverages similar mechanisms.

[1] Davoud Ataee Tarzanagh, Yingcong Li, Xuechen Zhang, and Samet Oymak. Max-margin token selection in attention mechanism

[2] Xiang Lisa Li and Percy Liang. Prefix-tuning: Optimizing continuous prompts for generation

[3] Brian Lester, Rami Al-Rfou, and Noah Constant. The power of scale for parameter-efficient prompt tuning.

About SNR Definition

Yes, the definition you wrote is correct.

Signal-Noise Orthogonality

We can understand this expression as removing the variance along the directions of $\mu_1$ and $\mu_2$ by subtracting the terms $\frac{\mu_1 \mu_1^\top}{\rho^2}​$​ and $\frac{\mu_2 \mu_2^\top}{\rho}$​​. So there is no vector component along the direction of $\mu_1$ or $\mu_2$ in noise vectors and the inner product between signal and noise is 0.

Accuracy vs n/d Plot

Yes, see Figure 4b in the Appendix A.4 of the updated version.

Neglect Cross Term

Assumption item 2 states that $d >> n^2$, and implies that noise tokens from different training samples are nearly orthogonal. This assumption appears in many prior works on benign overfitting (see line 190 for references), and we agree that it is an interesting theoretical challenge to relax this assumption. Experimentally, Figure 3b (page 13) shows results for different values of dimension d when the number of samples n is fixed. We observe that benign overfitting occurs even for $d=1000=2n<<n^2$.

Normalized Noise

Then there is only a scale difference in our results and the SNR threshold remains the same, i.e. $\rho = \Omega(1/\sqrt{n})$ to achieve benign overfitting.

SNR- self-attention vs convolutional

Here we compare our results with Kou et al. [5] which also considers a signal-noise data model with label flipping noise. Their SNR threshold separating benign and harmful overfitting is $||\mu||/\sqrt{d} = \Theta(1/n^{1/2} d^{1/4})$, which is smaller than ours ($\Theta(1/n^{1/2})$). Notably, they only analyze the case where SNR is small ($||\mu||/\sqrt{d} = O(1/n^{1/2})$), while we do not require an upper bound for SNR. The difference arises from the distinct architectures of CNNs and attention models, leading to different mechanisms for overfitting training data. In CNNs, signal and noise patches are learned together because the output is the sum of their contributions. In contrast, due to the optimal token selection mechanism of the attention layer, our model can focus on either the signal or the noise token for each sample, based on which has a higher softmax probability. This enables our analysis to handle cases with larger SNR, which is not possible in Kou et al.'s framework.

[1] Davoud Ataee Tarzanagh, Yingcong Li, Xuechen Zhang, and Samet Oymak. Max-margin token selection in attention mechanism.

[2] Jiarui Jiang, Wei Huang, Miao Zhang, Taiji Suzuki, and Liqiang Nie. Unveil benign overfit- ting for transformer in vision.

[3] Heejune Sheen, Siyu Chen, Tianhao Wang, and Harrison H. Zhou. Implicit regularization of gradient flow on one-layer softmax attention.

[4] Bhavya Vasudeva, Puneesh Deora, and Christos Thrampoulidis. Implicit bias and fast convergence rates for self-attention.

[5] Yiwen Kou, Zixiang Chen, Yuanzhou Chen, and Quanquan Gu. Benign overfitting in two-layer relu convolutional neural networks.

Generalizing to Self-Attention and Multi-Token Settings

Self-Attention Models: While our work focuses on a simplified prompt-attention model, we believe the results can generalize to self-attention settings. For self-attention models, we can consider the model $v^{\top} X^{\top} S(X W x_1)$ where $x_1$ is the first token. Specifically, the optimization behavior may differ due to the additional interactions in self-attention. However, as demonstrated by the numerical results from new experiments that we conducted (see Fig.5, page 67 in the new version), the two core findings from our paper—optimal token selection and the benign overfitting phenomenon—remain consistent in this setting. Additionally, we have some primary conjectures for theoretical analysis. The analysis of v (in all of the results) can directly follow our analysis. As for the analysis of W, we need to reconsider its convergence direction and from our empirical results we think it will also converge to its max-margin solution (W-SVM), i.e. the minimizer of certain norm under the constraints:
    $(\mu_i - \xi_i) W x_1 \ge 1 (i \in C)$
    $(\xi_i - \mu_i) W x_1 \ge 1 (i \in N)$

Multi-Token Settings: regarding the extension to multi-token settings, we conducted new experiments and showed that our results hold also in this case – see Fig. 6 on page 67 of the new version. The key idea presented in our paper remains consistent: the attention layer focuses on optimal tokens to achieve benign overfitting. The main difference lies in the optimal token selection mechanism. As shown in Figure 6(c), the optimal tokens for noisy samples can be a combination of noise tokens in this case. Once the optimal tokens are determined, benign or harmful overfitting follows similarly.

The figure also demonstrates that, in the multiple tokens setting, without additional assumptions the interaction between noise tokens becomes more complex, making it harder to analyze their attention probabilities. Existing works on multiple tokens setting all need additional assumptions. For example, [1] assumes a strict data generation rule (Assumption C) connecting data and the induced max-margin. [2] requires the second noise token to be much larger than others (Condition 4.1.(10)). and [3] fixes the linear prediction head v and assumes identical token scores for all noise tokens. We aim to explore this more general setting in future work.

Further steps for GD

In our setup, both clean and noisy samples share the same signal token (the first token), which differs from other works that often assume nearly orthogonal tokens across samples [1,4].

To illustrate the difficulty, consider a scenario where the model correctly classifies clean samples but not noisy samples (which indeed happened after one iteration of GD in our analysis). In this case, the loss of the noisy samples dominates the loss of clean samples, causing the model to prioritize fitting the noisy samples. Since clean and noisy samples have the same signal token, albeit in opposite directions, additional iterations of GD may cause the attention head and softmax weights to reduce the focus on the signal token. This undesirable behavior can ultimately degrade test accuracy.

This, along with other technical difficulties, especially analyzing the gradient with respect to the softmax weight p (with label-noisy data), makes even two steps of GD a challenging task. We hope future work can build upon our techniques to extend the analysis to further iterations.

We thank the reviewers for their comments and feedback. Below, we address each concern and provide clarifications. We will incorporate the suggested references into the final version of our paper and correct the noted typographical error in line 89.

Regarding the strong assumptions, we emphasize that, to the best of our knowledge, this is the first work to study benign overfitting in transformers with label noise. Therefore, it is natural to begin with simplified assumptions to better understand the behavior of transformers in this context. These assumptions provide a tractable framework for studying softmax attention with label noise, serving as a foundation for future work to explore more general settings. Even with these assumptions, the problem is highly non-trivial to analyze, as evident in our proofs.

We emphasize that Theorems 6-8, which show that the max-margin solution exhibits benign overfitting, can reasonably be viewed as a form of 'fully converged' analysis, correspond to training with weight decay.

Initialization at zero

We use zero initialization for simplicity, but the results extend for example to Gaussian initialization with sufficiently small variance. This follows from the model's smoothness, ensuring that the loss and gradient under Gaussian initialization closely match these under zero initialization. Thus, the model exhibits benign overfitting after two iterations also with Gaussian initialization, as confirmed by additional experiments that we conducted – see Fig. 8 in Appendix A.4 in the updated version of the paper. If required, we can formally prove this in the final version.

We appreciate the question regarding the extension to more general data models.

Multiple tokens setting

We conducted some new experiments which appear in Fig. 6, Page 67 in the updated pdf. We empirically showed that our results hold also in this case. The key idea presented in our paper remains consistent: the attention layer focuses on optimal tokens to achieve benign overfitting.

Regarding the extension of the proof of the GD, we think it should be possible, but we still need to verify a few technical details. The proof strategy is very similar and is based on analyzing the coefficients of the tokens in the attention head $v$, showing that the softmax weight $p$ assigns a high softmax probability to the signal token for clean samples, and a low softmax probability to the signal token for noisy samples (instead of showing a high softmax probability for a specific noisy token, as in the case of two tokens).

Regarding the max-margin part, the situation is more complicated. The main difference lies in the optimal token selection mechanism: as shown in Figure 6(c), the optimal tokens for noisy samples can be a combination of noise tokens in this case. Therefore, in the multiple tokens setting, without additional assumptions, the interaction between noise tokens becomes more complex, making it harder to analyze their attention probabilities. Existing works on multiple tokens setting all need additional assumptions. For example, [1] assumes a strict data generation rule (Assumption C) connecting data and the induced max-margin. [2] requires the second noise token to be much larger than others (Condition 4.1.(10)). and [3] fixes the linear prediction head v and assumes identical token scores for all noise tokens. We aim to explore this more general setting in future work. We note that once the optimal tokens are determined, benign or harmful overfitting follows similarly to our current proof.

Near orthogonality of data points

This assumption is used crucially in a number of works on benign overfitting ([4], [5], [6]). But we agree that this condition might be relaxable and this should be investigated further in the future.

Comparison to linear model

The essential difference lies in the token selection mechanism of the softmax layer, which induces significantly different dynamics and implicit bias. In the attention model, the softmax layer learns from the data and assigns probabilities to the first-layer output. While the optimization problem for the output layer resembles linear interpolation in the form, the softmax layer will automatically focus most of the attention on the signal token for clean samples ($\lambda \rightarrow1$) and the noise token for non-clean samples ($\lambda \rightarrow 0$). In contrast, linear interpolation lacks this token selection mechanism, leading to simultaneous learning of signal and noise tokens.

Fully Converged Solutions

Indeed, it is reasonable to think of theorem 6-8, which state that the max margin solution exhibits benign overfitting, as a kind of "fully converged" learning rule (which corresponds to training with weight decay). Regarding convergence of GD, this is indeed an intriguing question for future research. Additionally, we refer to the response to Q2 from Reviewer 7cCk, which offers some intuition on the challenges of analyzing further steps of gradient descent.

[1] Davoud Ataee Tarzanagh, Yingcong Li, Xuechen Zhang, and Samet Oymak. Max-margin token selection in attention mechanism

[2] Jiarui Jiang, Wei Huang, Miao Zhang, Taiji Suzuki, and Liqiang Nie. Unveil benign overfit- ting for transformer in vision

[3] Heejune Sheen, Siyu Chen, Tianhao Wang, and Harrison H. Zhou. Implicit regularization of gradient flow on one-layer softmax attention, 2024.

[4] Yuan Cao, Zixiang Chen, Misha Belkin, and Quanquan Gu. Benign overfitting in two-layer convolutional neural networks

[5] Yiwen Kou, Zixiang Chen, Yuanzhou Chen, and Quanquan Gu. Benign overfitting in two-layer relu convolutional neural networks

[6] Xuran Meng, Difan Zou, and Yuan Cao. Benign overfitting in two-layer relu convolutional neural networks for xor data

We thank the reviewers for their comments and feedback.

First, the prior works that the reviewer mentioned (Tarzanagh et al. and Cao et al.) are conceptually very different than our work. Indeed, Tarzanagh et al. studied the implicit bias of a single-head attention model, and did not consider generalization or benign overfitting. Cao et al. studied convolutional networks, and thus did not consider the effect of the attention mechanism which is central in our work.

The novelty of this work lies in exploring how attention mechanisms handle noisy training data and when it exhibits benign overfitting. Specifically, we investigate how attention mechanisms can effectively attend to noise tokens in noisy-labeled examples while maintaining strong generalization, a structural behavior absent in CNNs and prior studies on benign overfitting. This analysis provides fresh insights into the interplay between attention mechanisms and label noise, advancing the theoretical understanding of the overfitting behavior in attention-based models.

At the technical level, almost all of our proofs required techniques that are significantly different from prior work. Since we analyzed softmax attention-based model, there is virtually no technical similarity to Cao et al. which studied CNNs (interestingly, some assumptions repeat in several works on benign overfitting in different settings, but it does not indicate a technical similarity). Regarding Tarzanagh et al. [2], which studied implicit bias in attention mechanisms, we emphasize that our proof is significantly different from theirs. Here we highlight several specific distinctions:

1. The Proof of Theorem 4:

This proof is novel and analyzes the gradient of a softmax model with noisy labels, optimizing both the attention head v and softmax weights p, unlike [2] which assumes a fixed v in its GD results.

2. The Proof of Theorem 6:

While inspired by Tarzanagh et al. (specifically, Theorem 5 on page 7 of their arXiv version), our proof diverges significantly. Tarzanagh et al.'s proof relies on Assumption C (page 7 in their arXiv version), which appears somewhat unnatural. Our work extends their results in three key aspects:

• Specific Natural Distributions: We show a specific natural distribution (the one studied in Cao et al.) that satisfies Assumption C with high probability. See Proposition 15 (Section 5, Line 419) and its very non-trivial proof (pages 26–42), which is analog to Assumption C from Tarzanagh et al.

• Non-Asymptotic Results: We generalize Theorem 5 from Tarzanagh et al. to include non-asymptotic results. See Theorem 16 (Line 427), which provides explicit rates for the margin as a function of the norm constraints (r and R).

• Implicit Bias implies Benign Overfitting: We demonstrate that the implicit bias of attention leads to benign overfitting.

3. Theorem 8:

We analyze the max-margin solution under joint norm minimization of v and p, corresponding to GD with weight decay. This approach is distinct from [2] and other works, which impose separate norm constraints.

Regarding the reviewer’s comment on the simplified settings and assumptions. All existing works on benign overfitting consider very simple models (even analyzing benign overfitting in linear models is already challenging) and strong assumptions on the data. Also, all existing works on implicit bias in attention-based models require very strong assumptions. Our assumptions are not stronger than prior works, and analyzing benign overfitting in our setting is already very challenging. We believe that exploring simplified settings provides valuable insights into the overfitting behavior in deep learning.