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Unveil Benign Overfitting for Transformer in Vision: Training Dynamics, Convergence, and Generalization

Jiarui Jiang, Wei Huang, Miao Zhang, Taiji Suzuki, Liqiang Nie

TL;DR

This work delves deeply into the benign overfitting perspective of transformers in vision by developing techniques that address the challenges posed by softmax and the interdependent nature of multiple weights in transformer optimization and achieves generalization in post-training.

Abstract

Transformers have demonstrated great power in the recent development of large foundational models. In particular, the Vision Transformer (ViT) has brought revolutionary changes to the field of vision, achieving significant accomplishments on the experimental side. However, their theoretical capabilities, particularly in terms of generalization when trained to overfit training data, are still not fully understood. To address this gap, this work delves deeply into the benign overfitting perspective of transformers in vision. To this end, we study the optimization of a Transformer composed of a self-attention layer with softmax followed by a fully connected layer under gradient descent on a certain data distribution model. By developing techniques that address the challenges posed by softmax and the interdependent nature of multiple weights in transformer optimization, we successfully characterized the training dynamics and achieved generalization in post-training. Our results establish a sharp condition that can distinguish between the small test error phase and the large test error regime, based on the signal-to-noise ratio in the data model. The theoretical results are further verified by experimental simulation. To the best of our knowledge, this is the first work to characterize benign overfitting for Transformers.

Unveil Benign Overfitting for Transformer in Vision: Training Dynamics, Convergence, and Generalization

TL;DR

This work delves deeply into the benign overfitting perspective of transformers in vision by developing techniques that address the challenges posed by softmax and the interdependent nature of multiple weights in transformer optimization and achieves generalization in post-training.

Abstract

Transformers have demonstrated great power in the recent development of large foundational models. In particular, the Vision Transformer (ViT) has brought revolutionary changes to the field of vision, achieving significant accomplishments on the experimental side. However, their theoretical capabilities, particularly in terms of generalization when trained to overfit training data, are still not fully understood. To address this gap, this work delves deeply into the benign overfitting perspective of transformers in vision. To this end, we study the optimization of a Transformer composed of a self-attention layer with softmax followed by a fully connected layer under gradient descent on a certain data distribution model. By developing techniques that address the challenges posed by softmax and the interdependent nature of multiple weights in transformer optimization, we successfully characterized the training dynamics and achieved generalization in post-training. Our results establish a sharp condition that can distinguish between the small test error phase and the large test error regime, based on the signal-to-noise ratio in the data model. The theoretical results are further verified by experimental simulation. To the best of our knowledge, this is the first work to characterize benign overfitting for Transformers.
Paper Structure (62 sections, 30 theorems, 613 equations, 4 figures, 1 table)

This paper contains 62 sections, 30 theorems, 613 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Benign Overfiting in Deep Learning
  4. Optimization of Transformers
  5. Problem Setup
  6. Main Results
  7. Comparison with other results.
  8. Proof Sketch
  9. Vectorized Q & K and scalarized V
  10. Dealing with the softmax function
  11. 1. How $\bm{W}_Q$ and $\bm{W}_K$ affect $\bm{W}_V$:
  12. 2. How $\bm{W}_V$ affects $\bm{W}_Q$ and $\bm{W}_K$:
  13. Three-Stage Decoupling
  14. Stage 1:
  15. Stage 2:
  16. ...and 47 more sections

Key Result

Theorem 4.1

Under Condition cond, if $N \cdot \mathrm{SNR}^2 = \Omega(1)$, then with probability at least $1 - d^{-1}$, there exist $T = \Theta ( \eta^{-1} \epsilon^{-1} \Vert \bm{\mu} \Vert_2^{-2} \Vert \bm{w}_O \Vert_2^{-2} )$ such that:

Figures (4)

  • Figure 1: (a) is a heatmap of test loss on synthetic data across various signal-to-noise ratios ($\mathrm{SNR}$) and sample sizes (N). High test losses are indicated with yellow, while low test losses are indicated with purple. (b) is a heatmap that applies a cutoff value 0.2. It categorizes values below 0.2 as 0 (purple), and above 0.2 as 1 (yellow). The expression for the red curves in (a) and (b) is $N \cdot \mathrm{SNR}^2 = 1000$.
  • Figure 2: MNIST experiments
  • Figure 3: Training Dynamics Under Benign Overfitting Regime
  • Figure 4: Training Dynamics Under Harmful Overfitting Regime

Theorems & Definitions (52)

  • Definition 3.1: Data Generation Model
  • Theorem 4.1: Benign Overfiting
  • Theorem 4.2: Harmful Overfiting
  • Definition 5.1: Vectorized Q & K
  • Definition 5.2: Scalarized V
  • Lemma 5.1: Dynamics of $\gamma$ and $\rho$
  • Lemma 5.2: V's Beginning of Learning Signals
  • Lemma 5.3: Dynamics of QKV in Stage 2
  • Lemma 5.4
  • Lemma 5.5
  • ...and 42 more