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Benign Overfitting in Single-Head Attention
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Shows that a single-head softmax attention classifier trained on label-noisy data exhibits benign overfitting after two gradient descent steps and under max-margin/min-norm interpolation when SNR is Omega(1/sqrt(n)), with a matching harmful-overfitting lower bound.
Problem
Whether and when attention models, the basic building block of Transformers, exhibit benign overfitting (perfectly fitting noisy labels while generalizing well) in classification, and how this depends on the signal-to-noise ratio.
Approach
Analyzes joint optimization of the attention head v and softmax attention vector p (with a fixed query) on a signal-plus-noise token distribution with label flipping, proving that GD after two steps interpolates noisy data while attending to signal tokens for clean samples and noise tokens for noisy ones, without assuming attention-probability behavior during training.
Key results
- GD with logistic loss reaches 100% training accuracy after two iterations, with test error bounded by eta plus exponentially small terms (theory), and attention focuses on signal tokens for clean samples and noise tokens for noisy samples.
- Min-norm/max-margin interpolators exhibit benign overfitting when SNR = Omega(1/sqrt(n)).
- When SNR is O(1/sqrt(n)) in the two-token case, the max-margin interpolator has clean test error at least 1/16, i.e., harmful overfitting.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train with logistic-loss gradient descent from zero initialization for 2 iterations | Single-head softmax attention · Noisy training data | training accuracy: after the second iteration, the model interpolates the training data, achieving a training accuracy of 1 | Helped |
| Train with logistic-loss gradient descent from zero initialization for 2 iterations | Single-head softmax attention · Clean test distribution | test error: ≤ exp(-√n/2) + exp(-C₁ηd/( cρ^0.1 n^1.5)) | Helped |
| Use a sufficiently large SNR, SNR = Ω(1/√n) | Norm-constrained max-margin interpolator · Noisy test distribution | test error: η + exp(-Ω(d/ n^2))+exp(-Ω(((1 - ζ)/φ - log(n)/R)^2)) | Helped |
| Use SNR ≤ O(1/√n) | Norm-constrained max-margin interpolator · Clean test distribution, two tokens | test error: ≥ 1/16 | Hurt |
| Reduce the gradient-descent step size (ablation) | Single-head softmax attention · Noisy training and test data | iterations to benign overfitting: about 150 iterations | Hurt |
| Increase SNR above the threshold SNR² = 2.1/n (ablation) | Single-head softmax attention trained with gradient descent · Test accuracy after reaching 100% training accuracy | test accuracy: SNR² = 2.1/n | Helped |
Surprising findings
- Benign overfitting occurs after only two GD steps, with a larger step size than needed for slower convergence.
Limitations
- Analysis limited to two GD iterations and a constant number of tokens T
- Requires very high dimension d relative to n (d >= C n^2 log(n/delta))
- Min-norm results rely on norm-minimization rather than GD dynamics
Experimental setup
Datasets: synthetic Gaussian token data with label flipping
Benchmarks: synthetic test accuracy (2000 test samples)
Metrics: train accuracy test accuracy softmax attention probability
Builds on: benign overfitting in two-layer CNNs (Cao et al.) min-norm interpolation max-margin implicit bias of attention (Tarzanagh et al.)
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