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Generalization at the Edge of Stability

NeurIPS 2026 · Learning theory and algorithms › Kernels and regression theory · approach: Generalization theory of deep networks · 0 citations (OpenAlex)

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Frames SGD at the edge of stability as a random dynamical system, defines a Lyapunov-style "sharpness dimension" over the random pullback attractor, proves a worst-case generalization bound in terms of it, and validates it on MLPs, grokking, and GPT-2.

Problem

Generalization of neural networks trained with large learning rates at the edge of stability, where the optimizer is locally unstable and chaotic, so pointwise Hessian sharpness/flatness measures are argued to be inadequate.

Approach

Model stochastic optimizers as random dynamical systems whose pullback attractor is a fractal set; define RDS sharpness as expected log singular values of the one-step Jacobian over the attractor and a Lyapunov-inspired sharpness dimension (SD) from them; bound worst-case generalization gap over the attractor by SD (via covering numbers and mutual information / random-set stability).

Key results

• On a 3-layer MLP on MNIST (12,960 parameters), the full Hessian-spectrum sharpness dimension achieves the strongest correlation with generalization gap among compared complexity measures, outperforming trajectory-based PH-Dim and E_alpha and the top RDS sharpness lambda_1.
• On a 5-layer MLP (278,800 parameters), SD-SLQ outperforms other indicators on Kendall tau when the full learning-rate and batch-size grid is considered.
• On GPT-2 (124M) trained on WikiText-2, classical sharpness shows a strong negative correlation with loss gaps under AdamW, while SD-KDE and SD-PS show the most consistent positive correlations across optimizers.

Surprising findings

• Classical sharpness (largest Hessian eigenvalue) gives a strong negative correlation with loss gap under AdamW on GPT-2, contrary to the flat-minima conjecture.
• Trajectory-based topological indices (PH-Dim, E_alpha) show weak or unstable correlation with generalization in the edge-of-stability regime.

Limitations

• Requires estimating the full Hessian spectrum, which remains computationally demanding for very large models.
• Theory assumes non-singularity, integrability and subexponential spatial variation of the cocycle, and a globally Lipschitz bounded loss.
• Proofs rely on a mutual-information term, with a set-stability variant proposed as an alternative.

Experimental setup

Datasets: MNIST WikiText-2 modular arithmetic mod 97

Metrics: Kendall tau granulated Kendall coefficient generalization gap correlation loss gap correlation

Baselines: Hessian trace sharpness (largest Hessian eigenvalue) PH-Dim alpha-weighted lifetime sum

Builds on: Lyapunov dimension theory edge of stability random set stability

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