Conservation Law Breaking at the Edge of Stability: A Spectral Theory of Non-Convex Neural Network Optimization
Why does gradient descent reliably find good solutions in non-convex neural network optimization, despite the landscape being NP-hard in the worst case? We show that gradient flow on L-layer ReLU networks without bias preserves L-1 conservation laws C_l = ||W_{l+1}||_F^2 - ||W_l||_F^2, confining trajectories to lower-dimensional manifolds. Under discrete gradient descent, these laws break with total drift scaling as eta^alpha where alpha is approximately 1.1-1.6 depending on architecture, loss function, and width. We decompose this drift exactly as eta^2 * S(eta), where the gradient imbalance sum S(eta) admits a closed-form spectral crossover formula with mode coefficients c_k proportiona...
| Authors | Daniel Nobrega Medeiros |
|---|---|
| Primary category | cs.LG |
| Published | 8 April 2026 |
| Last updated at source | 8 April 2026 |
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