Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track
Zhiyuan Li, Tianhao Wang, Dingli Yu
We prove the Fast Equilibrium Conjecture proposed by Li et al., (2020), i.e., stochastic gradient descent (SGD) on a scale-invariant loss (e.g., using networks with various normalization schemes) with learning rate $\eta$ and weight decay factor $\lambda$ mixes in function space in $\mathcal{\tilde{O}}(\frac{1}{\lambda\eta})$ steps, under two standard assumptions: (1) the noise covariance matrix is non-degenerate and (2) the minimizers of the loss form a connected, compact and analytic manifold. The analysis uses the framework of Li et al., (2021) and shows that for every $T>0$, the iterates of SGD with learning rate $\eta$ and weight decay factor $\lambda$ on the scale-invariant loss converge in distribution in $\Theta\left(\eta^{-1}\lambda^{-1}(T+\ln(\lambda/\eta))\right)$ iterations as $\eta\lambda\to 0$ while satisfying $\eta \le O(\lambda)\le O(1)$. Moreover, the evolution of the limiting distribution can be described by a stochastic differential equation that mixes to the same equilibrium distribution for every initialization around the manifold of minimizers as $T\to\infty$.