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Fast Mixing of Stochastic Gradient Descent with Normalization and Weight Decay
Zhiyuan Li · Tianhao Wang · Dingli Yu

Tue Nov 29 02:00 PM -- 04:00 PM (PST) @ Hall J #823
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 $T\to\infty$.

Author Information

Zhiyuan Li (Stanford University)
Tianhao Wang (Yale University)
Dingli Yu (Princeton University)

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