Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD

• URL: http://arxiv.org/abs/2302.06570v1
• Date: Mon, 13 Feb 2023 18:13:36 GMT
• Title: Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD
• Authors: Matthew Faw, Litu Rout, Constantine Caramanis, Sanjay Shakkottai
• Abstract summary: This work considers the problem of finding first-order stationary point of a non atau function with potentially constant smoothness using a gradient. We develop a technique that allows us to prove $mathcalO(fracmathrmpolylog(T)sigmatT)$ convergence rates without assuming uniform bounds on the noise.
• Score: 38.221784575853796
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work considers the problem of finding a first-order stationary point of a non-convex function with potentially unbounded smoothness constant using a stochastic gradient oracle. We focus on the class of $(L_0,L_1)$-smooth functions proposed by Zhang et al. (ICLR'20). Empirical evidence suggests that these functions more closely captures practical machine learning problems as compared to the pervasive $L_0$-smoothness. This class is rich enough to include highly non-smooth functions, such as $\exp(L_1 x)$ which is $(0,\mathcal{O}(L_1))$-smooth. Despite the richness, an emerging line of works achieves the $\widetilde{\mathcal{O}}(\frac{1}{\sqrt{T}})$ rate of convergence when the noise of the stochastic gradients is deterministically and uniformly bounded. This noise restriction is not required in the $L_0$-smooth setting, and in many practical settings is either not satisfied, or results in weaker convergence rates with respect to the noise scaling of the convergence rate. We develop a technique that allows us to prove $\mathcal{O}(\frac{\mathrm{poly}\log(T)}{\sqrt{T}})$ convergence rates for $(L_0,L_1)$-smooth functions without assuming uniform bounds on the noise support. The key innovation behind our results is a carefully constructed stopping time $\tau$ which is simultaneously "large" on average, yet also allows us to treat the adaptive step sizes before $\tau$ as (roughly) independent of the gradients. For general $(L_0,L_1)$-smooth functions, our analysis requires the mild restriction that the multiplicative noise parameter $\sigma_1 < 1$. For a broad subclass of $(L_0,L_1)$-smooth functions, our convergence rate continues to hold when $\sigma_1 \geq 1$. By contrast, we prove that many algorithms analyzed by prior works on $(L_0,L_1)$-smooth optimization diverge with constant probability even for smooth and strongly-convex functions when $\sigma_1 > 1$.

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