Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises: High-Probability Bound, In-Expectation Rate and Initial Distance Adaptation
This addresses a crucial gap in optimization theory for practitioners dealing with nonsmooth objectives and heavy-tailed data, providing robust and adaptive algorithms.
The paper tackles stochastic nonsmooth convex optimization under heavy-tailed noise, establishing optimal high-probability and refined in-expectation convergence rates for convex and strongly convex functions, with parameter-free adaptation to noise levels.
Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $σ^{p}$ for some $σ\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $σ$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $σ$. Furthermore, an initial distance adaptive convergence rate is provided if $σ$ is assumed to be known.