Sharp High-Probability Statistical Rates for Heavy-Tailed Nonsmooth Convex Optimization

Abstract

We establish sharp high-probability statistical rates for nonsmooth stochastic convex optimization under heavy-tailed noise, closing the confidence-dependence gap in both general convex and strongly convex settings. We consider composite objectives with a Lipschitz convex loss and a known proximal regularizer, accessed through fresh unbiased stochastic subgradients. The centered noise has conditional norm and directional $p$th moments bounded by $L^p$ and $s^p$, respectively, where $1<p\le2$ and $0<s\le L$. For $T$ oracle calls and failure probability $0<δ<1/10$, set $q=1-1/p$, $\ell=\log(3/δ)$, and $A=s(L^2/s^2)^q$. With initial distance bound $D$, the minimax objective error is $Θ(D(A+s\ell^q)/T^q)$ in the nonsaturated statistical regime. For a $μ$-strongly convex regularizer, the corresponding quadratic-regime rate is $Θ((A^2+s^2\ell^{2q})/(μT^{2q}))$. These rates require domination of the deterministic and finite-variance terms; matching lower bounds require $d\ge\lceil L^2/s^2\rceil$, while the upper bounds hold in every finite dimension. The new upper bounds attain known statistical lower-bound orders, replacing the previous confidence coefficient $s^{1/p}L^{1-1/p}$ by $s$, and its square by $s^2$ under strong convexity. The key is an online inequality that separates directional fluctuations from the total second moment created by rejection sampling. Localization preserves this separation under strong convexity without additional horizon logarithms. The results characterize oracle complexity with unrestricted internal computation.