Optimal rates for first-order stochastic convex optimization under Tsybakov noise condition

TL;DR

This paper establishes optimal convergence rates for stochastic convex optimization under a Tsybakov noise condition, revealing how the growth rate of the function near its minimum influences learning complexity.

Contribution

It introduces a unified framework linking the growth of convex functions around their minima to optimal learning rates, extending classical results and connecting convex optimization with active learning.

Findings

Optimal rate for function value learning: (T^{-rac{a7}{2a7-2}})

Optimal rate for minimizer learning: (T^{-rac{1}{2a7-2}})

Classical rates are special cases of the general framework.

Abstract

We focus on the problem of minimizing a convex function $f$ over a convex set $S$ given $T$ queries to a stochastic first order oracle. We argue that the complexity of convex minimization is only determined by the rate of growth of the function around its minimizer $x^*_{f,S}$, as quantified by a Tsybakov-like noise condition. Specifically, we prove that if $f$ grows at least as fast as $\|x-x^*_{f,S}\|^\kappa$ around its minimum, for some $\kappa > 1$, then the optimal rate of learning $f(x^*_{f,S})$ is $\Theta(T^{-\frac{\kappa}{2\kappa-2}})$. The classic rate $\Theta(1/\sqrt T)$ for convex functions and $\Theta(1/T)$ for strongly convex functions are special cases of our result for $\kappa \rightarrow \infty$ and $\kappa=2$, and even faster rates are attained for $\kappa <2$. We also derive tight bounds for the complexity of learning $x_{f,S}^*$, where the optimal rate is $\Theta(T^{-\frac{1}{2\kappa-2}})$. Interestingly, these precise rates for convex optimization also characterize the complexity of active learning and our results further strengthen the connections between the two fields, both of which rely on feedback-driven queries.