An optimal randomized incremental gradient method

TL;DR

This paper introduces a randomized primal-dual gradient method for large-scale convex optimization that reduces gradient evaluations significantly and achieves optimal complexity, with a new game-theoretic interpretation.

Contribution

Develops a novel randomized primal-dual gradient method with improved complexity bounds and a new game-theoretic perspective for convex optimization.

Findings

RPDG reduces gradient evaluations by ${ m O}(\sqrt{m})$ times compared to deterministic methods.

RPDG achieves optimal iteration complexity for finite-sum convex problems.

A new lower bound shows RPDG's complexity is near the theoretical limit.

Abstract

In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization problems using a primal-dual termination criterion. Our major contribution is to develop a randomized primal-dual gradient (RPDG) method, which needs to compute the gradient of only one randomly selected smooth component at each iteration, but can possibly achieve better complexity than PDG in terms of the total number of gradient evaluations. More specifically, we show that the total number of gradient evaluations performed by RPDG can be ${\cal O} (\sqrt{m})$ times smaller, both in expectation and with high probability, than those performed by deterministic optimal first-order methods under favorable situations. We also show that the complexity of the RPDG method is not improvable by developing a new lower complexity bound for a general class of randomized methods for solving large-scale finite-sum convex optimization problems. Moreover, through the development of PDG and RPDG, we introduce a novel game-theoretic interpretation for these optimal methods for convex optimization.