Primal–Dual Algorithms for Convex Optimization via Regret Minimization

Abstract

We consider a large-scale convex program with functional constraints, where interior point methods are intractable due to the problem size. The effective solution techniques for these problems permit only simple operations at each iteration, and thus are based on primal–dual first-order methods such as the Arrow–Hurwicz–Uzawa subgradient method, which utilize only gradient computations and projections at each iteration. Such primal–dual algorithms admit the interpretation of solving the associated saddle point problem arising from the Lagrange dual. We revisit these methods through the lens of regret minimization from online learning and present a flexible framework. While it is well known that two regret-minimizing algorithms can be used to solve a convex–concave saddle point problem at the standard rate of $O$ ( $1/\sqrt {T}$ ), our framework for primal–dual algorithms allows us to exploit structural properties such as smoothness and/or strong convexity and achieve better convergence rates in favorable cases. In particular, for non-smooth problems with strongly convex objectives, our primal–dual framework equipped with an appropriate modification of Nesterov’s dual averaging algorithm achieves $O$ ( $1/T$ ) convergence rate.