Primal-Dual Frank-Wolfe for Constrained Stochastic Programs with Convex and Non-convex Objectives

TL;DR

This paper introduces a primal-dual Frank-Wolfe algorithm for constrained stochastic programs with convex and non-convex objectives, providing convergence guarantees and applicability to network scheduling and distributed optimization.

Contribution

It proposes a low-complexity primal-dual Frank-Wolfe method that learns optimal decisions without prior distribution knowledge, applicable to both convex and non-convex problems.

Findings

Convergence guarantees for convex and non-convex objectives.

Algorithm performs well in opportunistic network scheduling.

Applicable to distributed non-convex stochastic optimization.

Abstract

We study constrained stochastic programs where the decision vector at each time slot cannot be chosen freely but is tied to the realization of an underlying random state vector. The goal is to minimize a general objective function subject to linear constraints. A typical scenario where such programs appear is opportunistic scheduling over a network of time-varying channels, where the random state vector is the channel state observed, and the control vector is the transmission decision which depends on the current channel state. We consider a primal-dual type Frank-Wolfe algorithm that has a low complexity update during each slot and that learns to make efficient decisions without prior knowledge of the probability distribution of the random state vector. We establish convergence time guarantees for the case of both convex and non-convex objective functions. We also emphasize application of the algorithm to non-convex opportunistic scheduling and distributed non-convex stochastic optimization over a connected graph.