Inexact Online Proximal-gradient Method for Time-varying Convex Optimization

TL;DR

This paper introduces an inexact online proximal-gradient algorithm for tracking evolving convex minimizers, analyzing its convergence and regret bounds under various convexity assumptions and computational precisions.

Contribution

It proposes a novel inexact online proximal-gradient method for time-varying convex optimization and provides convergence and regret analysis under practical inexact computations.

Findings

Convergence of error iterates for strongly convex functions.

Regret bounds depending on cumulative error and solution path length.

Guidelines for balancing computational resources and optimization performance.

Abstract

This paper considers an online proximal-gradient method to track the minimizers of a composite convex function that may continuously evolve over time. The online proximal-gradient method is inexact, in the sense that: (i) it relies on an approximate first-order information of the smooth component of the cost; and, (ii) the proximal operator (with respect to the non-smooth term) may be computed only up to a certain precision. Under suitable assumptions, convergence of the error iterates is established for strongly convex cost functions. On the other hand, the dynamic regret is investigated when the cost is not strongly convex, under the additional assumption that the problem includes feasibility sets that are compact. Bounds are expressed in terms of the cumulative error and the path length of the optimal solutions. This suggests how to allocate resources to strike a balance between performance and precision in the gradient computation and in the proximal operator.