Augmented Lagrangian Methods for Time-varying Constrained Online Convex Optimization

TL;DR

This paper introduces augmented Lagrangian methods for online convex optimization with time-varying constraints, achieving sublinear regret and constraint violation even with delayed feedback, demonstrated through various numerical examples.

Contribution

The paper develops a novel model-based augmented Lagrangian method for time-varying constrained OCO, extending it to handle delayed feedback without extra assumptions.

Findings

Achieves sublinear regret and constraint violation in time-varying constrained OCO.

Extends MALM to delayed feedback scenarios with theoretical guarantees.

Demonstrates effectiveness through online network resource allocation, logistic regression, and quadratic programming examples.

Abstract

In this paper, we consider online convex optimization (OCO) with time-varying loss and constraint functions. Specifically, the decision maker chooses sequential decisions based only on past information, meantime the loss and constraint functions are revealed over time. We first develop a class of model-based augmented Lagrangian methods (MALM) for time-varying functional constrained OCO (without feedback delay). Under standard assumptions, we establish sublinear regret and sublinear constraint violation of MALM. Furthermore, we extend MALM to deal with time-varying functional constrained OCO with delayed feedback, in which the feedback information of loss and constraint functions is revealed to decision maker with delays. Without additional assumptions, we also establish sublinear regret and sublinear constraint violation for the delayed version of MALM. Finally, numerical results for several examples of constrained OCO including online network resource allocation, online logistic regression and online quadratically constrained quadratical program are presented to demonstrate the efficiency of the proposed algorithms.