Optimal Regularized Online Allocation by Adaptive Re-Solving

TL;DR

This paper presents an adaptive dual-based algorithm framework for regularized online resource allocation that achieves optimal regret bounds and reduces computational costs through infrequent re-solving, applicable to non-concave rewards and complex constraints.

Contribution

It introduces a novel adaptive dual-based framework with approximate solutions, eliminating the log-log regret factor and enabling fast, flexible algorithms for complex online allocation problems.

Findings

Achieves optimal logarithmic regret under certain conditions.

Eliminates the log-log factor in regret bounds through dual analysis.

Reduces computational complexity with infrequent re-solving scheme.

Abstract

This paper introduces a dual-based algorithm framework for solving the regularized online resource allocation problems, which have potentially non-concave cumulative rewards, hard resource constraints, and a non-separable regularizer. Under a strategy of adaptively updating the resource constraints, the proposed framework only requests approximate solutions to the empirical dual problems up to a certain accuracy and yet delivers an optimal logarithmic regret under a locally second-order growth condition. Surprisingly, a delicate analysis of the dual objective function enables us to eliminate the notorious log-log factor in regret bound. The flexible framework renders renowned and computationally fast algorithms immediately applicable, e.g., dual stochastic gradient descent. Additionally, an infrequent re-solving scheme is proposed, which significantly reduces computational demands without compromising the optimal regret performance. A worst-case square-root regret lower bound is established if the resource constraints are not adaptively updated during dual optimization, which underscores the critical role of adaptive dual variable update. Comprehensive numerical experiments demonstrate the merits of the proposed algorithm framework.