Optimal Regularized Online Convex Allocation by Adaptive Re-Solving

This paper introduces a dual-based algorithm framework for solving the regularized online resource allocation problems, which have cumulative convex rewards, hard resource constraints, and a non-separable regularizer. Under a strategy of adaptively updating the resource constraints, the proposed framework only requests an approximate solution to the empirical dual problem up to a certain accuracy, and yet delivers an optimal logarithmic regret under a locally strongly convex assumption. Surprisingly, a delicate analysis of dual objective function enables us to eliminate the notorious loglog factor in regret bound. The flexible framework renders renowned and computationally fast algorithms immediately applicable, e.g., dual gradient descent and stochastic gradient descent. 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 and real data application demonstrate the merits of proposed algorithm framework.