Competitive Analysis via Regularization

We provide a framework for designing competitive online algorithms using regularization, a widely used technique in online learning, particularly in online convex optimization. An online algorithm that uses regularization serves requests by computing a solution, in each step, to an objective function involving a smooth convex regularization function. Applying the technique of regularization allows us to obtain new results in the domain of competitive analysis. We remark that competitive analysis and online learning are two widely studied frameworks for online decision-making settings. We show that even though there are significant differences in assumptions, goals, and techniques between the two fields, one can still benefit by introducing techniques from one field to the other. In our new framework we exhibit a general O(log m)-competitive deterministic algorithm for generating a fractional solution that satisfies a time-varying set of online covering and precedence constraints, where m is the number of variables. This framework allows to incorporate both service costs (over time) and setup costs into a host of applications. We then provide an O(log m log n)-competitive randomized algorithm for the online set cover problem with service cost, where m is the number of sets and n is the number of elements. This model allows for sets to be both added and deleted over time from a solution.