Robust Algorithms for Online Convex Problems via Primal-Dual

The importance of primal-dual methods in online optimization can hardly be overstated, and they give several of the state-of-the art results in both of the most common models for online algorithms: the adversarial and the stochastic/random order models. Here we try to provide a more unified analysis of primal-dual algorithms to better understand the mechanisms behind this important method. With this we are able of recover and extend in one goal several results of the literature.

In particular we obtain robust online algorithm for fairly general online convex problems: we consider the MIXED model where in some of the time steps the data is stochastic and in the others the data is adversarial. Both the quantity and location of the adversarial time steps are unknown to the algorithm. The guarantees of our algorithms interpolate between the (close to) best guarantees for each of the pure models. In particular, the presence of adversarial times does not degrade the guarantee relative to the stochastic part of the instance.

More concretely, we first consider online convex programming: in each time step a feasible set V t is revealed, and the algorithm needs to select v t ∈ V t to minimize the total cost ψ( t v t ), for a convex function ψ. Our robust primal-dual algorithm for this problem on the MIXED model recovers and extends, for example, a result of Gupta et al. [15] as well as the recent work on ℓ p -norm load balancing [29]. We also consider the problem of welfare maximization with convex production costs: in each time a customer presents a value c t and resource consumption vector a t , and the goal is to fractionally select customers to maximize the profit t c t x t -ψ( t a t x t ). Our robust primal-dual algorithm for this problem on the MIXED model recovers and extends the result of Azar et al. [3].

Given the ubiquity of primal-dual algorithms, we hope that the ideas of the analyses presented here will be useful in obtaining other robust algorithm in the MIXED or related models.