Combinatorial structures in online and convex optimization

Abstract

Motivated by bottlenecks in algorithms across online and convex optimization, we consider three fundamental questions over combinatorial polytopes. First, we study the minimization of separable strictly convex functions over polyhedra. This problem is motivated by first-order optimization methods whose bottleneck relies on the minimization of a (often) separable, convex metric, known as the Bregman divergence. We provide a conceptually simple algorithm, Inc-Fix, in the case of submodular base polyhedra. For cardinality-based submodular polytopes, we show that Inc-Fix can be speeded up to be the state-of-the-art method for minimizing uniform divergences. We show that the running time of Inc-Fix is independent of the convexity parameters of the objective function. The second question is concerned with the complexity of the parametric line search problem in the extended submodular polytope P: starting from a point inside P, how far can one move along a given direction while maintaining feasibility. This problem arises as a bottleneck in many algorithmic applications like the above-mentioned Inc-Fix algorithm and variants of the Frank-Wolfe method. One of the most natural approaches is to use the discrete Newton's method, however, no upper bound on the number of iterations for this method was known. We show a quadratic bound resulting in a factor of n6 reduction in the worst-case running time from the previous state-of-the-art. The analysis leads to interesting extremal questions on set systems and submodular functions. Next, we develop a general framework to simulate the well-known multiplicative weights update algorithm for online linear optimization over combinatorial strategies U in time polynomial in log /U/, using efficient approximate general counting oracles. We further show that efficient counting over the vertex set of any 0/1 polytope P implies efficient convex minimization over P. As a byproduct of this result, we can approximately decompose any point in a 0/1 polytope into a product distribution over its vertices. Finally, we compare the applicability and limitations of the above results in the context of finding Nash-equilibria in combinatorial two-player zero-sum games with bilinear loss functions. We prove structural results that can be used to find certain Nash-equilibria with a single separable convex minimization.