Abstract

We consider an online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially. We analyze two classes of primal-dual algorithms and provide bounds on their competitive ratios. Our analysis relies on a smooth surrogate of the objective function that needs to satisfy a new diminishing-returns property. We then formulate a convex optimization problem to directly optimize this competitive ratio bound; this design problem can be solved offline before the data start arriving. We give examples of computing the smooth surrogate for D-optimal and A-optimal experiment design. This is joint work with Reza Eghbali and James Saunderson.