Online Optimization with Memory and Competitive Control

This paper presents competitive algorithms for a novel class of online optimization problems with memory. We consider a setting where the learner seeks to minimize the sum of a hitting cost and a switching cost that depends on the previous p decisions. This setting generalizes Smoothed Online Convex Optimization. The proposed approach, Optimistic Regularized Online Balanced Descent, achieves a constant, dimension-free competitive ratio. Further, we show a connection between online optimization with memory and online control with adversarial disturbances. This connection, in turn, leads to a new constant-competitive policy for a rich class of online control problems. 2 [23, 30] . The goal of the online learner is to minimize its total cost over T rounds: cost(ALG) = T t=1 f t (y t ) + c(y t , y t-1 ).