  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.
