Non-stationary Stochastic Optimization under $L_{p,q}$-Variation Measures

We consider a non-stationary sequential stochastic optimization problem, in which the underlying cost functions change over time under a variation budget constraint. We propose an $L_{p,q}$-variation functional to quantify the change, which yields less variation for dynamic function sequences whose changes are constrained to short time periods or small subsets of input domain. Under the $L_{p,q}$-variation constraint, we derive both upper and matching lower regret bounds for smooth and strongly convex function sequences, which generalize previous results in Besbes et al. (2015). Furthermore, we provide an upper bound for general convex function sequences with noisy gradient feedback, which matches the optimal rate as $p\to\infty$. Our results reveal some surprising phenomena under this general variation functional, such as the curse of dimensionality of the function domain. The key technical novelties in our analysis include affinity lemmas that characterize the distance of the minimizers of two convex functions with bounded Lp difference, and a cubic spline based construction that attains matching lower bounds.