Non-stationary Stochastic Optimization with Local Spatial and Temporal Changes

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 captures local spatial and temporal variations of the sequence of functions. Under the $L_{p,q}$-variation functional 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). 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 an affinity lemma that characterizes the distance of the minimizers of two convex functions with bounded $L_p$ difference, and a cubic spline based construction that attains matching lower bounds.