Title: Non-stationary Stochastic Optimization under $L_{p,q}$-Variation Measures,
Abstract: 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.