Technical Note—Nonstationary Stochastic Optimization Under Lp,q-Variation Measures

Abstract

Most existing literature in stochastic optimization assumes that the underlying cost function does not change over time. However, in practice, cost functions may be nonstationary and change along the time horizon. In “Nonstationary Stochastic Optimization Under L_{p,q}-Variation Measures,” X. Chen, Y. Wang, and Y.-X. Wang study nonstationary sequential stochastic optimization problems and consider a general L_{p,q}-variation functional to quantify the nonstationarity of underlying cost functions. The L_{p,q}-variation functional generalizes a previously considered variation constraint and captures local temporal and spatial changes of cost functions. The matching regret upper and lower bounds are provided. The regret bound shows interesting phenomena under this general variation functional, such as the curse of dimensionality, which shares a similar spirit as in nonparametric statistics.