Online Optimization in Dynamic Environments

High-velocity streams of high-dimensional data pose significant ``big data'' analysis challenges across a range of applications and settings. Online learning and online convex programming play a significant role in the rapid recovery of important or anomalous information from these large datastreams. While recent advances in online learning have led to novel and rapidly converging algorithms, these methods are unable to adapt to nonstationary environments arising in real-world problems. This paper describes a dynamic mirror descent framework which addresses this challenge, yielding low theoretical regrets bounds and accurate, adaptive, and computationally efficient algorithms which are applicable to broad classes of problems. The methods are capable of learning and adapting to an underlying and possibly time-varying dynamical model. Empirical results in the context of dynamic texture analysis, sequential compressed sensing of a dynamic scene, and tracking self-exciting point processes support the core theoretical findings.