Dynamic Regret for Online Composite Optimization

Abstract: This paper investigates online composite optimization in dynamic environments, where each objective or loss function contains a time-varying nondifferentiable regularizer. To resolve it, an online proximal gradient algorithm is studied for two distinct scenarios, including convex and strongly convex objectives without the smooth condition. In both scenarios, unlike most of works, an extended version of the conventional path variation is employed to bound the considered performance metric, i.e., dynamic regret. In the convex scenario, a bound $\mathcal{O}(\sqrt{T^{1-\beta}D_\beta(T)+T})$ is obtained which is comparable to the best-known result, where $D_\beta(T)$ is the extended path variation with $\beta\in[0,1)$ and $T$ being the total number of rounds. In strongly convex case, a bound $\mathcal{O}(\log T(1+T^{-\beta}D_\beta(T)))$ on the dynamic regret is established. In the end, numerical examples are presented to support the theoretical findings.