Online Non-convex Learning in Dynamic Environments

This paper considers the problem of online learning with non-convex loss functions in dynamic environments. Recently, Suggala and Netrapalli [2020] demonstrated that follow the perturbed leader (FTPL) can achieve optimal regret for non-convex losses, but their results are limited to static environments. In this research, we examine dynamic environments and choose dynamic regret and adaptive regret to measure the performance. First, we propose an algorithm named FTPL-D by restarting FTPL periodically and establish O ( T 23 ( V T + 1) 13 ) dynamic regret with the prior knowledge of V T , which is the variation of loss functions. In the case that V T is unknown, we run multiple FTPL-D with different restarting parameters as experts and use a meta-algorithm to track the best one on the fly. To address the challenge of non-convexity, we utilize randomized sampling in the process of tracking experts. Next, we present a novel algorithm called FTPL-A that dynamically maintains a group of FTPL experts and combines them with an advanced meta-algorithm to obtain O ( √ τ log T ) adaptive regret for any interval of length τ . Moreover, we demonstrate that FTPL-A also attains an ˜ O ( T 23 ( V T + 1) 13 ) dynamic regret bound. Finally, we discuss the application to online constrained meta-learning and conduct experiments to verify the effectiveness of our methods.