We study a variant of online optimization in which the learner receives k-round delayed feedback about hitting cost and there is a multi-step nonlinear switching cost, i.e., costs depend on multiple previous actions in a nonlinear manner. Our main result shows that a novel Iterative Regularized Online Balanced Descent (iROBD) algorithm has a constant, dimension-free competitive ratio that is O(L2k), where L is the Lipschitz constant of the switching cost. Additionally, we provide lower bounds that illustrate the Lipschitz condition is required and the dependencies on k and L are tight. Finally, via reductions, we show that this setting is closely related to online control problems with delay, nonlinear dynamics, and adversarial disturbances, where iROBD directly offers constant-competitive online policies.
@article{arxiv.2111.00095,
title = {Online Optimization with Feedback Delay and Nonlinear Switching Cost},
author = {Weici Pan and Guanya Shi and Yiheng Lin and Adam Wierman},
journal= {arXiv preprint arXiv:2111.00095},
year = {2021}
}