Online Optimization with Feedback Delay and Nonlinear Switching Cost
This addresses challenges in online control with delays and nonlinear dynamics, offering competitive policies for adversarial disturbances, but it is incremental as it builds on existing online optimization frameworks.
The paper tackles the problem of online optimization with delayed feedback and nonlinear switching costs, showing that their iROBD algorithm achieves a constant, dimension-free competitive ratio of O(L^{2k}), where L is the Lipschitz constant, and provides tight lower bounds.
We study a variant of online optimization in which the learner receives $k$-round $\textit{delayed feedback}$ 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 $O(L^{2k})$, 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.