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 (𝐿 2𝑘 ), where 𝐿 is the Lipschitz constant of the nonlinear 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 constantcompetitive online policies. This extended abstract is an abridged version of [2]. CCS CONCEPTS• Computing methodologies → Online learning settings.

We study a variant of online optimization in which the learner receives k-rounddelayed 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.

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, dimensionfree 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.

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 nonlinear 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. This extended abstract is an abridged version of [2].