Bregman-style Online Convex Optimization with EnergyHarvesting Constraints

This paper considers online convex optimization (OCO) problems where decisions are constrained by available energy resources. A key scenario is optimal power control for an energy harvesting device with a finite capacity battery. The goal is to minimize a time-average loss function while keeping the used energy less than what is available. In this setup, the distribution of the randomly arriving harvestable energy (which is assumed to be i.i.d.) is unknown, the current loss function is unknown, and the controller is only informed by the history of past observations. A prior algorithm is known to achieve $O(\sqrtT )$ regret by using a battery with an $O(\sqrtT )$ capacity. This paper develops a new algorithm that maintains this asymptotic trade-off with the number of time steps T while improving dependency on the dimension of the decision vector from $O(\sqrtn )$ to $O(\sqrtłog(n) )$. The proposed algorithm introduces a separation of the decision vector into amplitude and direction components. It uses two distinct types of Bregman divergence, together with energy queue information, to make decisions for each component.