A Single Recipe for Online Submodular Maximization with Adversarial or Stochastic Constraints

In this paper, we consider an online optimization problem in which the reward functions are DR-submodular, and in addition to maximizing the total reward, the sequence of decisions must satisfy some convex constraints on average. Specifically, at each round t\in\{1,\dots,T\}, upon committing to an action x_t, a DR-submodular utility function f_t(\cdot) and a convex constraint function g_t(\cdot) are revealed, and the goal is to maximize the overall utility while ensuring the average of the constraint functions \frac{1}{T}\sum_{t 1} T g_t(x_t) is non-positive. Such cumulative constraints arise naturally in applications where the average resource consumption is required to remain below a prespecified threshold. We study this problem under an adversarial model and a stochastic model for the convex constraints, where the functions g_t can vary arbitrarily or according to an i.i.d. We propose a single algorithm which achieves sub-linear (with respect to T) regret as well as sub-linear constraint violation bounds in both settings, without prior knowledge of the regime.