Best of Both Worlds: Stochastic and Adversarial Convex Function Chasing

Convex function chasing (CFC) is an online optimization problem in which during each round $t$, a player plays an action $x_t$ in response to a hitting cost $f_t(x_t)$ and an additional cost of $c(x_t,x_{t-1})$ for switching actions. We study the CFC problem in stochastic and adversarial environments, giving algorithms that achieve performance guarantees simultaneously in both settings. Specifically, we consider the squared $\ell_2$-norm switching costs and a broad class of quadratic hitting costs for which the sequence of minimizers either forms a martingale or is chosen adversarially. This is the first work that studies the CFC problem using a stochastic framework. We provide a characterization of the optimal stochastic online algorithm and, drawing a comparison between the stochastic and adversarial scenarios, we demonstrate that the adversarial-optimal algorithm exhibits suboptimal performance in the stochastic context. Motivated by this, we provide a best-of-both-worlds algorithm that obtains robust adversarial performance while simultaneously achieving near-optimal stochastic performance.