Abstract
We consider the problem of Online Convex Optimization (OCO) with two-point bandit feedback in an adversarial environment.
In this setting, a player attempts to minimize a sequence of adversarially generated convex loss functions, while only observing the value of each function at two points.
While it is well-known that two-point feedback allows for gradient estimation, achieving tight high-probability regret bounds for strongly convex functions still remained open as highlighted by \citet{agarwal2010optimal}. The primary challenge lies in the heavy-tailed nature of bandit gradient estimators, which makes standard concentration analysis difficult.
In this paper, we resolve this open challenge by providing the first high-probability regret bound of O(d(\log T + \log(1/\delta))/\mu) for \mu-strongly convex losses. Our result is minimax optimal with respect to both the time horizon T and the dimension d.
