Optimal High-Probability Regret for Online Convex Optimization with Two-Point Bandit Feedback

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 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( T + (1/δ))/μ) for μ-strongly convex losses. Our result is minimax optimal with respect to both the time horizon T and the dimension d.

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