Make the Minority Great Again: First-Order Regret Bound for Contextual Bandits

Regret bounds in online learning compare the player’s performance to L∗, the optimal performance in hindsight with a fixed strategy. Typically such bounds scale with the square root of the time horizon T . The more refined concept of first-order regret bound replaces this with a scaling √ L∗, which may be much smaller than √ T . It is well known that minor variants of standard algorithms satisfy first-order regret bounds in the full information and multi-armed bandit settings. In a COLT 2017 open problem Agarwal et al. [2017], Agarwal, Krishnamurthy, Langford, Luo, and Schapire raised the issue that existing techniques do not seem sufficient to obtain first-order regret bounds for the contextual bandit problem. In the present paper, we resolve this open problem by presenting a new strategy based on augmenting the policy space.