Efficient First-Order Contextual Bandits: Prediction, Allocation, and Triangular Discrimination

A recurring theme in statistical learning, online learning, and beyond is that faster convergence rates are possible for problems with low noise, often quantified by the performance of the best hypothesis; such results are known as first-order or small-loss guarantees. While first-order guarantees are relatively well understood in statistical and online learning, adapting to low noise in contextual bandits (and more broadly, decision making) presents major algorithmic challenges. In a COLT 2017 open problem, Agarwal, Krishnamurthy, Langford, Luo, and Schapire asked whether first-order guarantees are even possible for contextual bandits and---if so---whether they can be attained by efficient algorithms. We give a resolution to this question by providing an optimal and efficient reduction from contextual bandits to online regression with the logarithmic (or, cross-entropy) loss. Our algorithm is simple and practical, readily accommodates rich function classes, and requires no distributional assumptions beyond realizability.