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. In a large-scale empirical evaluation, we find that our approach typically outperforms comparable non-first-order methods. On the technical side, we show that the logarithmic loss and an information-theoretic quantity called the triangular discrimination play a fundamental role in obtaining first-order guarantees, and we combine this observation with new refinements to the regression oracle reduction framework of Foster and Rakhlin. The use of triangular discrimination yields novel results even for the classical statistical learning model, and we anticipate that it will find broader use.

本文探讨了如何在low noise的情况下, 通过logarithmic loss和triangular discrimination达到contextual bandits问题中的first-order guarantees，取得了很好的效果和结果

高效的一阶上下文臂状多臂老虎机：预测、分配和三角矩阵判别