  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.
