  We study stochastic linear optimization problem with bandit feedback. The set
of arms take values in an $N$-dimensional space and belong to a bounded
polyhedron described by finitely many linear inequalities. We provide a lower
bound for the expected regret that scales as $\Omega(N\log T)$. We then provide
a nearly optimal algorithm and show that its expected regret scales as
$O(N\log^{1+\epsilon}(T))$ for an arbitrary small $\epsilon >0$. The algorithm
alternates between exploration and exploitation intervals sequentially where
deterministic set of arms are played in the exploration intervals and greedily
selected arm is played in the exploitation intervals. We also develop an
algorithm that achieves the optimal regret when sub-Gaussianity parameter of
the noise term is known. Our key insight is that for a polyhedron the optimal
arm is robust to small perturbations in the reward function. Consequently, a
greedily selected arm is guaranteed to be optimal when the estimation error
falls below some suitable threshold. Our solution resolves a question posed by
Rusmevichientong and Tsitsiklis (2011) that left open the possibility of
efficient algorithms with asymptotic logarithmic regret bounds. We also show
that the regret upper bounds hold with probability $1$. Our numerical
investigations show that while theoretical results are asymptotic the
performance of our algorithms compares favorably to state-of-the-art algorithms
in finite time as well.

Machine Learning