Stochastic convex optimization with bandit feedback

Alekh Agarwal, Dean P. Foster, Daniel Hsu, Sham M. Kakade, Alexander Rakhlin

Published 2011-07-08, updated 2011-10-08Version 2

This paper addresses the problem of minimizing a convex, Lipschitz function $f$ over a convex, compact set $\xset$ under a stochastic bandit feedback model. In this model, the algorithm is allowed to observe noisy realizations of the function value $f(x)$ at any query point $x \in \xset$. The quantity of interest is the regret of the algorithm, which is the sum of the function values at algorithm's query points minus the optimal function value. We demonstrate a generalization of the ellipsoid algorithm that incurs $\otil(\poly(d)\sqrt{T})$ regret. Since any algorithm has regret at least $\Omega(\sqrt{T})$ on this problem, our algorithm is optimal in terms of the scaling with $T$.