Part of Advances in Neural Information Processing Systems 24 (NIPS 2011)

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

This paper addresses the problem of minimizing a convex, Lipschitz function $f$ over a convex, compact set $X$ 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 $x \in X$. We demonstrate a generalization of the ellipsoid algorithm that incurs $O(\poly(d)\sqrt{T})$ regret. Since any algorithm has regret at least $\Omega(\sqrt{T})$ on this problem, our algorithm is optimal in terms of the scaling with $T$.

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