Stochastic convex optimization with bandit feedback

TL;DR

This paper introduces a generalized ellipsoid algorithm for stochastic convex optimization with bandit feedback, achieving optimal regret bounds of ()\u00b7 extasciitilde()()()()()()()()()()()()()()() regret, matching the lower bound and thus being optimal.

Contribution

The paper generalizes the ellipsoid algorithm to stochastic convex optimization with bandit feedback, achieving optimal regret bounds.

Findings

Achieves ()\u00b7 extasciitilde() regret bounds.

Provides a theoretically optimal algorithm for the problem.

Extends ellipsoid method to stochastic bandit setting.

Abstract

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 $T$.