Zeroth Order Non-convex optimization with Dueling-Choice Bandits

We consider a novel setting of zeroth order non-convex optimization, where in addition to querying the function value at a given point, we can also duel two points and get the point with the larger function value. We refer to this setting as optimization with dueling-choice bandits since both direct queries and duels are available for optimization. We give the COMP-GP-UCB algorithm based on GP-UCB (Srinivas et al., 2009), where instead of directly querying the point with the maximum Upper Confidence Bound (UCB), we perform a constrained optimization and use comparisons to filter out suboptimal points. COMP-GP-UCB comes with theoretical guarantee of O(fracPhi√T) on simple regret where T is the number of direct queries and Phi is an improved information gain corresponding to a comparison based constraint set that restricts the search space for the optimum. In contrast, in the direct query only setting, Phi depends on the entire domain. Finally, we present experimental results to show the efficacy of our algorithm.