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The set is not convex, but isoperimetric. With the numbering of R3, [2] (which we cite for the novel analysis of optimization via sampling) gives a20 guarantee for optimization, not sampling. In some applications, the Renyi divergence is the natural measure of convergence and more suitable than KL. Our bounds are the first polynomial bounds in this36 generality. We agree that it would be very interesting to determine the precise polynomial dependencies on all the37 parameters.