Optimal Rates for Zero-Order Convex Optimization: The Power of Two Function Evaluations

We consider derivative-free algorithms for stochastic and nonstochastic convex optimization problems that use only function values rather than gradients. Focusing on nonasymptotic bounds on convergence rates, we show that if pairs of function values are available, algorithms for d-dimensional optimization that use gradient estimates based on random perturbations suffer a factor of at most √d in convergence rate over traditional stochastic gradient methods. We establish such results for both smooth and nonsmooth cases, sharpening previous analyses that suggested a worse dimension dependence, and extend our results to the case of multiple (m ≥ 2) evaluations. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, establishing the sharpness of our achievable results up to constant (sometimes logarithmic) factors.