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Zeroth-order optimization (ZO) algorithms have been recently used to solveblack-box or simulation-based learning and control problems, where the gradientof the objective function cannot be easily computed but can be approximatedusing the objective function values. Many existing ZO algorithms adopttwo-point feedback schemes due to their fast convergence rate compared toone-point feedback schemes. However, two-point schemes require two evaluationsof the objective function at each iteration, which can be impractical inapplications where the data are not all available a priori, e.g., in onlineoptimization. In this paper, we propose a novel one-point feedback scheme thatqueries the function value once at each iteration and estimates the gradientusing the residual between two consecutive points. When optimizing adeterministic Lipschitz function, we show that the query complexity of ZO withthe proposed one-point residual feedback matches that of ZO with the existingtwo-point schemes. Moreover, the query complexity of the proposed algorithm canbe improved when the objective function has Lipschitz gradient. Then, forstochastic bandit optimization problems where only noisy objective functionvalues are given, we show that ZO with one-point residual feedback achieves thesame convergence rate as that of two-point scheme with uncontrollable datasamples. We demonstrate the effectiveness of the proposed one-point residualfeedback via extensive numerical experiments.

A New One-Point Residual-Feedback Oracle For Black-Box Learning and Control