Boosting One-Point Derivative-Free Online Optimization via Residual Feedback

Zeroth-order optimization (ZO) typically relies on two-point feedback to estimate the gradient of the objective function. Nevertheless, two-point feedback cannot be used for online optimization with time-varying objective functions, where only a single query of the function value is possible at each time step. In this work, we propose a new one-point feedback method for online optimization that estimates the gradient using the residual between two feedback points at consecutive time instants. Moreover, we develop regret bounds for ZO with residual feedback for constrained convex and unconstrained nonconvex online optimization problems. Specifically, for both deterministic and stochastic problems and for both Lipschitz and smooth objective functions, we show that using residual feedback can produce gradient estimates with much smaller variance compared to conventional one-point feedback methods. As a result, our regret bounds are much tighter compared to existing regret bounds for ZO with conventional one-point feedback, which suggests that ZO with residual feedback can better track the optimizer of online optimization problems. Additionally, our regret bounds rely on weaker assumptions than those used in conventional one-point feedback methods. Numerical experiments show that ZO with residual feedback significantly outperforms existing one-point feedback methods.