acb3e20075b0a2dfa3565f06681578e5-Paper-Conference.pdf

This paper investigates convex-concave minimax optimization problems where only the function value access is allowed. We introduce a class of Hessianaware quantum zeroth-order methods that can find the ǫ-saddle point within O(d2/3ǫ 2/3) function value oracle calls. This represents an improvement of d1/3ǫ 1/3 over the O(dǫ 1) upper bound of classical zeroth-order methods, where d denotes the problem dimension. We extend these results to µ-stronglyconvex µ-strongly-concave minimax problems using a restart strategy, and show a speedup of d1/3µ 1/3 compared to classical zeroth-order methods. The acceleration achieved by our methods stems from the construction of efficient quantum estimators for the Hessian and the subsequent design of efficient Hessian-aware algorithms. In addition, we apply such ideas to non-convex optimization, leading to a reduction in the query complexity compared to classical methods.