Quantum Speedups for Minimax Optimization and Beyond

Chengchang Liu, Zongqi Wan, Jialin Zhang, Xiaoming Sun, John C.S. Lui

Keywords: minimax optimization, quantum computing

Abstract:

This paper investigates convex-concave minimax optimization problems where only the function value access is allowed.
We introduce a class of Hessian-aware quantum zeroth-order methods that can find the $\epsilon$-saddle point within $\tilde{\mathcal{O}}(d^{2/3}\epsilon^{-2/3})$ function value oracle calls.
This represents an improvement of $d^{1/3}\epsilon^{-1/3}$ over the $\mathcal{O}(d\epsilon^{-1})$ upper bound of classical zeroth-order methods, where $d$ denotes the problem dimension.
We extend these results to $\mu$-strongly-convex $\mu$-strongly-concave minimax problems using a restart strategy, and show a speedup of $d^{1/3}\mu^{-1/3}$ 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.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 5882