Fast Zeroth-Order Convex Optimization with Quantum Gradient Methods

Junhyung Lyle Kim, Brandon Augustino, Dylan Herman, Enrico Fontana, Jacob Watkins, Marco Pistoia, Shouvanik Chakrabarti

Keywords: quantum computing, convex optimization, quantum gradient methods, semidefinite programming, linear programming, zero-sum games

Abstract: We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.

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

Submission Number: 9575