Quantum Speedups of Optimizing Approximately Convex Functions with Applications to Logarithmic Regret Stochastic Convex Bandits

We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set cal Ksubseteqℝn and a function Fcolonℝn→ℝ such that there exists a convex function fcolonK→ℝ satisfying sup_x∈cal K|F(x)-f(x)|≤ ε/n, our quantum algorithm finds an x*∈cal K such that F(x*)-min_x∈cal K F(x)≤ε using tildeO(n3) quantum evaluation queries to F. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with tildeO(n5log2 T) regret, an exponential speedup in T compared to the classical Omega(√T) lower bound. Technically, we achieve quantum speedup in n by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in T for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.