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

Published: 31 Oct 2022, 18:00, Last Modified: 14 Jan 2023, 16:18NeurIPS 2022 AcceptReaders: Everyone
Keywords: Approximately convex functions, quantum computing, stochastic convex bandits, logarithmic regret
TL;DR: We initiate the study of quantum algorithms for optimizing approximately convex functions and prove polynomial quantum speedup in dimension n. This can applied to zeroth-order stochastic convex bandits with exponential speedup in iteration number T.
Abstract: We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set $\mathcal{K}\subseteq\mathbb{R}^{n}$ and a function $F\colon\mathbb{R}^{n}\to\mathbb{R}$ such that there exists a convex function $f\colon\mathcal{K}\to\mathbb{R}$ satisfying $\sup_{x\in\mathcal{K}}|F(x)-f(x)|\leq \epsilon/n$, our quantum algorithm finds an $x^{*}\in\mathcal{K}$ such that $F(x^{*})-\min_{x\in\mathcal{K}} F(x)\leq\epsilon$ using $\tilde{O}(n^{3})$ 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 $\tilde{O}(n^{5}\log^{2} T)$ regret, an exponential speedup in $T$ compared to the classical $\Omega(\sqrt{T})$ 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.
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