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

• URL: http://arxiv.org/abs/2209.12897v1
• Date: Mon, 26 Sep 2022 03:19:40 GMT
• Title: Quantum Speedups of Optimizing Approximately Convex Functions with Applications to Logarithmic Regret Stochastic Convex Bandits
• Authors: Tongyang Li, Ruizhe Zhang
• Abstract summary: A quantum algorithm finds an $x*incal K$ such that $F(x*)-min_xincal K F(x)leqepsilon$ using $tildeO(n3)$ quantum evaluation queries to $F$. As an application, we give a quantum function algorithm for zeroth-order convex bandits with $tildeO(n5log2 T)$ regret, an exponential speedup in $T$ compared to the classical $
• Score: 8.682187438614296
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set ${\cal K}\subseteq\mathbb{R}^{n}$ 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 $\sup_{x\in{\cal K}}|F(x)-f(x)|\leq \epsilon/n$, our quantum algorithm finds an $x^{*}\in{\cal K}$ such that $F(x^{*})-\min_{x\in{\cal K}} F(x)\leq\epsilon$ 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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