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arXiv:2406.03006

Quantum Algorithms and Lower Bounds for Finite-Sum Optimization

Yexin Zhang, Zhang, Chenyi, Cong Fang, Liwei Wang, Tongyang Li

Published Jun 5, 2024• 0 citations
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Abstract

Finite-sum optimization has wide applications in machine learning, covering important problems such as support vector machines, regression, etc. In this paper, we initiate the study of solving finite-sum optimization problems by quantum computing. Specifically, let $f_1,\ldots,f_n\colon\mathbb{R}^d\to\mathbb{R}$ be $\ell$-smooth convex functions and $ψ\colon\mathbb{R}^d\to\mathbb{R}$ be a $μ$-strongly convex proximal function. The goal is to find an $ε$-optimal point for $F(\mathbf{x})=\frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x})+ψ(\mathbf{x})$. We give a quantum algorithm with complexity $\tilde{O}\big(n+\sqrt{d}+\sqrt{\ell/μ}\big(n^{1/3}d^{1/3}+n^{-2/3}d^{5/6}\big)\big)$, improving the classical tight bound $\tildeΘ\big(n+\sqrt{n\ell/μ}\big)$. We also prove a quantum lower bound $\tildeΩ(n+n^{3/4}(\ell/μ)^{1/4})$ when $d$ is large enough. Both our quantum upper and lower bounds can extend to the cases where $ψ$ is not necessarily strongly convex, or each $f_i$ is Lipschitz but not necessarily smooth. In addition, when $F$ is nonconvex, our quantum algorithm can find an $ε$-critial point using $\tilde{O}(n+\ell(d^{1/3}n^{1/3}+\sqrt{d})/ε^2)$ queries.

DOI
10.48550/arxiv.2406.03006

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Finite-sum optimization has wide applications in machine learning, covering important problems such as support vector machines, regression, etc. In this paper, we initiate the study of solving finite-sum optimization problems by quantum computing. Specifically, let f1,…,fn ⁣:Rd→Rf_1,\ldots,f_n\colon\mathbb{R}^d\to\mathbb{R}f1​,…,fn​:Rd→R be ℓ\ellℓ-smooth convex functions and ψ ⁣:Rd→Rψ\colon\mathbb{R}^d\to\mathbb{R}ψ:Rd→R be a μμμ-strongly convex proximal function. The goal is to find an εεε-optimal point for F(x)=1n∑i=1nfi(x)+ψ(x)F(\mathbf{x})=\frac{1}{n}\sum_{i=1}^n f_i(\mathbf{x})+ψ(\mathbf{x})F(x)=n1​∑i=1n​fi​(x)+ψ(x). We give a quantum algorithm with complexity O~(n+d+ℓ/μ(n1/3d1/3+n−2/3d5/6))\tilde{O}\big(n+\sqrt{d}+\sqrt{\ell/μ}\big(n^{1/3}d^{1/3}+n^{-2/3}d^{5/6}\big)\big)O~(n+d​+ℓ/μ​(n1/3d1/3+n−2/3d5/6)), improving the classical tight bound Θ~(n+nℓ/μ)\tildeΘ\big(n+\sqrt{n\ell/μ}\big)Θ~(n+nℓ/μ​). We also prove a quantum lower bound Ω~(n+n3/4(ℓ/μ)1/4)\tildeΩ(n+n^{3/4}(\ell/μ)^{1/4})Ω~(n+n3/4(ℓ/μ)1/4) when ddd is large enough. Both our quantum upper and lower bounds can extend to the cases where ψψψ is not necessarily strongly convex, or each fif_ifi​ is Lipschitz but not necessarily smooth. In addition, when FFF is nonconvex, our quantum algorithm can find an εεε-critial point using O~(n+ℓ(d1/3n1/3+d)/ε2)\tilde{O}(n+\ell(d^{1/3}n^{1/3}+\sqrt{d})/ε^2)O~(n+ℓ(d1/3n1/3+d​)/ε2) queries.
The problem of minimizing the maximum of NNN convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring O(Nε−2/3+ε−8/3)O(Nε^{-2/3} + ε^{-8/3})O(Nε−2/3+ε−8/3) queries to a first-order oracle to compute an εεε-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of NNN convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of O~(Nε−5/3+ε−8/3)\tilde{O}(\sqrt{N}ε^{-5/3} + ε^{-8/3})O~(N​ε−5/3+ε−8/3). On the other hand, we prove that quantum algorithms must take Ω~(Nε−2/3)\tildeΩ(\sqrt{N}ε^{-2/3})Ω~(N​ε−2/3) queries to a first order quantum oracle, showing that our dependence on NNN is optimal up to poly-logarithmic factors.
Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding εεε-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to ppp-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is Ω(ε−1+pp)Ω\big(ε^{-\frac{1+p}{p}}\big)Ω(ε−p1+p​) regarding the first setting, and Ω(ε−4)Ω(ε^{-4})Ω(ε−4) regarding the second setting (or Ω(ε−3)Ω(ε^{-3})Ω(ε−3) if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding εεε-stationary points of nonconvex functions with ppp-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.
We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of Rn\mathbb{R}^nRn with evaluation and membership queries is Ω~(n)\tildeΩ(n)Ω~(n), nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an n×nn \times nn×n real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.
This chapter covers applications of quantum computing in the area of continuous optimization, including both convex and nonconvex optimization. We discuss quantum algorithms for computing Nash equilibria for zero-sum games and for solving linear, second-order, and semidefinite programs. These algorithms are based on quantum implementations of the multiplicative weights update method or interior point methods. We also discuss general quantum algorithms for convex optimization which can provide a speedup in cases where the objective function is much easier to evaluate than the gradient of the objective function. Finally, we cover quantum algorithms for escaping saddle points and finding local minima in nonconvex optimization problems.
Recent results suggest that quantum computers possess the potential to speed up nonconvex optimization problems. However, a crucial factor for the implementation of quantum optimization algorithms is their robustness against experimental and statistical noises. In this paper, we systematically study quantum algorithms for finding an εεε-approximate second-order stationary point (εεε-SOSP) of a ddd-dimensional nonconvex function, a fundamental problem in nonconvex optimization, with noisy zeroth- or first-order oracles as inputs. We first prove that, up to noise of O(ε10/d5)O(ε^{10}/d^5)O(ε10/d5), accelerated perturbed gradient descent with quantum gradient estimation takes O(log⁡d/ε1.75)O(\log d/ε^{1.75})O(logd/ε1.75) quantum queries to find an εεε-SOSP. We then prove that perturbed gradient descent is robust to the noise of O(ε6/d4)O(ε^6/d^4)O(ε6/d4) and O(ε/d0.5+ζ)O(ε/d^{0.5+ζ})O(ε/d0.5+ζ) for ζ>0ζ>0ζ>0 on the zeroth- and first-order oracles, respectively, which provides a quantum algorithm with poly-logarithmic query complexity. We then propose a stochastic gradient descent algorithm using quantum mean estimation on the Gaussian smoothing of noisy oracles, which is robust to O(ε1.5/d)O(ε^{1.5}/d)O(ε1.5/d) and O(ε/d)O(ε/\sqrt{d})O(ε/d​) noise on the zeroth- and first-order oracles, respectively. The quantum algorithm takes O(d2.5/ε3.5)O(d^{2.5}/ε^{3.5})O(d2.5/ε3.5) and O(d2/ε3)O(d^2/ε^3)O(d2/ε3) queries to the two oracles, giving a polynomial speedup over the classical counterparts. Moreover, we characterize the domains where quantum algorithms can find an εεε-SOSP with poly-logarithmic, polynomial, or exponential number of queries in ddd, or the problem is information-theoretically unsolvable even by an infinite number of queries. In addition, we prove an Ω(ε−12/7)Ω(ε^{-12/7})Ω(ε−12/7) lower bound in εεε for any randomized classical and quantum algorithm to find an εεε-SOSP using either noisy zeroth- or first-order oracles.
We study stochastic optimization with heavy-tailed gradient noise. We first propose a novel quantum mean estimator for multivariate heavy-tailed random variables that achieves lower query complexity than optimal classical estimators in the low-dimensional regime. We further develop an unbiased quantum mean estimator by applying a generalized multi-level Monte Carlo technique. We prove quantum lower bounds showing that, when the dimension ddd of the random vector is small and can be viewed as a constant, our quantum estimators are optimal up to logarithmic factors. We further derive stronger dimension-dependent lower bounds for tail index p>4/3p>4/3p>4/3, showing that a nontrivial dependence on the dimension is unavoidable in the low-dimensional regime. Based on these estimators, we propose a quantum normalized stochastic gradient descent method (QNSGD\texttt{QNSGD}QNSGD), which finds an εεε-stationary point using O~(d ε−5p−42p−2)\tilde{\mathcal{O}}\big(\sqrt d\,ε^{-\frac{5p-4}{2p-2}}\big)O~(d​ε−2p−25p−4​) queries to the quantum stochastic gradient oracle. For a convex objective function, we propose a quantum projected stochastic gradient descent method (QPSGD\texttt{QPSGD}QPSGD), which computes a solution with εεε-optimal solution using O~(d ε−3p−22p−2+ε−2)\tilde{\mathcal{O}}\big(\sqrt d\,ε^{-\frac{3p-2}{2p-2}}+ε^{-2}\big)O~(d​ε−2p−23p−2​+ε−2) queries in expectation. These sharper bounds improve upon the classical lower bounds Ω(ε−3p−2p−1)Ω\big(ε^{-\frac{3p-2}{p-1}}\big)Ω(ε−p−13p−2​) for nonconvex problems and Ω(ε−pp−1)Ω\big(ε^{-\frac{p}{p-1}}\big)Ω(ε−p−1p​) for convex problems in the low-dimensional regimes d≲ε−pp−1d\lesssimε^{-\frac{p}{p-1}}d≲ε−p−1p​ and d≲ε−2−pp−1d\lesssimε^{-\frac{2-p}{p-1}}d≲ε−p−12−p​, respectively.