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arXiv:2010.01801 · Leibniz international proceedings in informatics

No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization

Ankit Garg, Robin Kothari, Praneeth Netrapalli, Suhail Sherif

Published Oct 5, 2020• 1 citation
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Abstract

We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function $f:\mathbb{R}^n \to \mathbb{R}$ and its (sub)gradient. Our goal is to find an $ε$-approximate minimum of $f$ starting from a point that is distance at most $R$ from the true minimum. If $f$ is $G$-Lipschitz, then the classic gradient descent algorithm solves this problem with $O((GR/ε)^{2})$ queries. Importantly, the number of queries is independent of the dimension $n$ and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension $n$. In this paper we reprove the randomized lower bound of $Ω((GR/ε)^{2})$ using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using $O(GR/ε)$ quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need $Ω((GR/ε)^2)$ queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.

DOI
10.4230/lipics.itcs.2021.53

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We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:ℝⁿ → ℝ and its (sub)gradient. Our goal is to find an ε-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ε)²) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the randomized lower bound of Ω((GR/ε)²) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ε) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ε)²) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
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.
This work investigates the possibility of quantum speedups for continuous optimization through quantum Hamiltonian simulation. We establish the first rigorous query complexity bounds for unconstrained convex optimization via digital quantum annealing, based on the non-adiabatic Quantum Hamiltonian Descent (QHD) framework. In the process, we derive rigorous resource estimates for digital quantum simulation of Schrödinger operators depending only on input simulation parameters, given black-box evaluation access to separable, Lipschitz continuous potential b(t) f(x). We apply these simulation bounds to assess the complexity of high-dimensional convex optimization. Our annealing schedule achieves arbitrarily fast convergence rates in the evolution time, with computational time determined solely by the cost of discretization. We show that a G-Lipschitz convex function can be optimized to an error of ε with 𝒪̃(d^{1.5} G² R²/ε²) queries, given a starting point that is Euclidean distance R from optimal. Under reasonable assumptions such as the complexity of simulating Schrödinger operators, we show that Ω̃(d/ε²) queries are necessary. This suggests QHD does not offer improvements over classical methods in the noiseless zeroth order setting. However, we show that the QHD algorithm can tolerate 𝒪̃(ε³ /d^{1.5} G² R²) noise in function evaluation, and as a result, provides a super-quadratic query advantage over the best existing noise-tolerant classical algorithms in the high-dimensional setting. We leverage this to design a quantum algorithm for stochastic convex optimization that offers a super-quadratic speedup over all known classical and quantum algorithms in this regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization obtained through a dynamical algorithm.
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 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.
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.