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We investigate sublinear classical and quantum algorithms for matrix games, a fundamental problem in optimization and machine learning, with provable guarantees. Given a matrix $Ainmathbb{R}^{ntimes d}$, sublinear algorithms for the matrix game $min_{xinmathcal{X}}max_{yinmathcal{Y}} y^{top} Ax$ were previously known only for two special cases: (1) $mathcal{Y}$ being the $ell_{1}$-norm unit ball, and (2) $mathcal{X}$ being either the $ell_{1}$- or the $ell_{2}$-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed $qin (1,2]$, we solve the matrix game where $mathcal{X}$ is a $ell_{q}$-norm unit ball within additive error $epsilon$ in time $tilde{O}((n+d)/{epsilon^{2}})$. We also provide a corresponding sublinear quantum algorithm that solves the same task in time $tilde{O}((sqrt{n}+sqrt{d})textrm{poly}(1/epsilon))$ with a quadratic improvement in both $n$ and $d$. Both our classical and quantum algorithms are optimal in the dimension parameters $n$ and $d$ up to poly-logarithmic factors. Finally, we propose sublinear classical and quantum algorithms for the approximate Caratheodory problem and the $ell_{q}$-margin support vector machines as applications.
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We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin runs in $tilde{O}(n+d)$ time. We design sublinear quantum algorithms for the same task running in $tilde{O}(sqrt{n} +sqrt{d})$ time, a quadratic improvement in both $n$ and $d$. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines. As a side result, we also give sublinear quantum algorithms for approximating the equilibria of $n$-dimensional matrix zero-sum games with optimal complexity $tilde{Theta}(sqrt{n})$.
We present classical and quantum algorithms based on spectral methods for a problem in tensor principal component analysis. The quantum algorithm achieves a quartic speedup while using exponentially smaller space than the fastest classical spectral algorithm, and a super-polynomial speedup over classical algorithms that use only polynomial space. The classical algorithms that we present are related to, but slightly different from those presented recently in Ref. 1. In particular, we have an improved threshold for recovery and the algorithms we present work for both even and odd order tensors. These results suggest that large-scale inference problems are a promising future application for quantum computers.
We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A in mathbb{C}^{m times n}$ be a rank-$k$ matrix, and $b in mathbb{C}^m$ be a vector. We present two algorithms: a sampling algorithm that provides a sample from $A^{-1}b$ and a query algorithm that outputs an estimate of an entry of $A^{-1}b$, where $A^{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $O(mathrm{poly}(k, kappa, |A|_F, 1/epsilon),mathrm{polylog}(m, n))$, where $kappa$ is the condition number of $A$ and $epsilon$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.
We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of classical algorithms. (1) We prove that every (quantum or classical) one-local algorithm achieves on $D$-regular graphs of girth $> 5$ a maximum cut of at most $1/2 + C/sqrt{D}$ for $C=1/sqrt{2} approx 0.7071$. This is the first such result showing that one-local algorithms achieve a value bounded away from the true optimum for random graphs, which is $1/2 + P_*/sqrt{D} + o(1/sqrt{D})$ for $P_* approx 0.7632$. (2) We show that there is a classical $k$-local algorithm that achieves a value of $1/2 + C/sqrt{D} - O(1/sqrt{k})$ for $D$-regular graphs of girth $> 2k+1$, where $C = 2/pi approx 0.6366$. This is an algorithmic version of the existential bound of Lyons and is related to the algorithm of Aizenman, Lebowitz, and Ruelle (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-loc
We present a randomized primal-dual algorithm that solves the problem $min_{x} max_{y} y^top A x$ to additive error $epsilon$ in time $mathrm{nnz}(A) + sqrt{mathrm{nnz}(A)n}/epsilon$, for matrix $A$ with larger dimension $n$ and $mathrm{nnz}(A)$ nonzero entries. This improves the best known exact gradient methods by a factor of $sqrt{mathrm{nnz}(A)/n}$ and is faster than fully stochastic gradient methods in the accurate and/or sparse regime $epsilon le sqrt{n/mathrm{nnz}(A)}$. Our results hold for $x,y$ in the simplex (matrix games, linear programming) and for $x$ in an $ell_2$ ball and $y$ in the simplex (perceptron / SVM, minimum enclosing ball). Our algorithm combines Nemirovskis conceptual prox-method and a novel reduced-variance gradient estimator based on sampling from the difference between the current iterate and a reference point.
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