Aaron Sidford

Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford et al.

In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses very little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple (non-recursive) update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope that the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.

Yin Tat Lee, Aaron Sidford

In this paper we present a new algorithm for solving linear programs that requires only $\tilde{O}(\sqrt{rank(A)}L)$ iterations to solve a linear program with $m$ constraints, $n$ variables, and constraint matrix $A$, and bit complexity $L$. Each iteration of our method consists of solving $\tilde{O}(1)$ linear systems and additional nearly linear time computation. Our method improves upon the previous best iteration bound by factor of $\tildeΩ((m/rank(A))^{1/4})$ for methods with polynomial time computable iterations and by $\tildeΩ((m/rank(A))^{1/2})$ for methods which solve at most $\tilde{O}(1)$ linear systems in each iteration. Our method is parallelizable and amenable to linear algebraic techniques for accelerating the linear system solver. As such, up to polylogarithmic factors we either match or improve upon the best previous running times in both depth and work for different ratios of $m$ and $rank(A)$. Moreover, our method matches up to polylogarithmic factors a theoretical limit established by Nesterov and Nemirovski in 1994 regarding the use of a "universal barrier" for interior point methods, thereby resolving a long-standing open question regarding the running time of polynomial time interior point methods for linear programming.

Annie Marsden, Vatsal Sharan, Aaron Sidford et al.

We show that any memory-constrained, first-order algorithm which minimizes $d$-dimensional, $1$-Lipschitz convex functions over the unit ball to $1/\mathrm{poly}(d)$ accuracy using at most $d^{1.25 - δ}$ bits of memory must make at least $\tildeΩ(d^{1 + (4/3)δ})$ first-order queries (for any constant $δ\in [0, 1/4]$). Consequently, the performance of such memory-constrained algorithms are a polynomial factor worse than the optimal $\tilde{O}(d)$ query bound for this problem obtained by cutting plane methods that use $\tilde{O}(d^2)$ memory. This resolves a COLT 2019 open problem of Woodworth and Srebro.

Jonathan A. Kelner, Jerry Li, Allen Liu et al.

Sparse recovery is one of the most fundamental and well-studied inverse problems. Standard statistical formulations of the problem are provably solved by general convex programming techniques and more practical, fast (nearly-linear time) iterative methods. However, these latter "fast algorithms" have previously been observed to be brittle in various real-world settings. We investigate the brittleness of fast sparse recovery algorithms to generative model changes through the lens of studying their robustness to a "helpful" semi-random adversary, a framework which tests whether an algorithm overfits to input assumptions. We consider the following basic model: let $\mathbf{A} \in \mathbb{R}^{n \times d}$ be a measurement matrix which contains an unknown subset of rows $\mathbf{G} \in \mathbb{R}^{m \times d}$ which are bounded and satisfy the restricted isometry property (RIP), but is otherwise arbitrary. Letting $x^\star \in \mathbb{R}^d$ be $s$-sparse, and given either exact measurements $b = \mathbf{A} x^\star$ or noisy measurements $b = \mathbf{A} x^\star + ξ$, we design algorithms recovering $x^\star$ information-theoretically optimally in nearly-linear time. We extend our algorithm to hold for weaker generative models relaxing our planted RIP assumption to a natural weighted variant, and show that our method's guarantees naturally interpolate the quality of the measurement matrix to, in some parameter regimes, run in sublinear time. Our approach differs from prior fast iterative methods with provable guarantees under semi-random generative models: natural conditions on a submatrix which make sparse recovery tractable are NP-hard to verify. We design a new iterative method tailored to the geometry of sparse recovery which is provably robust to our semi-random model. We hope our approach opens the door to new robust, efficient algorithms for natural statistical inverse problems.

Jonathan A. Kelner, Jerry Li, Allen Liu et al.

We give a new framework for solving the fundamental problem of low-rank matrix completion, i.e., approximating a rank-$r$ matrix $\mathbf{M} \in \mathbb{R}^{m \times n}$ (where $m \ge n$) from random observations. First, we provide an algorithm which completes $\mathbf{M}$ on $99\%$ of rows and columns under no further assumptions on $\mathbf{M}$ from $\approx mr$ samples and using $\approx mr^2$ time. Then, assuming the row and column spans of $\mathbf{M}$ satisfy additional regularity properties, we show how to boost this partial completion guarantee to a full matrix completion algorithm by aggregating solutions to regression problems involving the observations. In the well-studied setting where $\mathbf{M}$ has incoherent row and column spans, our algorithms complete $\mathbf{M}$ to high precision from $mr^{2+o(1)}$ observations in $mr^{3 + o(1)}$ time (omitting logarithmic factors in problem parameters), improving upon the prior state-of-the-art [JN15] which used $\approx mr^5$ samples and $\approx mr^7$ time. Under an assumption on the row and column spans of $\mathbf{M}$ we introduce (which is satisfied by random subspaces with high probability), our sample complexity improves to an almost information-theoretically optimal $mr^{1 + o(1)}$, and our runtime improves to $mr^{2 + o(1)}$. Our runtimes have the appealing property of matching the best known runtime to verify that a rank-$r$ decomposition $\mathbf{U}\mathbf{V}^\top$ agrees with the sampled observations. We also provide robust variants of our algorithms that, given random observations from $\mathbf{M} + \mathbf{N}$ with $\|\mathbf{N}\|_{F} \le Δ$, complete $\mathbf{M}$ to Frobenius norm distance $\approx r^{1.5}Δ$ in the same runtimes as the noiseless setting. Prior noisy matrix completion algorithms [CP10] only guaranteed a distance of $\approx \sqrt{n}Δ$.

Moses Charikar, Zhihao Jiang, Kirankumar Shiragur et al.

We provide an efficient unified plug-in approach for estimating symmetric properties of distributions given $n$ independent samples. Our estimator is based on profile-maximum-likelihood (PML) and is sample optimal for estimating various symmetric properties when the estimation error $ε\gg n^{-1/3}$. This result improves upon the previous best accuracy threshold of $ε\gg n^{-1/4}$ achievable by polynomial time computable PML-based universal estimators [ACSS21, ACSS20]. Our estimator reaches a theoretical limit for universal symmetric property estimation as [Han21] shows that a broad class of universal estimators (containing many well known approaches including ours) cannot be sample optimal for every $1$-Lipschitz property when $ε\ll n^{-1/3}$.

Arun Jambulapati, Jerry Li, Christopher Musco et al.

We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite $\mathbf{K} \in \mathbb{R}^{d \times d}$ with $\mathrm{nnz}(\mathbf{K})$ nonzero entries, computes an $ε$-optimal diagonal preconditioner in time $\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(κ^\star,ε^{-1}))$, where $κ^\star$ is the optimal condition number of the rescaled matrix. We give an algorithm which, given $\mathbf{M} \in \mathbb{R}^{d \times d}$ that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in $\mathbf{M}$ in $\widetilde{O}(d^2)$ time. Our diagonal preconditioning results improve state-of-the-art runtimes of $Ω(d^{3.5})$ attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of $Ω(d^ω)$ where $ω> 2.3$ is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.

Yujia Jin, Christopher Musco, Aaron Sidford et al.

We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $ε$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $(1\pm2^{-Ω(1/ε)})$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $2^{O(1/ε)}$ random walks initiated at uniformly random nodes in the graph. As a strengthening of our main result, we show that improving the dependence on $1/ε$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $ε$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $2^{Ω(1/ε)}$ random walks of length $2^{Ω(1/ε)}$ started at random nodes.

Yujia Jin, Ishani Karmarkar, Christopher Musco et al.

We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $n$-node undirected graph. We provide a randomized algorithm that, with $O(nε^{-2})$ queries to a degree and neighbor oracle and in $O(nε^{-3})$ time, estimates the spectrum up to $ε$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(nε^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $ε$, a $2^{O(ε^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(nε^{-2})$-query and $O(nε^{-2})$-time algorithm for computing $O(nε^{-2})$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph).

Arun Jambulapati, Aaron Sidford, Kevin Tian

We develop a new parallel algorithm for minimizing Lipschitz, convex functions with a stochastic subgradient oracle. The total number of queries made and the query depth, i.e., the number of parallel rounds of queries, match the prior state-of-the-art, [CJJLLST23], while improving upon the computational depth by a polynomial factor for sufficiently small accuracy. When combined with previous state-of-the-art methods our result closes a gap between the best-known query depth and the best-known computational depth of parallel algorithms. Our method starts with a ball acceleration framework of previous parallel methods, i.e., [CJJJLST20, ACJJS21], which reduce the problem to minimizing a regularized Gaussian convolution of the function constrained to Euclidean balls. By developing and leveraging new stability properties of the Hessian of this induced function, we depart from prior parallel algorithms and reduce these ball-constrained optimization problems to stochastic unconstrained quadratic minimization problems. Although we are unable to prove concentration of the asymmetric matrices that we use to approximate this Hessian, we nevertheless develop an efficient parallel method for solving these quadratics. Interestingly, our algorithms can be improved using fast matrix multiplication and use nearly-linear work if the matrix multiplication exponent is 2.

Yair Carmon, Arun Jambulapati, Yujia Jin et al.

We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(Nε^{-2})$ queries to a first-order oracle to compute an $ε$-suboptimal point and $\tilde{O}(Nε^{-1})$ queries if the $f_i$ are $O(1/ε)$-smooth. We develop methods with improved complexity bounds of $\tilde{O}(Nε^{-2/3} + ε^{-8/3})$ in the non-smooth case and $\tilde{O}(Nε^{-2/3} + \sqrt{N}ε^{-1})$ in the $O(1/ε)$-smooth case. Our methods consist of a recently proposed ball optimization oracle acceleration algorithm (which we refine) and a careful implementation of said oracle for the softmax function. We also prove an oracle complexity lower bound scaling as $Ω(Nε^{-2/3})$, showing that our dependence on $N$ is optimal up to polylogarithmic factors.

Michael B. Cohen, Aaron Sidford, Kevin Tian

We show that standard extragradient methods (i.e. mirror prox and dual extrapolation) recover optimal accelerated rates for first-order minimization of smooth convex functions. To obtain this result we provide a fine-grained characterization of the convergence rates of extragradient methods for solving monotone variational inequalities in terms of a natural condition we call relative Lipschitzness. We further generalize this framework to handle local and randomized notions of relative Lipschitzness and thereby recover rates for box-constrained $\ell_\infty$ regression based on area convexity and complexity bounds achieved by accelerated (randomized) coordinate descent for smooth convex function minimization.

Nima Anari, Moses Charikar, Kirankumar Shiragur et al.

In this paper we provide a new efficient algorithm for approximately computing the profile maximum likelihood (PML) distribution, a prominent quantity in symmetric property estimation. We provide an algorithm which matches the previous best known efficient algorithms for computing approximate PML distributions and improves when the number of distinct observed frequencies in the given instance is small. We achieve this result by exploiting new sparsity structure in approximate PML distributions and providing a new matrix rounding algorithm, of independent interest. Leveraging this result, we obtain the first provable computationally efficient implementation of PseudoPML, a general framework for estimating a broad class of symmetric properties. Additionally, we obtain efficient PML-based estimators for distributions with small profile entropy, a natural instance-based complexity measure. Further, we provide a simpler and more practical PseudoPML implementation that matches the best-known theoretical guarantees of such an estimator and evaluate this method empirically.

Daniel Levy, Yair Carmon, John C. Duchi et al.

We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $χ^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $χ^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $χ^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.

Arun Jambulapati, Jerry Li, Christopher Musco et al.

The convergence rates of iterative methods for solving a linear system $\mathbf{A} x = b$ typically depend on the condition number of the matrix $\mathbf{A}$. Preconditioning is a common way of speeding up these methods by reducing that condition number in a computationally inexpensive way. In this paper, we revisit the decades-old problem of how to best improve $\mathbf{A}$'s condition number by left or right diagonal rescaling. We make progress on this problem in several directions. First, we provide new bounds for the classic heuristic of scaling $\mathbf{A}$ by its diagonal values (a.k.a. Jacobi preconditioning). We prove that this approach reduces $\mathbf{A}$'s condition number to within a quadratic factor of the best possible scaling. Second, we give a solver for structured mixed packing and covering semidefinite programs (MPC SDPs) which computes a constant-factor optimal scaling for $\mathbf{A}$ in $\widetilde{O}(\text{nnz}(\mathbf{A}) \cdot \text{poly}(κ^\star))$ time; this matches the cost of solving the linear system after scaling up to a $\widetilde{O}(\text{poly}(κ^\star))$ factor. Third, we demonstrate that a sufficiently general width-independent MPC SDP solver would imply near-optimal runtimes for the scaling problems we consider, and natural variants concerned with measures of average conditioning. Finally, we highlight connections of our preconditioning techniques to semi-random noise models, as well as applications in reducing risk in several statistical regression models.

Nima Anari, Moses Charikar, Kirankumar Shiragur et al.

In this paper we consider the problem of computing the likelihood of the profile of a discrete distribution, i.e., the probability of observing the multiset of element frequencies, and computing a profile maximum likelihood (PML) distribution, i.e., a distribution with the maximum profile likelihood. For each problem we provide polynomial time algorithms that given $n$ i.i.d.\ samples from a discrete distribution, achieve an approximation factor of $\exp\left(-O(\sqrt{n} \log n) \right)$, improving upon the previous best-known bound achievable in polynomial time of $\exp(-O(n^{2/3} \log n))$ (Charikar, Shiragur and Sidford, 2019). Through the work of Acharya, Das, Orlitsky and Suresh (2016), this implies a polynomial time universal estimator for symmetric properties of discrete distributions in a broader range of error parameter. We achieve these results by providing new bounds on the quality of approximation of the Bethe and Sinkhorn permanents (Vontobel, 2012 and 2014). We show that each of these are $\exp(O(k \log(N/k)))$ approximations to the permanent of $N \times N$ matrices with non-negative rank at most $k$, improving upon the previous known bounds of $\exp(O(N))$. To obtain our results on PML, we exploit the fact that the PML objective is proportional to the permanent of a certain Vandermonde matrix with $\sqrt{n}$ distinct columns, i.e. with non-negative rank at most $\sqrt{n}$. As a by-product of our work we establish a surprising connection between the convex relaxation in prior work (CSS19) and the well-studied Bethe and Sinkhorn approximations.

Yujia Jin, Aaron Sidford

Given a data matrix $\mathbf{A} \in \mathbb{R}^{n \times d}$, principal component projection (PCP) and principal component regression (PCR), i.e. projection and regression restricted to the top-eigenspace of $\mathbf{A}$, are fundamental problems in machine learning, optimization, and numerical analysis. In this paper we provide the first algorithms that solve these problems in nearly linear time for fixed eigenvalue distribution and large n. This improves upon previous methods which have superlinear running times when both the number of top eigenvalues and inverse gap between eigenspaces is large. We achieve our results by applying rational approximations to reduce PCP and PCR to solving asymmetric linear systems which we solve by a variant of SVRG. We corroborate these findings with preliminary empirical experiments.

Oliver Hinder, Aaron Sidford, Nimit S. Sohoni

In this paper, we provide near-optimal accelerated first-order methods for minimizing a broad class of smooth nonconvex functions that are strictly unimodal on all lines through a minimizer. This function class, which we call the class of smooth quasar-convex functions, is parameterized by a constant $γ\in (0,1]$, where $γ= 1$ encompasses the classes of smooth convex and star-convex functions, and smaller values of $γ$ indicate that the function can be "more nonconvex." We develop a variant of accelerated gradient descent that computes an $ε$-approximate minimizer of a smooth $γ$-quasar-convex function with at most $O(γ^{-1} ε^{-1/2} \log(γ^{-1} ε^{-1}))$ total function and gradient evaluations. We also derive a lower bound of $Ω(γ^{-1} ε^{-1/2})$ on the worst-case number of gradient evaluations required by any deterministic first-order method, showing that, up to a logarithmic factor, no deterministic first-order method can improve upon ours.

Arun Jambulapati, Aaron Sidford, Kevin Tian

Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings. We give an algorithm which solves this problem to additive $ε$ with $\tilde{O}(1/ε)$ parallel depth, and $\tilde{O}\left(n^2/ε\right)$ work. Barring a breakthrough on a long-standing algorithmic open problem, this is optimal for first-order methods. Blanchet et. al. '18, Quanrud '19 obtained similar runtimes through reductions to positive linear programming and matrix scaling. However, these reduction-based algorithms use complicated subroutines which may be deemed impractical due to requiring solvers for second-order iterations (matrix scaling) or non-parallelizability (positive LP). The fastest practical algorithms run in time $\tilde{O}(\min(n^2 / ε^2, n^{2.5} / ε))$ (Dvurechensky et. al. '18, Lin et. al. '19). We bridge this gap by providing a parallel, first-order, $\tilde{O}(1/ε)$ iteration algorithm without worse dependence on dimension, and provide preliminary experimental evidence that our algorithm may enjoy improved practical performance. We obtain this runtime via a primal-dual extragradient method, motivated by recent theoretical improvements to maximum flow (Sherman '17).

Yair Carmon, John C. Duchi, Aaron Sidford et al.

We show that a simple randomized sketch of the matrix multiplicative weight (MMW) update enjoys (in expectation) the same regret bounds as MMW, up to a small constant factor. Unlike MMW, where every step requires full matrix exponentiation, our steps require only a single product of the form $e^A b$, which the Lanczos method approximates efficiently. Our key technique is to view the sketch as a $\textit{randomized mirror projection}$, and perform mirror descent analysis on the $\textit{expected projection}$. Our sketch solves the online eigenvector problem, improving the best known complexity bounds by $Ω(\log^5 n)$. We also apply this sketch to semidefinite programming in saddle-point form, yielding a simple primal-dual scheme with guarantees matching the best in the literature.

Neha Gupta, Aaron Sidford

In this paper, we obtain improved running times for regression and top eigenvector computation for numerically sparse matrices. Given a data matrix $A \in \mathbb{R}^{n \times d}$ where every row $a \in \mathbb{R}^d$ has $\|a\|_2^2 \leq L$ and numerical sparsity at most $s$, i.e. $\|a\|_1^2 / \|a\|_2^2 \leq s$, we provide faster algorithms for these problems in many parameter settings. For top eigenvector computation, we obtain a running time of $\tilde{O}(nd + r(s + \sqrt{r s}) / \mathrm{gap}^2)$ where $\mathrm{gap} > 0$ is the relative gap between the top two eigenvectors of $A^\top A$ and $r$ is the stable rank of $A$. This running time improves upon the previous best unaccelerated running time of $O(nd + r d / \mathrm{gap}^2)$ as it is always the case that $r \leq d$ and $s \leq d$. For regression, we obtain a running time of $\tilde{O}(nd + (nL / μ) \sqrt{s nL / μ})$ where $μ> 0$ is the smallest eigenvalue of $A^\top A$. This running time improves upon the previous best unaccelerated running time of $\tilde{O}(nd + n L d / μ)$. This result expands the regimes where regression can be solved in nearly linear time from when $L/μ= \tilde{O}(1)$ to when $L / μ= \tilde{O}(d^{2/3} / (sn)^{1/3})$. Furthermore, we obtain similar improvements even when row norms and numerical sparsities are non-uniform and we show how to achieve even faster running times by accelerating using approximate proximal point [Frostig et. al. 2015] / catalyst [Lin et. al. 2015]. Our running times depend only on the size of the input and natural numerical measures of the matrix, i.e. eigenvalues and $\ell_p$ norms, making progress on a key open problem regarding optimal running times for efficient large-scale learning.

Rong Ge, Chi Jin, Sham M. Kakade et al.

This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009). We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-$k$ generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of $O(\frac{z k \sqrtκ}ρ \log(1/ε) \log \left(kκ/ρ\right))$ where $z$ is the total number of nonzero entries, $κ$ is the condition number and $ρ$ is the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.

Prateek Jain, Chi Jin, Sham M. Kakade et al.

This work provides improved guarantees for streaming principle component analysis (PCA). Given $A_1, \ldots, A_n\in \mathbb{R}^{d\times d}$ sampled independently from distributions satisfying $\mathbb{E}[A_i] = Σ$ for $Σ\succeq \mathbf{0}$, this work provides an $O(d)$-space linear-time single-pass streaming algorithm for estimating the top eigenvector of $Σ$. The algorithm nearly matches (and in certain cases improves upon) the accuracy obtained by the standard batch method that computes top eigenvector of the empirical covariance $\frac{1}{n} \sum_{i \in [n]} A_i$ as analyzed by the matrix Bernstein inequality. Moreover, to achieve constant accuracy, our algorithm improves upon the best previous known sample complexities of streaming algorithms by either a multiplicative factor of $O(d)$ or $1/\mathrm{gap}$ where $\mathrm{gap}$ is the relative distance between the top two eigenvalues of $Σ$. These results are achieved through a novel analysis of the classic Oja's algorithm, one of the oldest and most popular algorithms for streaming PCA. In particular, this work shows that simply picking a random initial point $w_0$ and applying the update rule $w_{i + 1} = w_i + η_i A_i w_i$ suffices to accurately estimate the top eigenvector, with a suitable choice of $η_i$. We believe our result sheds light on how to efficiently perform streaming PCA both in theory and in practice and we hope that our analysis may serve as the basis for analyzing many variants and extensions of streaming PCA.

Roy Frostig, Cameron Musco, Christopher Musco et al.

We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvement over the naive method of combining PCA with regression. To achieve our results, we first observe that ridge regression can be used to obtain a "smooth projection" onto the top principal components. We then sharpen this approximation to true projection using a low-degree polynomial approximation to the matrix step function. Step function approximation is a topic of long-term interest in scientific computing. We extend prior theory by constructing polynomials with simple iterative structure and rigorously analyzing their behavior under limited precision.

Yin Tat Lee, Aaron Sidford

In this paper, we consider the following inverse maintenance problem: given $A \in \mathbb{R}^{n\times d}$ and a number of rounds $r$, we receive a $n\times n$ diagonal matrix $D^{(k)}$ at round $k$ and we wish to maintain an efficient linear system solver for $A^{T}D^{(k)}A$ under the assumption $D^{(k)}$ does not change too rapidly. This inverse maintenance problem is the computational bottleneck in solving multiple optimization problems. We show how to solve this problem with $\tilde{O}(nnz(A)+d^ω)$ preprocessing time and amortized $\tilde{O}(nnz(A)+d^{2})$ time per round, improving upon previous running times for solving this problem. Consequently, we obtain the fastest known running times for solving multiple problems including, linear programming and computing a rounding of a polytope. In particular given a feasible point in a linear program with $d$ variables, $n$ constraints, and constraint matrix $A\in\mathbb{R}^{n\times d}$, we show how to solve the linear program in time $\tilde{O}(nnz(A)+d^{2})\sqrt{d}\log(ε^{-1}))$. We achieve our results through a novel combination of classic numerical techniques of low rank update, preconditioning, and fast matrix multiplication as well as recent work on subspace embeddings and spectral sparsification that we hope will be of independent interest.

Roy Frostig, Rong Ge, Sham M. Kakade et al.

We develop a family of accelerated stochastic algorithms that minimize sums of convex functions. Our algorithms improve upon the fastest running time for empirical risk minimization (ERM), and in particular linear least-squares regression, across a wide range of problem settings. To achieve this, we establish a framework based on the classical proximal point algorithm. Namely, we provide several algorithms that reduce the minimization of a strongly convex function to approximate minimizations of regularizations of the function. Using these results, we accelerate recent fast stochastic algorithms in a black-box fashion. Empirically, we demonstrate that the resulting algorithms exhibit notions of stability that are advantageous in practice. Both in theory and in practice, the provided algorithms reap the computational benefits of adding a large strongly convex regularization term, without incurring a corresponding bias to the original problem.

Michael B. Cohen, Yin Tat Lee, Cameron Musco et al.

Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time significantly. For theoretical performance guarantees, each row must be sampled with probability proportional to its statistical leverage score. Unfortunately, leverage scores are difficult to compute. A simple alternative is to sample rows uniformly at random. While this often works, uniform sampling will eliminate critical row information for many natural instances. We take a fresh look at uniform sampling by examining what information it does preserve. Specifically, we show that uniform sampling yields a matrix that, in some sense, well approximates a large fraction of the original. While this weak form of approximation is not enough for solving linear regression directly, it is enough to compute a better approximation. This observation leads to simple iterative row sampling algorithms for matrix approximation that run in input-sparsity time and preserve row structure and sparsity at all intermediate steps. In addition to an improved understanding of uniform sampling, our main proof introduces a structural result of independent interest: we show that every matrix can be made to have low coherence by reweighting a small subset of its rows.