arXiv subjects

Zeyuan Allen-Zhu

Publications and source records attributed to Zeyuan Allen-Zhu.

At least 55 records · Page 3Linked to original sources

arXiv2017

Faster Principal Component Regression and Stable Matrix Chebyshev Approximation

We solve principal component regression (PCR), up to a multiplicative accuracy $1+γ$, by reducing the problem to $\tilde{O}(γ^{-1})$ black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In contrast, previous result requires $\tilde{O}(γ^{-2})$ such black-box calls. We obtain this result by developing a general stable recurrence formula for matrix Chebyshev polynomials, and a degree-optimal polynomial approximation to the matrix sign function. Our techniques may be of independent interests, especially when designing iterative methods.

stat.ML↗

arXiv2017

Finding Approximate Local Minima Faster than Gradient Descent

We design a non-convex second-order optimization algorithm that is guaranteed to return an approximate local minimum in time which scales linearly in the underlying dimension and the number of training examples. The time complexity of our algorithm to find an approximate local minimum is even faster than that of gradient descent to find a critical point. Our algorithm applies to a general class of optimization problems including training a neural network and other non-convex objectives arising in machine learning.

math.OC↗

arXiv2017

First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate

We study streaming principal component analysis (PCA), that is to find, in $O(dk)$ space, the top $k$ eigenvectors of a $d\times d$ hidden matrix $\bf Σ$ with online vectors drawn from covariance matrix $\bf Σ$. We provide $\textit{global}$ convergence for Oja's algorithm which is popularly used in practice but lacks theoretical understanding for $k>1$. We also provide a modified variant $\mathsf{Oja}^{++}$ that runs $\textit{even faster}$ than Oja's. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank $k$, and on dimension $d$, up to poly-log factors. In addition, our convergence rate can be made gap-free, that is proportional to the approximation error and independent of the eigengap. In contrast, for general rank $k$, before our work (1) it was open to design any algorithm with efficient global convergence rate; and (2) it was open to design any algorithm with (even local) gap-free convergence rate in $O(dk)$ space.

math.OC↗

arXiv2017

Much Faster Algorithms for Matrix Scaling

We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this problem asks to find diagonal (scaling) matrices $X$ and $Y$ (if they exist), so that $X A Y$ $\varepsilon$-approximates a doubly stochastic, or more generally a matrix with prescribed row and column sums. We address the general scaling problem as well as some important special cases. In particular, if $A$ has $m$ nonzero entries, and if there exist $X$ and $Y$ with polynomially large entries such that $X A Y$ is doubly stochastic, then we can solve the problem in total complexity $\tilde{O}(m + n^{4/3})$. This greatly improves on the best known previous results, which were either $\tilde{O}(n^4)$ or $O(m n^{1/2}/\varepsilon)$. Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems.

cs.DS↗

arXiv2017

LazySVD: Even Faster SVD Decomposition Yet Without Agonizing Pain

We study $k$-SVD that is to obtain the first $k$ singular vectors of a matrix $A$. Recently, a few breakthroughs have been discovered on $k$-SVD: Musco and Musco [1] proved the first gap-free convergence result using the block Krylov method, Shamir [2] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [3] provided the fastest $O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$-time algorithm using alternating minimization. In this paper, we put forward a new and simple LazySVD framework to improve the above breakthroughs. This framework leads to a faster gap-free method outperforming [1], and the first accelerated and stochastic method outperforming [2]. In the $O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$ running-time regime, LazySVD outperforms [3] in certain parameter regimes without even using alternating minimization.

math.NA↗

arXiv2016

Doubly Accelerated Methods for Faster CCA and Generalized Eigendecomposition

We study $k$-GenEV, the problem of finding the top $k$ generalized eigenvectors, and $k$-CCA, the problem of finding the top $k$ vectors in canonical-correlation analysis. We propose algorithms $\mathtt{LazyEV}$ and $\mathtt{LazyCCA}$ to solve the two problems with running times linearly dependent on the input size and on $k$. Furthermore, our algorithms are DOUBLY-ACCELERATED: our running times depend only on the square root of the matrix condition number, and on the square root of the eigengap. This is the first such result for both $k$-GenEV or $k$-CCA. We also provide the first gap-free results, which provide running times that depend on $1/\sqrt{\varepsilon}$ rather than the eigengap.

math.OC↗

arXiv2016

Using Optimization to Solve Positive LPs Faster in Parallel

Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all known nearly-linear-time algorithms require $\tilde{O}(\varepsilon^{-4})$ iterations to converge to $1\pm \varepsilon$ approximate solutions. This $\varepsilon^{-4}$ dependence has not been improved since 1993, and limits the performance of parallel implementations for such algorithms. Moreover, previous algorithms and their analyses rely on update steps and convergence arguments that are combinatorial in nature and do not seem to arise naturally from an optimization viewpoint. In this paper, we leverage new insights from optimization theory to construct a novel algorithm that breaks the longstanding $\varepsilon^{-4}$ barrier. Our algorithm has a simple analysis and a clear motivation. Our work introduces a number of novel techniques, such as the combined application of gradient descent and mirror descent, and a truncated, smoothed version of the standard multiplicative weight update, which may be of independent interest.

cs.DS↗

arXiv2016

Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent

First-order methods play a central role in large-scale machine learning. Even though many variations exist, each suited to a particular problem, almost all such methods fundamentally rely on two types of algorithmic steps: gradient descent, which yields primal progress, and mirror descent, which yields dual progress. We observe that the performances of gradient and mirror descent are complementary, so that faster algorithms can be designed by LINEARLY COUPLING the two. We show how to reconstruct Nesterov's accelerated gradient methods using linear coupling, which gives a cleaner interpretation than Nesterov's original proofs. We also discuss the power of linear coupling by extending it to many other settings that Nesterov's methods cannot apply to.

cs.DS↗

arXiv2016

Exploiting the Structure: Stochastic Gradient Methods Using Raw Clusters

The amount of data available in the world is growing faster than our ability to deal with it. However, if we take advantage of the internal \emph{structure}, data may become much smaller for machine learning purposes. In this paper we focus on one of the fundamental machine learning tasks, empirical risk minimization (ERM), and provide faster algorithms with the help from the clustering structure of the data. We introduce a simple notion of raw clustering that can be efficiently computed from the data, and propose two algorithms based on clustering information. Our accelerated algorithm ClusterACDM is built on a novel Haar transformation applied to the dual space of the ERM problem, and our variance-reduction based algorithm ClusterSVRG introduces a new gradient estimator using clustering. Our algorithms outperform their classical counterparts ACDM and SVRG respectively.

cs.LG↗

arXiv2016

Variance Reduction for Faster Non-Convex Optimization

We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order non-convex optimization remain to be full gradient descent that converges in $O(1/\varepsilon)$ iterations for smooth objectives, and stochastic gradient descent that converges in $O(1/\varepsilon^2)$ iterations for objectives that are sum of smooth functions. We provide the first improvement in this line of research. Our result is based on the variance reduction trick recently introduced to convex optimization, as well as a brand new analysis of variance reduction that is suitable for non-convex optimization. For objectives that are sum of smooth functions, our first-order minibatch stochastic method converges with an $O(1/\varepsilon)$ rate, and is faster than full gradient descent by $Ω(n^{1/3})$. We demonstrate the effectiveness of our methods on empirical risk minimizations with non-convex loss functions and training neural nets.

math.OC↗

arXiv2016

Improved SVRG for Non-Strongly-Convex or Sum-of-Non-Convex Objectives

Many classical algorithms are found until several years later to outlive the confines in which they were conceived, and continue to be relevant in unforeseen settings. In this paper, we show that SVRG is one such method: being originally designed for strongly convex objectives, it is also very robust in non-strongly convex or sum-of-non-convex settings. More precisely, we provide new analysis to improve the state-of-the-art running times in both settings by either applying SVRG or its novel variant. Since non-strongly convex objectives include important examples such as Lasso or logistic regression, and sum-of-non-convex objectives include famous examples such as stochastic PCA and is even believed to be related to training deep neural nets, our results also imply better performances in these applications.

cs.LG↗

arXiv2016

Even Faster Accelerated Coordinate Descent Using Non-Uniform Sampling

Accelerated coordinate descent is widely used in optimization due to its cheap per-iteration cost and scalability to large-scale problems. Up to a primal-dual transformation, it is also the same as accelerated stochastic gradient descent that is one of the central methods used in machine learning. In this paper, we improve the best known running time of accelerated coordinate descent by a factor up to $\sqrt{n}$. Our improvement is based on a clean, novel non-uniform sampling that selects each coordinate with a probability proportional to the square root of its smoothness parameter. Our proof technique also deviates from the classical estimation sequence technique used in prior work. Our speed-up applies to important problems such as empirical risk minimization and solving linear systems, both in theory and in practice.

math.OC↗

arXiv2016

Optimal Black-Box Reductions Between Optimization Objectives

The diverse world of machine learning applications has given rise to a plethora of algorithms and optimization methods, finely tuned to the specific regression or classification task at hand. We reduce the complexity of algorithm design for machine learning by reductions: we develop reductions that take a method developed for one setting and apply it to the entire spectrum of smoothness and strong-convexity in applications. Furthermore, unlike existing results, our new reductions are OPTIMAL and more PRACTICAL. We show how these new reductions give rise to new and faster running times on training linear classifiers for various families of loss functions, and conclude with experiments showing their successes also in practice.

math.OC↗

arXiv2016

Optimization Algorithms for Faster Computational Geometry

We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined by $m$ hyperplanes, and the minimum enclosing ball (MinEB) of a set of $n$ points, both in $d$-dimensional space. We improve the running time of iterative algorithms on MaxIB from $\tilde{O}(m d α^3 / \varepsilon^3)$ to $\tilde{O}(md + m \sqrt{d} α/ \varepsilon)$, a speed-up up to $\tilde{O}(\sqrt{d} α^2 / \varepsilon^2)$, and MinEB from $\tilde{O}(n d / \sqrt{\varepsilon})$ to $\tilde{O}(nd + n \sqrt{d} / \sqrt{\varepsilon})$, a speed-up up to $\tilde{O}(\sqrt{d})$. Our improvements are based on a novel saddle-point optimization framework. We propose a new algorithm $\mathtt{L1L2SPSolver}$ for solving a class of regularized saddle-point problems, and apply a randomized Hadamard space rotation which is a technique borrowed from compressive sensing. Interestingly, the motivation of using Hadamard rotation solely comes from our optimization view but not the original geometry problem: indeed, it is not immediately clear why MaxIB or MinEB, as a geometric problem, should be easier to solve if we rotate the space by a unitary matrix. We hope that our optimization perspective sheds lights on solving other geometric problems as well.

cs.CG↗

arXiv2016

Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver

We study the design of polylogarithmic depth algorithms for approximately solving packing and covering semidefinite programs (or positive SDPs for short). This is a natural SDP generalization of the well-studied positive LP problem. Although positive LPs can be solved in polylogarithmic depth while using only $\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best known positive SDP solvers due to Jain and Yao require $O(\log^{14} n /\varepsilon^{13})$ parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations. However, the correctness of the convergence analyses in these works has been called into question, as they both rely on algebraic monotonicity properties that do not generalize to matrix algebra. In this paper, we propose a very simple algorithm based on the optimization framework proposed for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n / \varepsilon^2)$ iterations, matching that of the best LP solver. To surmount the obstacles encountered by previous approaches, our analysis requires a new matrix inequality that extends Lieb-Thirring's inequality, and a sign-consistent, randomized variant of the gradient truncation technique proposed in.

cs.DS↗

arXiv2015

Expanders via Local Edge Flips

Designing distributed and scalable algorithms to improve network connectivity is a central topic in peer-to-peer networks. In this paper we focus on the following well-known problem: given an $n$-node $d$-regular network for $d=Ω(\log n)$, we want to design a decentralized, local algorithm that transforms the graph into one that has good connectivity properties (low diameter, expansion, etc.) without affecting the sparsity of the graph. To this end, Mahlmann and Schindelhauer introduced the random "flip" transformation, where in each time step, a random pair of vertices that have an edge decide to `swap a neighbor'. They conjectured that performing $O(n d)$ such flips at random would convert any connected $d$-regular graph into a $d$-regular expander graph, with high probability. However, the best known upper bound for the number of steps is roughly $O(n^{17} d^{23})$, obtained via a delicate Markov chain comparison argument. Our main result is to prove that a natural instantiation of the random flip produces an expander in at most $O(n^2 d^2 \sqrt{\log n})$ steps, with high probability. Our argument uses a potential-function analysis based on the matrix exponential, together with the recent beautiful results on the higher-order Cheeger inequality of graphs. We also show that our technique can be used to analyze another well-studied random process known as the `random switch', and show that it produces an expander in $O(n d)$ steps with high probability.

cs.DS↗

arXiv2015

Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates

In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required $Ω(n^4)$ running time [BSS14, Zou12], our sparsification routine can be implemented in almost-quadratic running time $O(n^{2+\varepsilon})$. The fundamental conceptual novelty of our work is the leveraging of a strong connection between sparsification and a regret minimization problem over density matrices. This connection was known to provide an interpretation of the randomized sparsifiers of Spielman and Srivastava [SS11] via the application of matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we explain how matrix MWU naturally arises as an instance of the Follow-the-Regularized-Leader framework and generalize this approach to yield a larger class of updates. This new class allows us to accelerate the construction of linear-sized spectral sparsifiers, and give novel insights on the motivation behind Batson, Spielman and Srivastava [BSS14].

cs.LG↗

arXiv2015

Restricted Isometry Property for General p-Norms

The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an $m \times n$ matrix satisfies RIP of order $k$ for the $\ell_p$ norm, if $\|Ax\|_p \approx \|x\|_p$ for every vector $x$ with at most $k$ non-zero coordinates. For every $1 \leq p < \infty$ we obtain almost tight bounds on the minimum number of rows $m$ necessary for the RIP property to hold. Prior to this work, only the cases $p = 1$, $1 + 1 / \log k$, and $2$ were studied. Interestingly, our results show that the case $p = 2$ is a "singularity" point: the optimal number of rows $m$ is $\widetildeΘ(k^{p})$ for all $p\in [1,\infty)\setminus \{2\}$, as opposed to $\widetildeΘ(k)$ for $k=2$. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.

cs.DS↗