202779595
Principal Component Projection and Regression in Nearly Linear Time through Asymmetric SVRG
Given a n-by-d data matrix A, principal component projection (PCP) and principal component regression (PCR), i.e. projection and regression restricted to the top-eigenspace of 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 had superlinear running times when either the number of top eigenvalues or gap between the eigenspaces were large. We achieve our results by applying rational polynomial approximations to reduce the problem to solving asymmetric linear systems which we solve by a variant of SVRG. We corroborate these findings with preliminary empirical experiments.
9003176
Principal Component Projection Without Principal Component Analysis
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.
33484744
Faster Principal Component Regression and Stable Matrix Chebyshev Approximation
We solve principal component regression (PCR), up to a multiplicative accuracy $1+\gamma$, by reducing the problem to $\tilde{O}(\gamma^{-1})$ 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 $\tilde{O}(\gamma^{-2})$ 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.
12380645
Uniform approximation of sgn x by polynomials and entire functions
In 1877, E. I. Zolotarev [19, 2] found an explicit expression, in terms ofelliptic functions, of the rational function of given degree mwhich is uni-formly closest to sgn(x) on the union of two intervals [−1,−a] ∪[a,1]. Thisresult was subject to many generalizations, and it has applications in electricengineering.Surprisingly, to the best of our knowledge, the similar problem for poly-nomials was not solved yet, so we investigate it in this paper.For comparison, we mention here the results on the uniform approxima-tion of |x|
6761326
Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions
The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with arithmetic costs. Spectral divide-and-conquer algorithms, which recursively decouple the problem into two smaller subproblems, can achieve both requirements. Such algorithms can be constructed with the polar decomposition playing two key roles: it forms a bridge between the symmetric eigendecomposition and the SVD, and its connection to the matrix sign function naturally leads to spectral-decoupling. For computing the polar decomposition, the
 scaled Newton and QDWH iterations are two of the most popular algorithms, as they are backward stable and converge in at most nine and six iterations, respectively. Following this framework, we develop a higher-order variant of the QDWH iteration for the polar decomposition. The key idea of this algorithm comes from approximation theory: we use the best rational approximant for the scalar sign function due to Zolotarev in 1877. The algorithm exploits the extraordinary property enjoyed by the sign function
 that a high-degree Zolotarev function (best rational approximant) can be obtained by appropriately composing low-degree Zolotarev functions. This lets the algorithm converge in just \emph{two} iterations in double-precision arithmetic, with the whopping rate of convergence \emph{seventeen}. The resulting algorithms for the
 symmetric eigendecompositions and the SVD have higher arithmetic costs than the QDWH-based algorithms, but are better-suited for parallel computing and exhibit excellent numerical backward stability.
6587578
Accelerating Stochastic Gradient Descent using Predictive Variance Reduction
Stochastic gradient descent is popular for large scale optimization but has slow convergence asymptotically due to the inherent variance. To remedy this problem, we introduce an explicit variance reduction method for stochastic gradient descent which we call stochastic variance reduced gradient (SVRG). For smooth and strongly convex functions, we prove that this method enjoys the same fast convergence rate as those of stochastic dual coordinate ascent (SDCA) and Stochastic Average Gradient (SAG). However, our analysis is significantly simpler and more intuitive. Moreover, unlike SDCA or SAG, our method does not require the storage of gradients, and thus is more easily applicable to complex problems such as some structured prediction problems and neural network learning.
12469358
Stochastic Variance Reduction Methods for Saddle-Point Problems
We consider convex-concave saddle-point problems where the objective functions may be split in many components, and extend recent stochastic variance reduction methods (such as SVRG or SAGA) to provide the first large-scale linearly convergent algorithms for this class of problems which is common in machine learning. While the algorithmic extension is straightforward, it comes with challenges and opportunities: (a) the convex minimization analysis does not apply and we use the notion of monotone operators to prove convergence, showing in particular that the same algorithm applies to a larger class of problems, such as variational inequalities, (b) there are two notions of splits, in terms of functions, or in terms of partial derivatives, (c) the split does need to be done with convex-concave terms, (d) non-uniform sampling is key to an efficient algorithm, both in theory and practice, and (e) these incremental algorithms can be easily accelerated using a simple extension of the "catalyst" framework, leading to an algorithm which is always superior to accelerated batch algorithms.
1903720
Un-regularizing: approximate proximal point and faster stochastic algorithms for empirical risk minimization
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.
6163335
A Universal Catalyst for First-Order Optimization
We introduce a generic scheme for accelerating first-order optimization methods in the sense of Nesterov, which builds upon a new analysis of the accelerated proximal point algorithm. Our approach consists of minimizing a convex objective by approximately solving a sequence of well-chosen auxiliary problems, leading to faster convergence. This strategy applies to a large class of algorithms, including gradient descent, block coordinate descent, SAG, SAGA, SDCA, SVRG, Finito/MISO, and their proximal variants. For all of these methods, we provide acceleration and explicit support for non-strongly convex objectives. In addition to theoretical speed-up, we also show that acceleration is useful in practice, especially for ill-conditioned problems where we measure significant improvements.
5095654
Minimizing finite sums with the stochastic average gradient
We analyze the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method’s iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. The convergence rate is improved from $$O(1/\sqrt{k})$$O(1/k) to O(1 / k) in general, and when the sum is strongly-convex the convergence rate is improved from the sub-linear O(1 / k) to a linear convergence rate of the form $$O(\rho ^k)$$O(ρk) for $$\rho < 1$$ρ<1. Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. This extends our earlier work Le Roux et al. (Adv Neural Inf Process Syst, 2012), which only lead to a faster rate for well-conditioned strongly-convex problems. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.
16577977
Simplified neuron model as a principal component analyzer
A simple linear neuron model with constrained Hebbian-type synaptic modification is analyzed and a new class of unconstrained learning rules is derived. It is shown that the model neuron tends to extract the principal component from a stationary input vector sequence.
122159626
On stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix
Abstract In applications of signal processing and pattern recognition, eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed. Iterative methods are suggested and analyzed, in which no sample moments are used. Convergence is shown by stochastic approximation theory.
53236615
Low rank approximation and regression in input sparsity time
We design a new distribution over poly(r ε<sup>-1</sup>) x n matrices S so that for any fixed n x d matrix A of rank r, with probability at least 9/10, SAx<sub>2</sub> = (1 pm ε)Ax<sub>2</sub> simultaneously for all x ∈ R<sup>d</sup>. Such a matrix S is called a <i>subspace embedding</i>. Furthermore, SA can be computed in O(nnz(A)) + ~O(r<sup>2</sup>ε<sup>-2</sup>) time, where nnz(A) is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at least Ω(nd log d) time to achieve this property. We call our matrices S <i>sparse embedding matrices</i>.
 Using our sparse embedding matrices, we obtain the fastest known algorithms for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and l<sub>p</sub>-regression: to output an x' for which Ax'-b<sub>2</sub> ≤ (1+ε)min<sub>x</sub> Ax-b<sub>2</sub> for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A)) + ~O(d<sup>3</sup>ε<sup>-2</sup>) time, and another in O(nnz(A)log(1/ε)) + ~O(d<sup>3</sup>log(1/ε)) time. (Here ~O(f) = f ⋅ log<sup>O(1)</sup>(f).) to obtain a decomposition of an n x n matrix A into a product of an n x k matrix L, a k x k diagonal matrix D, and a n x k matrix W, for which F{A - L D W} ≤ (1+ε)F{A-A<sub>k</sub>}, where A<sub>k</sub> is the best rank-k approximation, our algorithm runs in O(nnz(A)) + ~O(nk<sup>2</sup> ε<sup>-4</sup>log n + k<sup>3</sup>ε<sup>-5</sup>log<sup>2</sup>n) time. to output an approximation to all leverage scores of an n x d input matrix A simultaneously, with constant relative error, our algorithms run in O(nnz(A) log n) + ~O(r<sup>3</sup>) time. to output an x' for which Ax'-b<sub>p</sub> ≤ (1+ε)min<sub>x</sub> Ax-b<sub>p</sub> for an n x d matrix A and an n x 1 column vector b, we obtain an algorithm running in O(nnz(A) log n) + poly(r ε<sup>-1</sup>) time, for any constant 1 ≤ p < ∞. We optimize the polynomial factors in the above stated running times, and show various tradeoffs. Finally, we provide preliminary experimental results which suggest that our algorithms are of interest in practice.
14426203
Even Faster SVD Decomposition Yet Without Agonizing Pain
We study k-SVD that is to obtain the first k singular vectors of a matrix $A$ approximately. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [1] provided the first gap-free theorem for the block Krylov method, Shamir [2] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [3] provided the fastest $O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$-type of algorithm using alternating minimization. 
In this paper, put forward a new framework for SVD and improve the above breakthroughs. We obtain faster gap-free convergence rate outperforming [1], we obtain the first accelerated AND stochastic method outperforming [2]. In the $O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$ running-time regime, we outperform [3] in certain parameter regimes without even using alternating minimization.
161623
Faster Eigenvector Computation via Shift-and-Invert Preconditioning
We give faster algorithms and improved sample complexities for estimating the top eigenvector of a matrix $\Sigma$ -- i.e. computing a unit vector $x$ such that $x^T \Sigma x \ge (1-\epsilon)\lambda_1(\Sigma)$: 
Offline Eigenvector Estimation: Given an explicit $A \in \mathbb{R}^{n \times d}$ with $\Sigma = A^TA$, we show how to compute an $\epsilon$ approximate top eigenvector in time $\tilde O([nnz(A) + \frac{d*sr(A)}{gap^2} ]* \log 1/\epsilon )$ and $\tilde O([\frac{nnz(A)^{3/4} (d*sr(A))^{1/4}}{\sqrt{gap}} ] * \log 1/\epsilon )$. Here $nnz(A)$ is the number of nonzeros in $A$, $sr(A)$ is the stable rank, $gap$ is the relative eigengap. By separating the $gap$ dependence from the $nnz(A)$ term, our first runtime improves upon the classical power and Lanczos methods. It also improves prior work using fast subspace embeddings [AC09, CW13] and stochastic optimization [Sha15c], giving significantly better dependencies on $sr(A)$ and $\epsilon$. Our second running time improves these further when $nnz(A) \le \frac{d*sr(A)}{gap^2}$. 
Online Eigenvector Estimation: Given a distribution $D$ with covariance matrix $\Sigma$ and a vector $x_0$ which is an $O(gap)$ approximate top eigenvector for $\Sigma$, we show how to refine to an $\epsilon$ approximation using $ O(\frac{var(D)}{gap*\epsilon})$ samples from $D$. Here $var(D)$ is a natural notion of variance. Combining our algorithm with previous work to initialize $x_0$, we obtain improved sample complexity and runtime results under a variety of assumptions on $D$. 
We achieve our results using a general framework that we believe is of independent interest. We give a robust analysis of the classic method of shift-and-invert preconditioning to reduce eigenvector computation to approximately solving a sequence of linear systems. We then apply fast stochastic variance reduced gradient (SVRG) based system solvers to achieve our claims.
14350287
Multiplying matrices faster than coppersmith-winograd
We develop an automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction. Using this approach we obtain a new improved bound on the matrix multiplication exponent ω<2.3727.
124717216
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
A handbook of mathematical functions that is designed to provide scientific investigations with a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems.
27648311
Leverage Score Sampling for Faster Accelerated Regression and ERM
Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a vector $b \in\mathbb{R}^{d}$, we show how to compute an $\epsilon$-approximate solution to the regression problem $ \min_{x\in\mathbb{R}^{d}}\frac{1}{2} \|\mathbf{A} x - b\|_{2}^{2} $ in time $ \tilde{O} ((n+\sqrt{d\cdot\kappa_{\text{sum}}})\cdot s\cdot\log\epsilon^{-1}) $ where $\kappa_{\text{sum}}=\mathrm{tr}\left(\mathbf{A}^{\top}\mathbf{A}\right)/\lambda_{\min}(\mathbf{A}^{T}\mathbf{A})$ and $s$ is the maximum number of non-zero entries in a row of $\mathbf{A}$. Our algorithm improves upon the previous best running time of $ \tilde{O} ((n+\sqrt{n \cdot\kappa_{\text{sum}}})\cdot s\cdot\log\epsilon^{-1})$. 
We achieve our result through a careful combination of leverage score sampling techniques, proximal point methods, and accelerated coordinate descent. Our method not only matches the performance of previous methods, but further improves whenever leverage scores of rows are small (up to polylogarithmic factors). We also provide a non-linear generalization of these results that improves the running time for solving a broader class of ERM problems.
51918825
Katyusha: the first direct acceleration of stochastic gradient methods
Nesterov's momentum trick is famously known for accelerating gradient descent, and has been proven useful in building fast iterative algorithms. However, in the stochastic setting, counterexamples exist and prevent Nesterov's momentum from providing similar acceleration, even if the underlying problem is convex. We introduce Katyusha, a direct, primal-only stochastic gradient method to fix this issue. It has a provably accelerated convergence rate in convex (off-line) stochastic optimization. The main ingredient is Katyusha momentum, a novel "negative momentum" on top of Nesterov's momentum that can be incorporated into a variance-reduction based algorithm and speed it up. Since variance reduction has been successfully applied to a growing list of practical problems, our paper suggests that in each of such cases, one could potentially give Katyusha a hug.
27655166
Efficiency of the Accelerated Coordinate Descent Method on Structured Optimization Problems
In this paper we prove a new complexity bound for a variant of Accelerated Coordinate Descent Method [7]. We show that this method often outperforms the standard Fast Gradient Methods (FGM, [3, 6]) on optimization problems with dense data. In many important situations, the computational expenses of oracle and method itself at each iteration of our scheme are perfectly balanced (both depend linearly on dimensions of the problem). As application examples, we consider unconstrained convex quadratic minimization, and the problems arising in Smoothing Technique [6]. On some special problem instances, the provable acceleration factor with respect to FGM can reach the square root of the number of variables. Our theoretical conclusions are confirmed by numerical experiments.
18314386
A geometric alternative to Nesterov's accelerated gradient descent
We propose a new method for unconstrained optimization of a smooth and strongly convex function, which attains the optimal rate of convergence of Nesterov’s accelerated gradient descent. The new algorithm has a simple geometric interpretation, loosely inspired by the ellipsoid method. We provide some numerical evidence that the new method can be superior to Nesterov’s accelerated gradient descent.
6023169
Accelerated proximal stochastic dual coordinate ascent for regularized loss minimization
We introduce a proximal version of the stochastic dual coordinate ascent method and show how to accelerate the method using an inner-outer iteration procedure. We analyze the runtime of the framework and obtain rates that improve state-of-the-art results for various key machine learning optimization problems including SVM, logistic regression, ridge regression, Lasso, and multiclass SVM. Experiments validate our theoretical findings.
62646318
NIST Handbook of Mathematical Functions
Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.
125291746
Lectures on Convex Optimization
Trying to find professional reading resources? We have lectures on convex optimization to read, not just read, however likewise download them and even read online. Locate this great book writtern by by now, just below, yeah just below. Obtain the data in the types of txt, zip, kindle, word, ppt, pdf, as well as rar. Once again, never miss out on to read online and also download this publication in our site here. Click the link. Are you looking to uncover lectures on convex optimization Digitalbook. Correct here it is possible to locate as well as download lectures on convex optimization Book. We've got ebooks for every single topic lectures on convex optimization accessible for download cost-free. Search the site also as find Jean Campbell eBook in layout. We also have a fantastic collection of information connected to this Digitalbook for you. As well because the best part is you could assessment as well as download for lectures on convex optimization eBook Our goal is always to offer you an assortment of cost-free ebooks too as aid resolve your troubles. We have got a considerable collection of totally free of expense Book for people from every single stroll of life. We have got tried our finest to gather a sizable library of preferred cost-free as well as paid files. GO TO THE TECHNICAL WRITING FOR AN EXPANDED TYPE OF THIS LECTURES ON CONVEX OPTIMIZATION, ALONG WITH A CORRECTLY FORMATTED VERSION OF THE INSTANCE MANUAL PAGE ABOVE.