arXiv subjects

Peter Richtarik

Publications and source records attributed to Peter Richtarik.

26 records · Page 2Linked to original sources

arXiv2018

SEGA: Variance Reduction via Gradient Sketching

We propose a randomized first order optimization method--SEGA (SkEtched GrAdient method)-- which progressively throughout its iterations builds a variance-reduced estimate of the gradient from random linear measurements (sketches) of the gradient obtained from an oracle. In each iteration, SEGA updates the current estimate of the gradient through a sketch-and-project operation using the information provided by the latest sketch, and this is subsequently used to compute an unbiased estimate of the true gradient through a random relaxation procedure. This unbiased estimate is then used to perform a gradient step. Unlike standard subspace descent methods, such as coordinate descent, SEGA can be used for optimization problems with a non-separable proximal term. We provide a general convergence analysis and prove linear convergence for strongly convex objectives. In the special case of coordinate sketches, SEGA can be enhanced with various techniques such as importance sampling, minibatching and acceleration, and its rate is up to a small constant factor identical to the best-known rate of coordinate descent.

math.OC↗

arXiv2018

Weighted Low-Rank Approximation of Matrices and Background Modeling

We primarily study a special a weighted low-rank approximation of matrices and then apply it to solve the background modeling problem. We propose two algorithms for this purpose: one operates in the batch mode on the entire data and the other one operates in the batch-incremental mode on the data and naturally captures more background variations and computationally more effective. Moreover, we propose a robust technique that learns the background frame indices from the data and does not require any training frames. We demonstrate through extensive experiments that by inserting a simple weight in the Frobenius norm, it can be made robust to the outliers similar to the $\ell_1$ norm. Our methods match or outperform several state-of-the-art online and batch background modeling methods in virtually all quantitative and qualitative measures.

cs.CV↗

arXiv2018

Randomized projection methods for convex feasibility problems: conditioning and convergence rates

Finding a point in the intersection of a collection of closed convex sets, that is the convex feasibility problem, represents the main modeling strategy for many computational problems. In this paper we analyze new stochastic reformulations of the convex feasibility problem in order to facilitate the development of new algorithmic schemes. We also analyze the conditioning problem parameters using certain (linear) regularity assumptions on the individual convex sets. Then, we introduce a general random projection algorithmic framework, which extends to the random settings many existing projection schemes, designed for the general convex feasibility problem. Our general random projection algorithm allows to project simultaneously on several sets, thus providing great flexibility in matching the implementation of the algorithm on the parallel architecture at hand. Based on the conditioning parameters, besides the asymptotic convergence results, we also derive explicit sublinear and linear convergence rates for this general algorithmic framework.

math.OC↗

arXiv2017

Online and Batch Supervised Background Estimation via L1 Regression

We propose a surprisingly simple model for supervised video background estimation. Our model is based on $\ell_1$ regression. As existing methods for $\ell_1$ regression do not scale to high-resolution videos, we propose several simple and scalable methods for solving the problem, including iteratively reweighted least squares, a homotopy method, and stochastic gradient descent. We show through extensive experiments that our model and methods match or outperform the state-of-the-art online and batch methods in virtually all quantitative and qualitative measures.

math.OC↗

arXiv2016

Matrix Completion under Interval Uncertainty

Matrix completion under interval uncertainty can be cast as matrix completion with element-wise box constraints. We present an efficient alternating-direction parallel coordinate-descent method for the problem. We show that the method outperforms any other known method on a benchmark in image in-painting in terms of signal-to-noise ratio, and that it provides high-quality solutions for an instance of collaborative filtering with 100,198,805 recommendations within 5 minutes.

math.OC↗

arXiv2016

Stochastic Dual Ascent for Solving Linear Systems

We develop a new randomized iterative algorithm---stochastic dual ascent (SDA)---for finding the projection of a given vector onto the solution space of a linear system. The method is dual in nature: with the dual being a non-strongly concave quadratic maximization problem without constraints. In each iteration of SDA, a dual variable is updated by a carefully chosen point in a subspace spanned by the columns of a random matrix drawn independently from a fixed distribution. The distribution plays the role of a parameter of the method. Our complexity results hold for a wide family of distributions of random matrices, which opens the possibility to fine-tune the stochasticity of the method to particular applications. We prove that primal iterates associated with the dual process converge to the projection exponentially fast in expectation, and give a formula and an insightful lower bound for the convergence rate. We also prove that the same rate applies to dual function values, primal function values and the duality gap. Unlike traditional iterative methods, SDA converges under no additional assumptions on the system (e.g., rank, diagonal dominance) beyond consistency. In fact, our lower bound improves as the rank of the system matrix drops. Many existing randomized methods for linear systems arise as special cases of SDA, including randomized Kaczmarz, randomized Newton, randomized coordinate descent, Gaussian descent, and their variants. In special cases where our method specializes to a known algorithm, we either recover the best known rates, or improve upon them. Finally, we show that the framework can be applied to the distributed average consensus problem to obtain an array of new algorithms. The randomized gossip algorithm arises as a special case.

math.NA↗

arXiv2015

Distributed Block Coordinate Descent for Minimizing Partially Separable Functions

In this work we propose a distributed randomized block coordinate descent method for minimizing a convex function with a huge number of variables/coordinates. We analyze its complexity under the assumption that the smooth part of the objective function is partially block separable, and show that the degree of separability directly influences the complexity. This extends the results in [Richtarik, Takac: Parallel coordinate descent methods for big data optimization] to a distributed environment. We first show that partially block separable functions admit an expected separable overapproximation (ESO) with respect to a distributed sampling, compute the ESO parameters, and then specialize complexity results from recent literature that hold under the generic ESO assumption. We describe several approaches to distribution and synchronization of the computation across a cluster of multi-core computers and provide promising computational results.

math.OC↗

arXiv2013

Separable Approximations and Decomposition Methods for the Augmented Lagrangian

In this paper we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey and Ruszczyński and the Parallel Coordinate Descent Method (PCDM) of Richtárik and Takáč. We show that the two methods are equivalent for feasibility problems up to the selection of a single step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least $8(L'/\bar{L})(ω-1)^2$ times better than the best known bound for DQAM, where $ω$ is the degree of partial separability and $L'$ and $\bar{L}$ are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian.

math.OC↗