Download or read book A Survey of Algorithms and Analysis for Stochastic Gradient Methods written by and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Statistical Analysis of Stochastic Gradient Based Algorithms written by Neil J. Bershad and published by . This book was released on 1991 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: The work explicitly reported here was performed during the period April 15, 1988 - October 15, 1990. Work performed during the period April 15, 1986 - April 15, 1988 has been previously presented in the annual reports for those years. AFOSR has supported research work primarily on 1) the stochastic behavior of LMS and stochastic gradient related adaptive algorithms and 2) comparative performance analysis of LMS and Recursive Least Squares (RLS) in non-stationary environments. Preliminary efforts covering the Constant Modulus Adaptive (CMA) algorithms and Infinite Impulse Response (IIR) adaptive filters have resulted in several publications 14,17,25 but are not discussed below. Preprints or reprints of all referenced publications have been provided to AFOSR.
Download or read book Stochastic Gradients Methods for Statistical Inference written by Tianyang Li (Ph. D.) and published by . This book was released on 2019 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical inference, such as hypothesis testing and calculating a confidence interval, is an important tool for accessing uncertainty in machine learning and statistical problems. Stochastic gradient methods, such as stochastic gradient descent (SGD), have recently been successfully applied to point estimation in large scale machine learning problems. In this work, we present novel stochastic gradient methods for statistical inference in large scale machine learning problems. Unregularized M -estimation using SGD. Using SGD with a fixed step size, we demonstrate that the average of such SGD sequences can be used for statistical inference, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. From a practical perspective, our SGD-based inference procedure is a first order method, and is well-suited for large scale problems. To show its merits, we apply it to both synthetic and real datasets, and demonstrate that its accuracy is comparable to classical statistical methods, while requiring potentially far less computation. Approximate Newton-based statistical inference using only stochastic gradients for unregularized M -estimation. We present a novel inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. In theory, our method efficiently computes the statistical error covariance in M -estimation for unregularized convex learning problems, without using exact second order information, or resampling the entire data set. In practice, we demonstrate the effectiveness of our framework on large-scale machine learning problems, that go even beyond convexity: as a highlight, our work can be used to detect certain adversarial attacks on neural networks. High dimensional linear regression statistical inference using only stochastic gra- dients. As an extension of the approximate Newton-based statistical inference algorithm for unregularized problems, we present a similar algorithm, using only stochastic gradients, for statistical inference in high dimensional linear regression, where the number of features is much larger than the number of samples. Stochastic gradient methods for time series analysis. We present a novel stochastic gradient descent algorithm for time series analysis, which correctly captures correlation structures in a time series dataset during optimization. Instead of uniformly sampling indices in vanilla SGD, we uniformly sample contiguous blocks of indices, where the block length depends on the dataset
Download or read book Introductory Lectures on Convex Optimization written by Y. Nesterov and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: It was in the middle of the 1980s, when the seminal paper by Kar markar opened a new epoch in nonlinear optimization. The importance of this paper, containing a new polynomial-time algorithm for linear op timization problems, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre diction of its high efficiency was supported by excellent computational results. This unusual fact dramatically changed the style and direc tions of the research in nonlinear optimization. Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments. In a new rapidly develop ing field, which got the name "polynomial-time interior-point methods", such a justification was obligatory. Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19]. Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students. The idea was to create a course which would reflect the new developments in the field. Actually, this was a major challenge. At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students. The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12].
Download or read book Optimization for Machine Learning written by Suvrit Sra and published by MIT Press. This book was released on 2012 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: An up-to-date account of the interplay between optimization and machine learning, accessible to students and researchers in both communities. The interplay between optimization and machine learning is one of the most important developments in modern computational science. Optimization formulations and methods are proving to be vital in designing algorithms to extract essential knowledge from huge volumes of data. Machine learning, however, is not simply a consumer of optimization technology but a rapidly evolving field that is itself generating new optimization ideas. This book captures the state of the art of the interaction between optimization and machine learning in a way that is accessible to researchers in both fields. Optimization approaches have enjoyed prominence in machine learning because of their wide applicability and attractive theoretical properties. The increasing complexity, size, and variety of today's machine learning models call for the reassessment of existing assumptions. This book starts the process of reassessment. It describes the resurgence in novel contexts of established frameworks such as first-order methods, stochastic approximations, convex relaxations, interior-point methods, and proximal methods. It also devotes attention to newer themes such as regularized optimization, robust optimization, gradient and subgradient methods, splitting techniques, and second-order methods. Many of these techniques draw inspiration from other fields, including operations research, theoretical computer science, and subfields of optimization. The book will enrich the ongoing cross-fertilization between the machine learning community and these other fields, and within the broader optimization community.
Download or read book Machine Learning Refined written by Jeremy Watt and published by Cambridge University Press. This book was released on 2020-01-09 with total page 597 pages. Available in PDF, EPUB and Kindle. Book excerpt: An intuitive approach to machine learning covering key concepts, real-world applications, and practical Python coding exercises.
Download or read book Convex Optimization written by Sébastien Bubeck and published by Foundations and Trends (R) in Machine Learning. This book was released on 2015-11-12 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. It begins with the fundamental theory of black-box optimization and proceeds to guide the reader through recent advances in structural optimization and stochastic optimization. The presentation of black-box optimization, strongly influenced by the seminal book by Nesterov, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. Special attention is also given to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging), and discussing their relevance in machine learning. The text provides a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization it discusses stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. It also briefly touches upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.
Download or read book Optimization Algorithms for Machine Learning written by Anant Raj and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: With the advent of massive datasets and increasingly complex tasks, modern machine learning systems pose several new challenges in terms of scalability to high dimensional data as well as to large datasets. In this thesis, we consider to study scalable descent methods such as coordinate descent and stochastic coordinate descent which are based on the stochastic approximation of full gradient. In the first part of the thesis, we propose faster and scalable coordinate based opti- mization which scales to high dimensional problems. As a first step to achieve scalable coordinate based descent approaches, we propose a new framework to derive screening rules for convex optimization problems based on duality gap which covers a large class of constrained and penalized optimization formulations. In later stages, we develop new approximately greedy coordinate selection strategy in coordinate descent for large-scale optimization. This novel coordinate selection strategy provavbly works better than uni- formly random selection, and can reach the efficiency of steepest coordinate descent (SCD) in the best case. In best case scenario, this may enable an acceleration of a factor of up to n, the number of coordinates. Having similar objective in mind, we further propose an adaptive sampling strategy for sampling in stochastic gradient based optimization. The proposed safe sampling scheme provably achieves faster convergence than any fixed deterministic sampling schemes for coordinate descent and stochastic gradient descent methods. Exploiting the connection between matching pursuit where a more generalized notion of directions is considered and greedy coordinate descent where all the moving directions are orthogonal, we also propose a unified analysis for both the approaches and extend it to get the accelerated rate. In the second part of this thesis, we focus on providing provably faster and scalable mini batch stochastic gradient descent (SGD) algorithms. Variance reduced SGD methods converge significantly faster than the vanilla SGD counterpart. We propose a variance reduce algorithm k-SVRG that addresses issues of SVRG [98] and SAGA[54] by making best use of the available memory and minimizes the stalling phases without progress. In later part of the work, we provide a simple framework which utilizes the idea of optimistic update to obtain accelerated stochastic algorithms. We obtain accelerated variance reduced algorithm as well as accelerated universal algorithm as a direct consequence of this simple framework. Going further, we also employ the idea of local sensitivity based importance sampling in an iterative optimization method and analyze its convergence while optimizing over the selected subset. In the final part of the thesis, we connect the dots between coordinate descent method and stochastic gradient descent method in the interpolation regime. We show that better stochastic gradient based dual algorithms with fast rate of convergence can be obtained to optimize the convex objective in the interpolation regime.
Download or read book Beyond the Worst Case Analysis of Algorithms written by Tim Roughgarden and published by Cambridge University Press. This book was released on 2021-01-14 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces exciting new methods for assessing algorithms for problems ranging from clustering to linear programming to neural networks.
Download or read book Gradient based Optimization and Implicit Regularization Over Non convex Landscapes written by Xiaoxia (Shirley) Wu and published by . This book was released on 2020 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Large-scale machine learning problems can be reduced to non-convex optimization problems if state-of-the-art models such as deep neural networks are applied. One of the most widely-used algorithms is the first-order iterative gradient-based algorithm, i.e., (stochastic) gradient descent method. Two main challenges arise from understanding the gradient-based algorithm over the non-convex landscapes: the convergence complexity and the algorithm's solutions. This thesis aims to tackle the two challenges by providing a theoretical framework and empirical investigation on three popular gradient-based algorithms, namely, adaptive gradient methods [39], weight normalization [138] and curriculum learning [18]. For convergence, the stepsize or learning rate plays a pivotal role in the iteration complexity. However, it depends crucially on the (generally unknown) Lipschitz smoothness constant and noise level on the stochastic gradient. A popular stepsize auto-tuning method is the adaptive gradient methods such as AdaGrad that update the learning rate on the fly according to the gradients received along the way; Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, non-convex functions; we show that it converges to a stationary point at the (log(N)/ √N) rate in the stochastic setting and at the optimal (1/N) rate in the batch (non-stochastic) setting. Extensive numerical experiments are provided to corroborate our theory. For the gradient-based algorithm solution, we study weight normalization (WN) methods in the setting of an over-parameterized linear regression problem where WN decouples the weight vector with a scale and a unit vector. We show that this reparametrization has beneficial regularization effects compared to gradient descent on the original objective. WN adaptively regularizes the weights and converges close to the minimum l2 norm solution, even for initializations far from zero. To further understand the stochastic gradient-based algorithm, we study the continuation method -- curriculum learning (CL) -- inspired by cognitive science that humans learn from simple to complex order. CL has proposed ordering examples during training based on their difficulty, while anti-CL proposed the opposite ordering. Both CL and anti-CL have been suggested as improvements to the standard i.i.d. training. We set out to investigate the relative benefits of ordered learning in three settings: standard-time, short-time, and noisy label training. We find that both orders have only marginal benefits for standard benchmark datasets. However, with limited training time budget or noisy data, curriculum, but not anti-curriculum ordering, can improve the performance
Download or read book Langevin Algorithms in Data Science written by Xiaoyu Wang and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Langevin algorithms are a collection of powerful optimization algorithms and Markov Chain Monte Carlo sampling algorithms that provide computational foundations for high dimensional data in machine learning. We first review variants of Langevin diffusions and the corresponding algorithms based on different discretization schemes for overdamped and non-reversible Langevin algorithms including underdamped Langevin algorithms. Meanwhile, we revisit Stochastic Gradient Descent (SGD) on saddle points and illustrate the necessity of Stochastic Gradient Langevin Dynamics (SGLD) driven by Brownian motion for global convergence. Then, we propose a more general Langevin-like dynamics and algorithms driven by L\'evy motion. As a sampling toolkit, we survey Langevin sampling algorithms for both logconcave and non-logconcave target distributions. As an optimization method, we review the non-asymptotic analysis on two very commonly used Langevin algorithms, Stochastic Gradient Langevin Dynamics (SGLD) and Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), and we then present the acceleration for global convergence by breaking reversibility of the Langevin diffusions. As a follow up work, we do a non-asymptotic analysis of the global convergence of the nonreversible Langevin algorithm for non-convex optimization with explicit constants, our result leads to non-asymptotic guarantees for both empirical and population risk minimization problems. Moreover, it suggests that by choosing an appropriate anti-symmetric matrix, the algorithm can outperform SGLD. As an application, we briefly introduce a Langevin algorithm to model gradient noise when using SGD optimizes the loss function in training neural networks with Gaussian noise injections. Then, we review the explicit effect of Gaussian noise data and additionally show the heavy tails and skewness of gradient noise are brought by the implicit effect which was marginalized out when studying the explicit effect. Finally, we quantitatively study the implicit bias induced by Gaussian noise injections for training neural networks by analyzing the weak error bound in terms of the heavy tails index and skewness parameter using carefully constructed auxiliary stochastic differential equations and designing an approximation scheme.
Download or read book Understanding Machine Learning written by Shai Shalev-Shwartz and published by Cambridge University Press. This book was released on 2014-05-19 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage.
Download or read book Stochastic Models Statistical Methods and Algorithms in Image Analysis written by Piero Barone and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume comprises a collection of papers by world- renowned experts on image analysis. The papers range from survey articles to research papers, and from theoretical topics such as simulated annealing through to applied image reconstruction. It covers applications as diverse as biomedicine, astronomy, and geophysics. As a result, any researcher working on image analysis will find this book provides an up-to-date overview of the field and in addition, the extensive bibliographies will make this a useful reference.
Download or read book Handbook of Big Data written by Peter Bühlmann and published by CRC Press. This book was released on 2016-02-22 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Big Data provides a state-of-the-art overview of the analysis of large-scale datasets. Featuring contributions from well-known experts in statistics and computer science, this handbook presents a carefully curated collection of techniques from both industry and academia. Thus, the text instills a working understanding of key statistical
Download or read book MODIFIED STOCHASTIC VARIANCE REDUCTION GRADIENT DESCENT ALGORITHM AND ITS APPLICATION written by Cai Fei and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: While machine learning is becoming an indispensable element in our modern society, various algorithms are developed to help decision makers solve complicated problems. A major theme of this study is to review and analyze popular algorithms with a focus on Stochastic Gradient (SG) based methods in large-scale machine learning problems. While SG has been the fundamental method playing an essential role in optimization problems, the algorithm has been further modified by various researchers for improved performances. Stochastic Gradient Descent with Variance Reduction (SVRG) is a method known for its low computation cost and fast convergence rate in solving convex optimization problems. However, in nonconvex settings, the existence of saddle points negatively influences the performance of the algorithm. While the practical problems in the real-world majorly lie in nonconvex settings, to further improve the performance of SVRG, a new algorithm is designed and discussed in this study. The new algorithm combines traditional SVRG with two additional features introduced by Perturbed Accelerated Gradient Descent (Perturbed AGD) to expedite algorithm from escaping from saddle points, which ultimately leads to convergence in nonconvex optimization. This study focuses on the elaboration of the modified SVRG algorithm and its implementation with a synthetic and an empirical dataset.
Download or read book Parallel and Distributed Computation Numerical Methods written by Dimitri Bertsekas and published by Athena Scientific. This book was released on 2015-03-01 with total page 832 pages. Available in PDF, EPUB and Kindle. Book excerpt: This highly acclaimed work, first published by Prentice Hall in 1989, is a comprehensive and theoretically sound treatment of parallel and distributed numerical methods. It focuses on algorithms that are naturally suited for massive parallelization, and it explores the fundamental convergence, rate of convergence, communication, and synchronization issues associated with such algorithms. This is an extensive book, which aside from its focus on parallel and distributed algorithms, contains a wealth of material on a broad variety of computation and optimization topics. It is an excellent supplement to several of our other books, including Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 1999), Dynamic Programming and Optimal Control (Athena Scientific, 2012), Neuro-Dynamic Programming (Athena Scientific, 1996), and Network Optimization (Athena Scientific, 1998). The on-line edition of the book contains a 95-page solutions manual.
Download or read book Stochastic Gradient Descent for Modern Machine Learning written by Rahul Kidambi and published by . This book was released on 2019 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tremendous advances in large scale machine learning and deep learning have been powered by the seemingly simple and lightweight stochastic gradient method. Variants of the stochastic gradient method (based on iterate averaging) are known to be asymptotically optimal (in terms of predictive performance). This thesis examines non-asymptotic issues surrounding the use of stochastic gradient descent (SGD) in practice with an aim to achieve its asymptotically optimal statistical properties. Focusing on the stochastic approximation problem of least squares regression, this thesis considers: 1. Understanding the benefits of tail-averaged SGD, and understanding how SGD's non-asymptotic behavior is influenced when faced with mis-specified problem instances. 2. Understand the parallelization properties of SGD, with a specific focus on mini-batching, model averaging and batch size doubling. Can this characterization shed light on algorithmic regimes (for e.g. largest instance dependent batch sizes) that admit linear parallelization speedups over vanilla SGD (with a batch size 1), thus presenting useful prescriptions that make best use of our hardware resources whilst not being wasteful of computation? As a byproduct of these results, can we understand how the learning rate behaves as a function of the batch size? 3. Similar to how momentum/acceleration schemes such as heavy ball momentum, or Nesterov's acceleration improve over standard batch gradient descent, can we formalize improvements achieved by accelerated methods when working with sampled stochastic gradients? Is there an algorithm that achieves this improvement over SGD? How does deterministic accelerated schemes such as heavy ball momentum, or say, Nesterov's acceleration work when used with sampled stochastic gradients? 4. This thesis considers the behavior of the final iterate of SGD (as opposed to a majority of efforts in the stochastic approximation literature which focus on iterate averaging) with varying stepsize schemes, including the standard polynomially decaying stepsizes and the practically preferred step decay scheme, with an aim to achieve minimax rates. The overarching goal of this section is to understand the behavior of SGD's final iterate owing to its widespread use in practical implementations for machine learning applications. Alongside the theory results that focus on the least squares regression, this thesis examines the general applicability of various results (in a qualitative sense) towards the problem of training multi-layer deep neural networks on benchmark datasets, and presents several useful implications when training deep learning models of practical interest.
