Book•
Fundamentals of Convex Analysis
25 Sep 2001-
TL;DR: In this paper, the authors define and define Convex functions, Sublinear Functions and Sublinearity and Support Functions of a Nonempty Set Correspondence between ConveX Sets and SubLinear Functions, and Subdifferentials of Finite Functions.
Abstract: Introduction: Notation, Elementary Results.- Convex Sets: Generalities Convex Sets Attached to a Convex Set Projection onto Closed Convex Sets Separation and Applications Conical Approximations of Convex Sets.- Convex Functions: Basic Definitions and Examples Functional Operations Preserving Convexity Local and Global Behaviour of a Convex Function First- and Second-Order Differentiation.- Sublinearity and Support Functions: Sublinear Functions The Support Function of a Nonempty Set Correspondence Between Convex Sets and Sublinear Functions.- Subdifferentials of Finite Convex Functions: The Subdifferential: Definitions and Interpretations Local Properties of the Subdifferential First Examples Calculus Rules with Subdifferentials Further Examples The Subdifferential as a Multifunction.- Conjugacy in Convex Analysis: The Convex Conjugate of a Function Calculus Rules on the Conjugacy Operation Various Examples Differentiability of a Conjugate Function.
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23 May 2011TL;DR: It is argued that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas.
Abstract: Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for l1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.
17,433 citations
Cites background from "Fundamentals of Convex Analysis"
...The algorithm solves problems in the form minimize f(x) + g(z) subject to Ax+Bz = c (9) with variables x ∈ Rn and z ∈ Rm, where A ∈ Rp×n, B ∈ Rp×m, and c ∈ Rp....
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Book•
16 Dec 2008TL;DR: The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.
Abstract: The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.
4,335 citations
Cites methods from "Fundamentals of Convex Analysis"
...Rockafellar [198] is a standard reference on convex analysis; see also the books by Hiriart-Urruty and Lemaréchal [109, 110], Boyd and Vandenberghe [36], and Bertsekas [21]....
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Book•
12 Dec 2012TL;DR: In this article, the authors focus on regret analysis in the context of multi-armed bandit problems, where regret is defined as the balance between staying with the option that gave highest payoff in the past and exploring new options that might give higher payoffs in the future.
Abstract: A multi-armed bandit problem - or, simply, a bandit problem - is a sequential allocation problem defined by a set of actions. At each time step, a unit resource is allocated to an action and some observable payoff is obtained. The goal is to maximize the total payoff obtained in a sequence of allocations. The name bandit refers to the colloquial term for a slot machine (a "one-armed bandit" in American slang). In a casino, a sequential allocation problem is obtained when the player is facing many slot machines at once (a "multi-armed bandit"), and must repeatedly choose where to insert the next coin. Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the 1930s, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In this book, the focus is on two extreme cases in which the analysis of regret is particularly simple and elegant: independent and identically distributed payoffs and adversarial payoffs. Besides the basic setting of finitely many actions, it also analyzes some of the most important variants and extensions, such as the contextual bandit model. This monograph is an ideal reference for students and researchers with an interest in bandit problems.
2,427 citations
Posted Content•
Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.
2,095 citations
01 Jan 2011
TL;DR: The basic properties of proximity operators which are relevant to signal processing and optimization methods based on these operators are reviewed and proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework.
Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.
1,942 citations
Additional excerpts
...used [13, 87, 114]....
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