Incremental proximal methods for large scale convex optimization

Bertsekas, Dimitri P.

doi:10.1007/s10107-011-0472-0

Cited by 282 publications

(326 citation statements)

References 53 publications

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“…Using the same principle, a distributed algorithm involving an additional consensus step has been proposed in [27]. Random iterations involving proximal and subgradient operators were considered in [28] and in [29]. In [29], the functions g(ξ, . )…”

Section: Related Workmentioning

confidence: 99%

Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators

Bianchi¹,

Hachem²

2016

J Optim Theory Appl

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The purpose of this paper is to study the dynamical behavior of the sequence produced by a forward-backward algorithm, involving two random maximal monotone operators and a sequence of decreasing step sizes. Defining a mean monotone operator as an Aumann integral, and assuming that the sum of the two mean operators is maximal (sufficient maximality conditions are provided), it is shown that with probability one, the interpolated process obtained from the iterates is an asymptotic pseudo trajectory in the sense of Benaïm and Hirsch of the differential inclusion involving the sum of the mean operators. The convergence of the empirical means of the iterates towards a zero of the sum of the mean operators is shown, as well as the convergence of the sequence itself to such a zero under a demipositivity assumption. These results find applications in a wide range of optimization problems or variational inequalities in random environments.

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Section: Related Workmentioning

confidence: 99%

Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators

Bianchi¹,

Hachem²

2016

J Optim Theory Appl

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“…Another closely related work is Bertsekas [Ber11]. It proposed an algorithmic framework that alternates incrementally between subgradient and proximal iterations for minimizing a cost function f = m i=1 f i , the sum of a large but finite number of convex components f i , over a constraint set X.…”

Section: • Sampling Schemes For Subgradients G(· V K ) or Component mentioning

confidence: 99%

“…This line of analysis is shared with incremental subgradient and proximal methods (see [NB00], [NB01], [Ber11]). However, here the technical details are more intricate because there are two types of iterations, which involve the two different stepsizes α k and β k .…”

Section: Assumptions and Preliminariesmentioning

confidence: 99%

“…For this algorithm, convergence rates and finite-sample error bounds have been derived in [WB12]. For minimization of general convex functions, convergence rate analysis is not available in the existing literature, except in special cases which involve no constraint sampling and/or more restrictive assumptions (see [NB00], [NB01], [Ber11], [Ned11], [WB12]). …”

Section: By Using Lemma 6(c) We Further Obtainmentioning

confidence: 99%

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Stochastic First-Order Methods with Random Constraint Projection

Wang

Bertsekas

2016

SIAM J. Optim.

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Abstract. We consider convex optimization problems with structures that are suitable for sequential treatment or online sampling. In particular, we focus on problems where the objective function is an expected value, and the constraint set is the intersection of a large number of simpler sets. We propose an algorithmic framework for stochastic first-order methods using random projection/proximal updates and random constraint updates, which contain as special cases several known algorithms as well as many new algorithms. To analyze the convergence of these algorithms in a unified manner, we prove a general coupled convergence theorem. It states that the convergence is obtained from an interplay between two coupled processes: progress toward feasibility and progress toward optimality. Under suitable stepsize assumptions, we show that the optimality error decreases at a rate of O(1/ √ k) and the feasibility error decreases at a rate of O(log k/k). We also consider a number of typical sampling processes for generating stochastic first-order information and random constraints, which are common in data-intensive applications, online learning, and simulation optimization. By using the coupled convergence theorem as a modular architecture, we are able to analyze the convergence of stochastic algorithms that use arbitrary combinations of these sampling processes.

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“…We emphasize that these assumptions hold for a variety of cost functions including regularized squared error loss, hinge loss, and logistic loss [6] and similar assumptions are widely used to analyze the convergence properties of incremental gradient methods in the literature [2,4,7,12,23]. Note that in contrast with many of these analyses, we do not assume that the component functions f i are convex.…”

Section: Assumption 32 (Strong Convexity)mentioning

confidence: 99%

Global convergence rate of incremental aggregated gradient methods for nonsmooth problems

Vanli

Gürbüzbalaban

Ozdaglar

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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Abstract. We focus on the problem of minimizing the sum of smooth component functions (where the sum is strongly convex) and a non-smooth convex function, which arises in regularized empirical risk minimization in machine learning and distributed constrained optimization in wireless sensor networks and smart grids. We consider solving this problem using the proximal incremental aggregated gradient (PIAG) method, which at each iteration moves along an aggregated gradient (formed by incrementally updating gradients of component functions according to a deterministic order) and taking a proximal step with respect to the non-smooth function. While the convergence properties of this method with randomized orders (in updating gradients of component functions) have been investigated, this paper, to the best of our knowledge, is the first study that establishes the convergence rate properties of the PIAG method for any deterministic order. In particular, we show that the PIAG algorithm is globally convergent with a linear rate provided that the step size is sufficiently small. We explicitly identify the rate of convergence and the corresponding step size to achieve this convergence rate. Our results improve upon the best known condition number dependence of the convergence rate of the incremental aggregated gradient methods used for minimizing a sum of smooth functions.

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Incremental proximal methods for large scale convex optimization

Cited by 282 publications

References 53 publications

Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators

Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators

Stochastic First-Order Methods with Random Constraint Projection

Global convergence rate of incremental aggregated gradient methods for nonsmooth problems

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