Sparse estimation in Ising Model via penalized Monte Carlo methods

Stochastic Learning for Sparse Discrete Markov Random Fields with Controlled Gradient Approximation Error

“…The sizes of the BPMNs generated in this paper are comparable to those in [Honorio, 2012a, Atchade et al, 2014, Miasojedow and Rejchel, 2016.…”

“…To efficiently solve (1), many efforts have been made in combining Gibbs sampling [Levin et al, 2009] and proximal gradient descent [Parikh et al, 2014] into SPG, a method that adopts the proximal gradient framework to update iterates, but uses Gibbs sampling as a stochastic oracle to approximate the gradient when the gradient information is needed [Honorio, 2012a, Atchade et al, 2014, Miasojedow and Rejchel, 2016. Specifically, Gibbs sampling with q chains running τ steps (Gibbs-τ ) can generate q samples for P θ (x).…”

“…In this paper, we consider stochastic proximal gradient (SPG; Honorio 2012a, Atchade et al 2014, Miasojedow andRejchel 2016), a stochastic learning framework for l 1 -regularized discrete MRFs. SPG hinges on a stochastic oracle for gradient approximation of the log-likelihood function (inexact inference).…”

“…Since SPG uses Gibbs sampling for gradient approximation, it can be viewed as an inexact proximal gradient method [Schmidt et al, 2011], whose success depends on whether the gradient approximation error can be effectively controlled. While previous works [Honorio, 2012a, Atchade et al, 2014, Miasojedow and Rejchel, 2016 have shown that the quality of the gradient approximation can be improved in the long run with increasingly demanding computational resources, such long term guarantees might not translate to satisfactory performance in practice (see Section 7). Therefore, it is desirable to estimate and control the gradient approximation error of SPG meticulously in each iteration so that a more refined approximation to the exact gradient will be rewarded with a higher gain of efficiency and accuracy in practice.…”

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We study the L 1 -regularized maximum likelihood estimator/estimation (MLE) problem for discrete Markov random fields (MRFs), where efficient and scalable learning requires both sparse regularization and approximate inference.To address these challenges, we consider a stochastic learning framework called stochastic proximal gradient (SPG; Honorio 2012a, Atchade et al 2014, Miasojedow andRejchel 2016). SPG is an inexact proximal gradient algorithm [Schmidt et al, 2011], whose inexactness stems from the stochastic oracle (Gibbs sampling) for gradient approximation -exact gradient evaluation is infeasible in general due to the NP-hard inference problem for discrete MRFs [Koller and Friedman, 2009].

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