[Paper Review] Sparse estimation in Ising Model via penalized Monte Carlo methods

Błażej Miasojedow, Wojciech Rejchel|arXiv (Cornell University)|Dec 22, 2016

Markov Chains and Monte Carlo Methods25 references3 citations

TL;DR

This paper proposes a Lasso-penalized Monte Carlo maximum likelihood method for sparse structure learning in high-dimensional Ising models, addressing the intractable normalizing constant via importance sampling-based MCMC approximation. Under mild regularity conditions, the method consistently identifies the true graphical structure with high probability, even in high-dimensional settings with $ d \gg n $, as supported by theoretical guarantees and numerical validation.

ABSTRACT

We consider a problem of model selection in high-dimensional binary Markov random fields. The usefulness of the Ising model in studying systems of complex interactions has been confirmed in many papers. The main drawback of this model is the intractable norming constant that makes estimation of parameters very challenging. In the paper we propose a Lasso penalized version of the Monte Carlo maximum likelihood method. We prove that our algorithm, under mild regularity conditions, recognizes the true dependence structure of the graph with high probability. The efficiency of the proposed method is also investigated via simulation studies.

Motivation & Objective

• Address model selection in high-dimensional binary Markov random fields where the number of variables $ d $ is comparable to or exceeds the sample size $ n $.
• Overcome the intractable normalizing constant in the Ising model, which hinders standard likelihood-based estimation.
• Develop a computationally feasible method that combines Lasso penalization with Monte Carlo approximation to achieve sparse graphical model estimation.
• Establish theoretical consistency of the method in recovering the true graph structure under mild regularity conditions.

Proposed method

• Use a Lasso penalty on the log-likelihood to induce sparsity in the estimated Ising model parameters.
• Approximate the intractable normalizing constant using importance sampling within a Monte Carlo Markov Chain (MCMC) framework.
• Employ a weighted empirical average of sufficient statistics from MCMC samples to estimate the gradient and Hessian of the log-likelihood.
• Apply concentration inequalities to bound the estimation error of the MCMC approximation, ensuring convergence to the true likelihood under mild mixing conditions.
• Use a two-stage procedure: first estimate the MCMC approximation of the penalized likelihood, then apply a thresholding rule based on the Lasso penalty to identify the edge set.
• Leverage theoretical bounds on the $ l_\infty $-norm of the difference between the MCMC-approximated and true score functions to control estimation error.

Experimental results

Research questions

• RQ1 Can a penalized Monte Carlo maximum likelihood approach consistently recover the true graphical structure in high-dimensional Ising models?
• RQ2 Does the use of importance sampling in MCMC estimation effectively handle the intractable normalizing constant without relying on pseudolikelihood approximations?
• RQ3 Under what conditions does the Lasso-penalized MCMC estimator achieve model selection consistency in high-dimensional settings?
• RQ4 How does the method's performance compare to existing approaches like pseudolikelihood or Gaussian graphical model adaptations in terms of structure recovery accuracy?

Key findings

• The proposed method achieves model selection consistency: with high probability, it correctly identifies the true set of edges in the Ising graph under mild regularity conditions.
• Theoretical analysis shows that the MCMC approximation of the score function converges to the true score function in $ l_\infty $-norm, with high probability, provided the MCMC sample size $ m $ is sufficiently large.
• The method is robust to the complexity of the true graph structure, as it does not rely on pseudolikelihood assumptions that fail for complex graphs.
• The Lasso penalty ensures sparsity, and the estimator achieves exact support recovery: true edges are detected with high probability, while non-edges are correctly excluded.
• Numerical studies confirm the method's efficiency and consistency in recovering the true graph structure across various high-dimensional scenarios.