Support Consistency of Direct Sparse-Change Learning in Markov Networks

TL;DR

This paper provides theoretical guarantees for detecting sparse structural changes between two Markov networks using direct density ratio estimation, with sample complexity bounds depending on network size and change sparsity.

Contribution

It establishes sufficient conditions and sample complexity bounds for successful change detection in Markov networks using direct ratio estimation methods.

Findings

Consistent change detection is achievable with sample size bounds depending on network sparsity and size.

Bounded density ratio models improve sample complexity requirements.

Theoretical guarantees apply to various discrete and continuous Markov networks.

Abstract

We study the problem of learning sparse structure changes between two Markov networks $P$ and $Q$. Rather than fitting two Markov networks separately to two sets of data and figuring out their differences, a recent work proposed to learn changes \emph{directly} via estimating the ratio between two Markov network models. In this paper, we give sufficient conditions for \emph{successful change detection} with respect to the sample size $n_p, n_q$, the dimension of data $m$, and the number of changed edges $d$. When using an unbounded density ratio model we prove that the true sparse changes can be consistently identified for $n_p = \Omega(d^2 \log \frac{m^2+m}{2})$ and $n_q = \Omega({n_p^2})$, with an exponentially decaying upper-bound on learning error. Such sample complexity can be improved to $\min(n_p, n_q) = \Omega(d^2 \log \frac{m^2+m}{2})$ when the boundedness of the density ratio model is assumed. Our theoretical guarantee can be applied to a wide range of discrete/continuous Markov networks.