Limits on Testing Structural Changes in Ising Models

TL;DR

This paper establishes fundamental information-theoretic limits on detecting sparse structural changes in Ising models, revealing that such detection is as sample-intensive as full model learning, especially for small changes.

Contribution

It provides the first theoretical bounds showing the difficulty of detecting sparse changes in Ising models, contrasting previous sparse recovery assumptions.

Findings

Sample complexity for change detection matches that of full model learning.

Detecting large changes is easier than detecting small, sparse changes.

Change detection in graphical models may require different approaches than structure learning.

Abstract

We present novel information-theoretic limits on detecting sparse changes in Ising models, a problem that arises in many applications where network changes can occur due to some external stimuli. We show that the sample complexity for detecting sparse changes, in a minimax sense, is no better than learning the entire model even in settings with local sparsity. This is a surprising fact in light of prior work rooted in sparse recovery methods, which suggest that sample complexity in this context scales only with the number of network changes. To shed light on when change detection is easier than structured learning, we consider testing of edge deletion in forest-structured graphs, and high-temperature ferromagnets as case studies. We show for these that testing of small changes is similarly hard, but testing of \emph{large} changes is well-separated from structure learning. These results imply that testing of graphical models may not be amenable to concepts such as restricted strong convexity leveraged for sparsity pattern recovery, and algorithm development instead should be directed towards detection of large changes.