Learning a Tree-Structured Ising Model in Order to Make Predictions

TL;DR

This paper introduces a method for learning tree-structured Ising models focused on prediction accuracy rather than exact graph recovery, demonstrating that fewer samples are needed for accurate predictions than for perfect graph reconstruction.

Contribution

The paper proposes a new prediction-focused learning approach for tree Ising models using a novel ssTV distance, with theoretical bounds showing fewer samples suffice for accurate predictions.

Findings

Fewer samples are needed for accurate predictions than for exact graph recovery.

The ssTV distance effectively measures prediction accuracy in learned models.

The approach achieves non-asymptotic bounds on sample complexity for prediction tasks.

Abstract

We study the problem of learning a tree Ising model from samples such that subsequent predictions made using the model are accurate. The prediction task considered in this paper is that of predicting the values of a subset of variables given values of some other subset of variables. Virtually all previous work on graphical model learning has focused on recovering the true underlying graph. We define a distance ("small set TV" or ssTV) between distributions $P$ and $Q$ by taking the maximum, over all subsets $\mathcal{S}$ of a given size, of the total variation between the marginals of $P$ and $Q$ on $\mathcal{S}$; this distance captures the accuracy of the prediction task of interest. We derive non-asymptotic bounds on the number of samples needed to get a distribution (from the same class) with small ssTV relative to the one generating the samples. One of the main messages of this paper is that far fewer samples are needed than for recovering the underlying tree, which means that accurate predictions are possible using the wrong tree.