Efficiently learning Ising models on arbitrary graphs (1411.6156v2)

Abstract: We consider the problem of reconstructing the graph underlying an Ising model from i.i.d. samples. Over the last fifteen years this problem has been of significant interest in the statistics, machine learning, and statistical physics communities, and much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models. Nevertheless, for learning Ising models on general graphs with $p$ nodes of degree at most $d$, it is not known whether or not it is possible to improve upon the $p^{d}$ computation needed to exhaustively search over all possible neighborhoods for each node. In this paper we show that a simple greedy procedure allows to learn the structure of an Ising model on an arbitrary bounded-degree graph in time on the order of $p^2$. We make no assumptions on the parameters except what is necessary for identifiability of the model, and in particular the results hold at low-temperatures as well as for highly non-uniform models. The proof rests on a new structural property of Ising models: we show that for any node there exists at least one neighbor with which it has a high mutual information. This structural property may be of independent interest.

• The paper introduces a novel greedy algorithm for efficiently learning the graph structure of Ising models on arbitrary bounded-degree graphs, achieving a computational complexity of O(p^2 log p) which is faster than previous methods for general graphs.
• The algorithm leverages the high mutual information between a node and its neighbors, iteratively constructing and pruning pseudo-neighborhoods to correctly identify the true graph structure with high probability given sufficient data.
• This work implies that structure learning for high-dimensional models is feasible without strong assumptions like correlation decay, although the proposed method currently faces limitations in scalability concerning the graph degree 'd'.

Efficiently Learning Ising Models on Arbitrary Graphs

The paper under consideration introduces a novel algorithm for learning the graph structure of Ising models on arbitrary bounded-degree graphs with computational efficiency. This research is pivotal in the domain of statistical mechanics, machine learning, and statistical inference as it tackles the challenge of structure learning without making assumptions about parameter uniformity or high temperatures.

Overview of Ising Models

In the context of this paper, Ising models represent a class of undirected graphical models that capture pairwise interactions between binary variables situated on the nodes of a graph. These models are widely employed in various fields such as physics, biology, social networks, and more, to model systems that exhibit local interactions. The central problem addressed in this work is the reconstruction of an underlying graph structure from independent and identically distributed (i.i.d.) samples, a task that is computationally intensive for general graphs.

Main Contribution and Results

The core contribution of the paper is the development and analysis of a greedy algorithm that efficiently learns the structure of Ising models from data. Specifically, the proposed approach is capable of reconstructing graphs on $p$ nodes of maximum degree $d$ in time $O(p^2 \log p)$, an improvement over previous exhaustive methods requiring time on the order of $p^d$. Notably, this computational complexity parallels the complexity involved in learning tree-structured graphical models.

The algorithm exploits a structural property of Ising models, whereby each node in the graph maintains a high mutual information association with at least one of its neighbors. Using this insight, the algorithm iteratively builds a pseudo-neighborhood by selecting nodes based on their conditional influences, and then prunes the pseudo-neighborhood to identify actual neighbors. Despite its greedy nature, this simple procedure is rigorously shown to yield the correct graph structure with high probability when enough samples are available.

Implications and Future Directions

From a theoretical standpoint, the results imply that structure learning for high-dimensional statistical models is feasible without reliance on strong assumptions like correlation decay, which are necessary for efficient sampling algorithms. This progress highlights a divergence between the complexity of learning the model structure and the complexity of other statistical tasks like sampling and partition function estimation.

Practically, the insights from this paper suggest potential applications in scenarios where efficient inference of interaction structures from data is paramount, such as in biological networks or communication systems. However, this work does observe limitations in terms of scalability with respect to the node degree $d$, as the sample complexity and computational cost exhibit a doubly-exponential dependency on $d$.

Future work could address these limitations by refining the sample complexity to depend more optimally on $d$ or even adapting the approach to environments with unknown graph parameters or the presence of latent variables. Moreover, extending the methodology to accommodate larger alphabets or other classes of models could broaden the applicability of these results.

In summation, the paper provides a substantive leap forward in understanding the structure learning problem in complex statistical models, balancing computational efficiency with robust theoretical guarantees. While challenges remain, the framework and results lay a foundation for ongoing advancements in learning within various high-dimensional contexts.