This work establishes a novel link between the problem of PAC-learning high-dimensional graphical models and the task of (efficient) counting and sampling of graph structures, using an online learning framework. The problem of efficiently counting and sampling graphical structures, such as spanning trees and acyclic orientations, has been a vibrant area of research in algorithms. We show that this rich algorithmic foundation can be leveraged to develop new algorithms for learning high-dimensional graphical models. We present the first efficient algorithm for (both realizable and agnostic) learning of Bayes nets with a chordal skeleton. In particular, we present an algorithm that, given integers $k,d > 0$, error parameter $\varepsilon > 0$, an undirected chordal graph $G$ on $n$ vertices, and sample access to a distribution $P^\ast$ on $[k]^n$; (1) returns a Bayes net $\widehat{P}$ with skeleton $G$ and indegree $d$, whose KL-divergence from $P^\ast$ is at most $\varepsilon$ more than the optimal KL-divergence between $P^\ast$ and any Bayes net with skeleton $G$ and indegree $d$, (2) uses $\widetilde{O}(n^3k^{d+1}/\varepsilon^2)$ samples from $P^\ast$ and runs in time $\mathrm{poly}(n,k,\varepsilon^{-1})$ for constant $d$. Prior results in this spirit were for only for trees ($d=1$, tree skeleton) via Chow-Liu, and in the realizable setting for polytrees (arbitrary $d$ but tree skeleton). Thus, our result significantly extends the state-of-the-art in learning Bayes net distributions. We also establish new results for learning tree and polytree distributions.

Humans have a powerful and mysterious capacity to reason. Working through a set of mental steps enables us to make inferences we would not be capable of making directly even though we get no additional data from the world. Similarly, when large language models generate intermediate steps (a chain of thought) before answering a question, they often produce better answers than they would directly. We investigate why and how chain-of-thought reasoning is useful in language models, testing the hypothesis that reasoning is effective when training data consists of overlapping local clusters of variables that influence each other strongly. These training conditions enable the chaining of accurate local inferences to estimate relationships between variables that were not seen together in training.

This paper studies the problem of entropy identity testing: given sample access to a distribution p and a fully described distribution q (both discrete distributions over a domain of size k), and the promise that either p = q or |H (p) H (q)| ε, where H () denotes the Shannon entropy, a tester needs to distinguish between the two cases with high probability.

Humans have a powerful and mysterious capacity to reason.

Neural Information Processing Systems http://nips.cc/

In this work, we study the fully observable discrete case where the structure of the network is given.

This paper studies the problem of entropy identity testing: given sample access to a distribution p and a fully described distribution q (both discrete distributions over a domain of size k), and the promise that either p = q or |H (p) H (q)| ε, where H () denotes the Shannon entropy, a tester needs to distinguish between the two cases with high probability.

We thank the reviewers for their thoughtful comments. An expander graph code allows simple, neurally plausible decoding to perform at par with BP . These expander codes can also be decoded by belief propagation (BP), but it's harder the other way around. We plan to follow this paper with another paper describing neuroscience applications. For space and coherence, this paper focuses on the conceptual theory without elaborating on applications.

It is often convenient to model high-dimensional datasets as probability distributions.