On the Sample Complexity of Learning Sum-Product Networks

TL;DR

This paper studies the sample complexity of PAC-learning sum-product networks (SPNs), showing that tree-structured Gaussian and discrete SPNs can be learned efficiently with a number of samples proportional to their parameters.

Contribution

It provides the first theoretical bounds on the sample complexity of learning SPNs, demonstrating linear growth with the number of parameters for tree-structured models.

Findings

Sample complexity grows at most linearly with parameters.

Tree-structured Gaussian SPNs can be learned with $ ilde{O}(rac{ed^2+k}{ ext{epsilon}^2})$ samples.

Results extend to discrete leaves using similar bounds.

Abstract

Sum-Product Networks (SPNs) can be regarded as a form of deep graphical models that compactly represent deeply factored and mixed distributions. An SPN is a rooted directed acyclic graph (DAG) consisting of a set of leaves (corresponding to base distributions), a set of sum nodes (which represent mixtures of their children distributions) and a set of product nodes (representing the products of its children distributions). In this work, we initiate the study of the sample complexity of PAC-learning the set of distributions that correspond to SPNs. We show that the sample complexity of learning tree structured SPNs with the usual type of leaves (i.e., Gaussian or discrete) grows at most linearly (up to logarithmic factors) with the number of parameters of the SPN. More specifically, we show that the class of distributions that corresponds to tree structured Gaussian SPNs with $k$ mixing weights and $e$ ($d$-dimensional Gaussian) leaves can be learned within Total Variation error $\epsilon$ using at most $\widetilde{O}(\frac{ed^2+k}{\epsilon^2})$ samples. A similar result holds for tree structured SPNs with discrete leaves. We obtain the upper bounds based on the recently proposed notion of distribution compression schemes. More specifically, we show that if a (base) class of distributions $\mathcal{F}$ admits an "efficient" compression, then the class of tree structured SPNs with leaves from $\mathcal{F}$ also admits an efficient compression.