On the Sample Complexity of Learning Sum-Product Networks
This work addresses the theoretical sample efficiency for learning SPNs, which is incremental as it builds on existing compression schemes to derive new bounds.
The paper tackles the problem of determining the sample complexity required to learn Sum-Product Networks (SPNs), a type of deep graphical model, and shows that for tree-structured SPNs with Gaussian or discrete leaves, the sample complexity grows linearly (up to logarithmic factors) with the number of parameters, specifically requiring at most $\widetilde{O}( rac{ed^2+k}{\epsilon^2})$ samples for Gaussian SPNs.
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 $ε$ using at most $\widetilde{O}(\frac{ed^2+k}{ε^2})$ 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.