Maximum A Posteriori Inference in Sum-Product Networks

TL;DR

This paper explores the computational complexity of MAP inference in sum-product networks, proving NP-hardness, and introduces both exact and approximate algorithms that outperform existing methods in speed and accuracy.

Contribution

It provides the first reduction of general MAP inference to the case without evidence and hidden variables, and presents new algorithms with practical efficiency and improved performance.

Findings

Exact MAP solver handles up to 1k variables and 150k arcs.

Approximate solver achieves better speed-accuracy trade-offs.

Experimental results show superior performance over existing methods.

Abstract

Sum-product networks (SPNs) are a class of probabilistic graphical models that allow tractable marginal inference. However, the maximum a posteriori (MAP) inference in SPNs is NP-hard. We investigate MAP inference in SPNs from both theoretical and algorithmic perspectives. For the theoretical part, we reduce general MAP inference to its special case without evidence and hidden variables; we also show that it is NP-hard to approximate the MAP problem to $2^{n^\epsilon}$ for fixed $0 \leq \epsilon < 1$, where $n$ is the input size. For the algorithmic part, we first present an exact MAP solver that runs reasonably fast and could handle SPNs with up to 1k variables and 150k arcs in our experiments. We then present a new approximate MAP solver with a good balance between speed and accuracy, and our comprehensive experiments on real-world datasets show that it has better overall performance than existing approximate solvers.