High Dimensional Discrete Integration over the Hypergrid

TL;DR

This paper improves approximation techniques for high-dimensional discrete integration over hypergrids, reducing the approximation factor significantly compared to previous methods, and demonstrates practical effectiveness through experiments.

Contribution

It introduces a novel method for better approximation factors in hypergrid integration problems, extending prior work to larger domains with minimal additional computational burden.

Findings

Achieves an approximation factor of 4+O(1/q^2) for hypergrid integration.

Method can be implemented with inequality constraints or unconstrained optimization.

Experimental results support theoretical guarantees.

Abstract

Recently Ermon et al. (2013) pioneered a way to practically compute approximations to large scale counting or discrete integration problems by using random hashes. The hashes are used to reduce the counting problem into many separate discrete optimization problems. The optimization problems then can be solved by an NP-oracle such as commercial SAT solvers or integer linear programming (ILP) solvers. In particular, Ermon et al. showed that if the domain of integration is $\{0,1\}^n$ then it is possible to obtain a solution within a factor of $16$ of the optimal (a 16-approximation) by this technique. In many crucial counting tasks, such as computation of partition function of ferromagnetic Potts model, the domain of integration is naturally $\{0,1,\dots, q-1\}^n, q>2$, the hypergrid. The straightforward extension of Ermon et al.'s method allows a $q^2$-approximation for this problem. For large values of $q$, this is undesirable. In this paper, we show an improved technique to obtain an approximation factor of $4+O(1/q^2)$ to this problem. We are able to achieve this by using an idea of optimization over multiple bins of the hash functions, that can be easily implemented by inequality constraints, or even in unconstrained way. Also the burden on the NP-oracle is not increased by our method (an ILP solver can still be used). We provide experimental simulation results to support the theoretical guarantees of our algorithms.