Parameter Learning for Log-supermodular Distributions

TL;DR

This paper introduces a new parameter estimation method for log-supermodular models using stochastic subgradient techniques, improving bounds on the log-partition function and demonstrating effectiveness in image denoising tasks.

Contribution

It proposes a novel approach to parameter learning in log-supermodular models based on perturb-and-MAP bounds and stochastic optimization, advancing probabilistic modeling of combinatorial problems.

Findings

Separable optimization bounds are inferior to perturb-and-MAP bounds.

The stochastic subgradient method effectively maximizes the lower-bound on log-likelihood.

The approach performs well in binary image denoising with missing data.

Abstract

We consider log-supermodular models on binary variables, which are probabilistic models with negative log-densities which are submodular. These models provide probabilistic interpretations of common combinatorial optimization tasks such as image segmentation. In this paper, we focus primarily on parameter estimation in the models from known upper-bounds on the intractable log-partition function. We show that the bound based on separable optimization on the base polytope of the submodular function is always inferior to a bound based on "perturb-and-MAP" ideas. Then, to learn parameters, given that our approximation of the log-partition function is an expectation (over our own randomization), we use a stochastic subgradient technique to maximize a lower-bound on the log-likelihood. This can also be extended to conditional maximum likelihood. We illustrate our new results in a set of experiments in binary image denoising, where we highlight the flexibility of a probabilistic model to learn with missing data.