Parameter Learning for Log-supermodular Distributions
This work addresses parameter learning for probabilistic models related to combinatorial optimization, with incremental improvements in efficiency for tasks like image processing.
The paper tackled parameter estimation in log-supermodular distributions on binary variables by comparing bounds on the intractable log-partition function and using stochastic subgradient methods to maximize a lower-bound on log-likelihood, demonstrating results in binary image denoising experiments.
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.