Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions

TL;DR

This paper introduces Implicit MLE, a versatile framework for end-to-end learning with discrete exponential family distributions and neural networks, avoiding the need for smooth relaxations and enabling differentiation through combinatorial solvers.

Contribution

The paper presents I-MLE, a novel approach that unifies various differentiation methods for discrete distributions and combinatorial problems, including a new class of noise distributions for marginals.

Findings

I-MLE often outperforms existing relaxation-based methods.

It simplifies to maximum likelihood in certain combinatorial learning settings.

The framework is broadly applicable to models combining discrete distributions and neural components.

Abstract

Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.