Generalized Gumbel-Softmax Gradient Estimator for Various Discrete Random Variables

Estimating the gradients of stochastic nodes is one of the crucial research questions in the deep generative modeling community, which enables the gradient descent optimization on neural network parameters. This estimation problem becomes further complex when we regard the stochastic nodes to be discrete because pathwise derivative techniques cannot be applied. Hence, the stochastic gradient estimation of discrete distributions requires either a score function method or continuous relaxation of the discrete random variables. This paper proposes a general version of the Gumbel-Softmax estimator with continuous relaxation, and this estimator is able to relax the discreteness of probability distributions including more diverse types, other than categorical and Bernoulli. In detail, we utilize the truncation of discrete random variables and the Gumbel-Softmax trick with a linear transformation for the relaxed reparameterization. The proposed approach enables the relaxed discrete random variable to be reparameterized and to backpropagated through a large scale stochastic computational graph. Our experiments consist of (1) synthetic data analyses, which show the efficacy of our methods; and (2) applications on VAE and topic model, which demonstrate the value of the proposed estimation in practices

Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions

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.Comment: NeurIPS 2021 camera-ready; repo: https://github.com/nec-research/tf-iml