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The Gumbel-max trick is a method to draw a sample from a categorical distribution, given by its unnormalized (log-)probabilities. Over the past years, the machine learning community has proposed several extensions of this trick to facilitate, e.g., drawing multiple samples, sampling from structured domains, or gradient estimation for error backpropagation in neural network optimization. The goal of this survey article is to present background about the Gumbel-max trick, and to provide a structured overview of its extensions to ease algorithm selection. Moreover, it presents a comprehensive outline of (machine learning) literature in which Gumbel-based algorithms have been leveraged, reviews commonly-made design choices, and sketches a future perspective.
Gradient estimation in models with discrete latent variables is a challenging problem, because the simplest unbiased estimators tend to have high variance. To counteract this, modern estimators either introduce bias, rely on multiple function evaluations, or use learned, input-dependent baselines. Thus, there is a need for estimators that require minimal tuning, are computationally cheap, and have low mean squared error. In this paper, we show that the variance of the straight-through variant of the popular Gumbel-Softmax estimator can be reduced through Rao-Blackwellization without increasing the number of function evaluations. This provably reduces the mean squared error. We empirically demonstrate that this leads to variance reduction, faster convergence, and generally improved performance in two unsupervised latent variable models.
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