This paper introduces the Gumbel-Softmax distribution, a continuous approximation of categorical distributions that enables differentiable sampling and efficient gradient estimation for stochastic neural networks with discrete latent variables. The Gumbel-Softmax distribution allows for smooth annealing into a categorical distribution and provides a differentiable alternative to non-differentiable categorical samples. This enables the use of backpropagation to compute gradients, which is essential for training models with categorical latent variables. The Gumbel-Softmax estimator outperforms existing gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables. It also enables significant speedups in semi-supervised classification by avoiding costly marginalization over unobserved categorical latent variables. The Gumbel-Softmax estimator is particularly effective for both Bernoulli and categorical variables, and can be used in conjunction with the Straight-Through Gumbel-Softmax estimator for scenarios where discrete sampling is required. The Gumbel-Softmax distribution is derived from the Gumbel-Max trick, which provides a way to sample from a categorical distribution using continuous relaxation. The distribution is defined as a continuous distribution over the simplex that can approximate samples from a categorical distribution. The density of the Gumbel-Softmax distribution is derived and shown to be a continuous relaxation of the categorical distribution. The paper also compares the Gumbel-Softmax estimator to other gradient estimation techniques, including score function-based estimators and path derivative estimators. It shows that the Gumbel-Softmax estimator outperforms these methods in terms of variance and computational efficiency. The paper also discusses the application of the Gumbel-Softmax estimator in semi-supervised learning, where it enables efficient training of models with categorical latent variables without the need for costly marginalization. The paper concludes that the Gumbel-Softmax estimator is a powerful tool for training stochastic neural networks with discrete latent variables, and that it provides a simple, differentiable approximate sampling mechanism for categorical variables that can be integrated into neural networks and trained using standard backpropagation.This paper introduces the Gumbel-Softmax distribution, a continuous approximation of categorical distributions that enables differentiable sampling and efficient gradient estimation for stochastic neural networks with discrete latent variables. The Gumbel-Softmax distribution allows for smooth annealing into a categorical distribution and provides a differentiable alternative to non-differentiable categorical samples. This enables the use of backpropagation to compute gradients, which is essential for training models with categorical latent variables. The Gumbel-Softmax estimator outperforms existing gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables. It also enables significant speedups in semi-supervised classification by avoiding costly marginalization over unobserved categorical latent variables. The Gumbel-Softmax estimator is particularly effective for both Bernoulli and categorical variables, and can be used in conjunction with the Straight-Through Gumbel-Softmax estimator for scenarios where discrete sampling is required. The Gumbel-Softmax distribution is derived from the Gumbel-Max trick, which provides a way to sample from a categorical distribution using continuous relaxation. The distribution is defined as a continuous distribution over the simplex that can approximate samples from a categorical distribution. The density of the Gumbel-Softmax distribution is derived and shown to be a continuous relaxation of the categorical distribution. The paper also compares the Gumbel-Softmax estimator to other gradient estimation techniques, including score function-based estimators and path derivative estimators. It shows that the Gumbel-Softmax estimator outperforms these methods in terms of variance and computational efficiency. The paper also discusses the application of the Gumbel-Softmax estimator in semi-supervised learning, where it enables efficient training of models with categorical latent variables without the need for costly marginalization. The paper concludes that the Gumbel-Softmax estimator is a powerful tool for training stochastic neural networks with discrete latent variables, and that it provides a simple, differentiable approximate sampling mechanism for categorical variables that can be integrated into neural networks and trained using standard backpropagation.