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Structured latent variables allow incorporating meaningful prior knowledgeinto deep learning models. However, learning with such variables remainschallenging because of their discrete nature. Nowadays, the standard learningapproach is to define a latent variable as a perturbed algorithm output and touse a differentiable surrogate for training. In general, the surrogate putsadditional constraints on the model and inevitably leads to biased gradients.To alleviate these shortcomings, we extend the Gumbel-Max trick to definedistributions over structured domains. We avoid the differentiable surrogatesby leveraging the score function estimators for optimization. In particular, wehighlight a family of recursive algorithms with a common feature we callstochastic invariant. The feature allows us to construct reliable gradientestimates and control variates without additional constraints on the model. Inour experiments, we consider various structured latent variable models andachieve results competitive with relaxation-based counterparts.

Leveraging Recursive Gumbel-Max Trick for Approximate Inference in Combinatorial Spaces