# What are the limmitations of reparametrization gradients for discrete random variables? (Gumbel-softmax)

We know that one approach for re-parametrizing gradients for variational inference is taking the Gumbel-softmax estimator proposed in [1] and [2].

In [3], that is a talk on SVI, D. Blei at around 29:31 mentions that there is no re-parametrization for discrete variables for VI (also on the slides). The talk is one year after the papers. Since I find it highly unlikely that something like that would have gotten under the radar of a leading expert what am I missing?

More concretely, the questions I have are the following:

1. Is the Gumbel softmax trick considered to have solved in a satisfying manner the issue of low-variance gradients in SVI with re-parametrization or not?
2. Is the statement in the talk simply inaccurate?

This is not to be perceived as a criticism of the talk - I highly recommend it by the way. I am truly asking to clarify my confusion on the subject.

[1]: Categorical Reparametrization with Gumbel-Softmax (https://arxiv.org/pdf/1611.01144.pdf)

[2]: The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables (https://arxiv.org/abs/1611.00712)

[3]: Talk by D. Blei on SVI: https://www.youtube.com/watch?v=-H2N4tVDK7I

I think in [1], at first Gumbel-max trick is used which is an exact method to obtain samples from the categorical distribution. But, Gumbel-max trick (eqn 1 in paper [1]) introduces arg-max which needs is approximated by using the softmax function (eqn. 2 in paper [1]). Basically, this approximation enables automatic differentiation but also introduces a hyperparameter $$\tau$$ which determines the variance. If $$\tau$$ is small, the arg-max approximation is accurate but it leads to gradients with high variance. Conversely, if $$\tau$$ is large, the arg-max approximation may become inaccurate (see figure 1 and section 2.1 in paper [1]). The paper introduces a novel method that solves gradient backpropagation problem for discrete latent variables but I think the solution is not perfect (as there is an approximation for arg-max)