# Regarding L0 sparsification of DNNs proposed by Louizos, Kingma and Welling

I am reading the paper on $$\ell_0$$ regularization of DNNs by Louizos, Welling and Kingma (2017) (Link to arxiv).

In Section 2.1 the authors define the cost function as follows: $$\mathcal{R}\left( \tilde{\theta}, \pi \right) = \mathbb{E}_{q(z|\pi)}\left[ \frac{1}{N} \left(\sum_{i=1}^N \mathcal{L}\left(h\left( x_i, \tilde{\theta}\circ Z\right), y_i \right) \right)\right] + \lambda\sum_{i=1}^{|\tilde{\theta}|}\pi_i.$$

In the above display, $$\tilde{\theta}$$ are the weights, $$Z$$ is a random vector of the same dimension as $$\tilde{\theta}$$ consisting of independent Bernoulli components $$q(Z_i|\pi) \sim Bernoulli (\pi_i)$$, and $$\circ$$ is the element-wise product.

The authors then state the following:

the first term is problematic for $$\pi$$ due to the discrete nature of $$Z$$, which does not allow for efficient gradient based optimization.

I am not sure I understand this. Denoting the first term by $$\mathcal{R}_1 = \sum_{i=1}^N \frac{1}{N}R_i$$ ($$R_i$$ defined below), and using the notation $$\pi_z = \prod \pi_i^{z_i} (1-\pi_i)^{1-z_i}$$ and $$\mathcal{Z}$$ for the set of all possible values of $$Z$$, we should have $$R_i := \mathbb{E}_{q(z|\pi)}\left[ \mathcal{L}\left( h\left(x_i, \tilde{\theta}\circ z\right), y_i \right) \right] = \sum_{z \in \mathcal{Z}}\pi_z\mathcal{L}\left( h\left(x_i, \tilde{\theta}\circ z\right), y_i \right)$$

So, it seems to me that the gradient of $$R_i$$ with respect to $$\pi_j$$ can be obtained as $$\frac{d R_i}{d\pi_j} = \sum_{z \in \mathcal{Z}}\mathcal{L}\left( h\left(x_i, \tilde{\theta}\circ z\right), y_i \right) \frac{d\pi_z}{d\pi_j}$$ and $$\frac{d\pi_z}{d\pi_j} = \frac{\pi_z}{\pi_j}$$ if $$z_j=1$$ and $$-\frac{\pi_z}{1-\pi_j}$$ if $$z_j=0$$. So, it appears that we can obtain the derivative of the first term with respect to $$\pi_j$$ as well.

My question is the following:

If my above calculation is correct, then the derivatives $$\frac{d\mathcal{R}_1}{d\pi_j}$$ can be computed, and we can perform SGD on the cost function $$\mathcal{R}(\tilde{\theta}, \pi)$$. But the authors claim that it cannot be obtained and hence they introduce the `hard concrete' distribution etc. to construct a differentiable cost function.