You can use the straight through operator!

During the forward pass you sample a discrete outcome given your NN weights to get an error for backprop. During the backward pass you directly propogate through the weights.

This GradTree paper[1] does a good job covering how to do discrete gradient-based optimization (i.e. NNs w/ discrete representations).

Another option is to use a GFlowNet[2]. Then you have a NN policy that takes discrete actions like you're playing an RL game. You're not back-propogating through something that isn't continuous, but you're utilizing a NN to make informed decisions about a problem with a discrete representation.

[1] GradTree (https://arxiv.org/pdf/2305.03515.pdf) [2] GFlowNet (https://arxiv.org/abs/2111.09266)