I noticed this remark in DeepMind's VQ-VAE paper:

One could also use the subgradient through the quantisation operation, but this simple estimator worked well for the initial experiments in this paper.

How would one use the subgradient here? The quantization operation is discontinuous. Isn't the subgradient of the objective function empty because of this?

Typically, the subgradient is defined for convex functions. And convex functions are continuous.

However, DeepMind's VQ-VAE paper defines a model with a discontinuous vector quantization (VQ) layer, resulting in a discontinuous objective function. Still, the authors remark:

One could also use the subgradient through the quantisation operation, but this simple estimator worked well for the initial experiments in this paper.

Is there a more general definition of the subgradient that would make sense here?

I noticed this remark in DeepMind's VQ-VAE paper:

One could also use the subgradient through the quantisation operation, but this simple estimator worked well for the initial experiments in this paper.

How would one use the subgradient here? The quantization operation is discontinuous and non-convex. Isn't the subgradient empty?

I noticed this remark in DeepMind's VQ-VAE paper:

One could also use the subgradient through the quantisation operation, but this simple estimator worked well for the initial experiments in this paper.

How would one use the subgradient here? The quantization operation is discontinuous. Isn't the subgradient of the objective function empty because of this?