[ENH] MDNRegressor (Mixture Density Network)

[joshdunnlime]

Reference Issues/PRs

No issue opened.

What does this implement/fix? Explain your changes.

A new regressor implementation of Mixture Density Network (MDN) as per Bishop 1994 with noise regularisation as per Rothfuss 2019.
It also includes optional passing of pytorch activation functions and optimizers.

It implements a fully vectorised NormalMixture distribution where each rows weights are individually learnt and applied. It also implements a custom vectorised bisection for fast _ppf method calls.

Does your contribution introduce a new dependency? If yes, which one?

Yes, soft deps: Pytorch and pytorch-optimizer.

What should a reviewer concentrate their feedback on?

NormalMixture distribution. This adds slightly opinionated design choice to the API.

Did you add any tests for the change?

Yes. Standard library param tests for dist and est. Additional test for est coming.

Any other comments?

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[fkiraly]

Thanks! Nice!

• question: why implement a NormalMixture when Mixture is available and should be capable of representing mixture of normals by a composition of Normal and Mixture?
• if we add NormalMixture, it should also be added in the API reference for distribution

[joshdunnlime]

My understanding of Mixture is that it only has global weights for each sub-dist. MDNR could use this, however, it would mean that for each prediction row, it would use a new instance of Mixture (computationally this would be crippling). MixtureNormal has performant, vectorised implementation of most of the class methods so is actually pretty quick!

You could argue that we could call it Normal(Row|Instance|HeteroWeighted)Mixture but I think it is documented clearly enough to avoid confusion.

[fkiraly]

Ah, I see. Yes, that makes sense.

Do you want to open an issue that wishes for an abstract mixture that can have row-wise different mixture weights? That could be extended from Mixture, if weights are not just a list but a matrix.

[joshdunnlime]

I did consider the following:

1. Extending Mixture to include "instance" and "global" weights.
2. Create a generic InstanceWeightedMixture class.
3. A base class BaseInstanceWeightedMixture.

The downsides were:

1. Far too much work. IMHO these are two quite different classes to the point I think this would just be a wrapped around a Mixture and InstanceWeightedMixture.

2. InstanceWeightedMixture which accepts a mixture of different dists comes with a lot of complications. The primary one is that the NormalMixture has very well defined (and mostly exact) methods for all of the standard methods. My understanding of weighted mixtures of distributions stops a normals so I have no idea what is feasible here. The vectorisation is clearly very well defined for normal distributions in scipy, and assumptions on the support for root-finding in _ppf were fairly straightforward. Implemenation-wise, I believe it serves to keep NormalMixture even if we have 3) as the speed-ups and exactness are significant.

3. Happy to open a Issue. My knowledge of anything beyond normal mixtures of dists is effectively zero so I doubt I'd be up to a PR.

[joshdunnlime]

Just as a side note: The natural extension to MDNs is probably something Flow/Kernel based, as opposed to extending this with non-Normal dists.

[fkiraly]

Should be fine now, but one request:

the estimator MDNRegressor does not actually depend on pytorch-optimizer, except if the string "SOAP" is passed. But that only does aliasing, and you are doing soft dependency checking there is no intrinsic dependency. Hence I would remove pytorch-optimizer from the dependency set.

I would also suggest to replace the try/except for dependency checking with _check_soft_dependencies (severity="none")

[fkiraly]

how about "deep learning based regressors" instead?