Gpytorch: Document various ways GPs can be used in batch mode

[Balandat]

Batch mode (fitting and predicting) is currently a "hidden" feature as it's not very well documented and does not have a tutorial.

We should add documentation and also unit tests to make sure this works as expected (this was the cause for a lot of bugs in the past).

[Balandat]

I guess this is particularly important for batch-training of GPs, b/c the standard pattern

loss = -mll(output, train_y)
loss.backward()

will fail.

If none of the hyperparameters are shared, then a straightforward way to do this is to use loss= -mll(output, train_y).sum(), but things get murkier if some parameters are shared across batches since the losses of the individual batches will not be independent w.r.t. to the model parameters.

[Balandat]

This is related to #130

[gpleiss]

I'll work on an example notebook when I go through and fix all the other notebooks.

[gpleiss]

I'm also working on better batch mode tests r.n. as part of the batch mode PR

[Balandat]

Thanks a lot for the comprehensive tests in your PR. We should probably make sure to write up some documentation about this - maybe even a little feature matrix showing which of the following can be combined, and what the expected train/test input dimensions and resulting output dimensions are:

• batch training (shared hyperparameters)
• batch training (separate hyperparameters)
• batch evaluation (arbitrary input batch size)
• single-task vs. multi-task
• approximate GPs

[gpleiss]

I'll work on it!

[h-jia]

Also problem with me. Hope this can be fixed. Thanks

[gpleiss]

There are now some docs about batch GPs: https://gpytorch.readthedocs.io/en/latest/batch_gps.html

Closing this for now, though I'm happy for any suggestions on how to improve the docs!