How to get the derivative of an entry in a covariance matrix wrt a kernel parameter?

[bcolloran]

I'm trying to get the derivative of an entry of a covariance matrix wrt a kernel parameter, and I'm having a bit of trouble after banging on it for a few hours. I've pasted a toy example below, but the results I get don't match my Mathematica sanity check (I expect the derivative to be 2.4176, but I'm getting 0.515859 with this code).

I thought maybe I was doing something wrong with the raw/transformed parameters, but I've looked around in the code base and that doesn't seem to be it (though I could have totally misunderstood...), so I suspect I'm misunderstanding something basic about Torch, Gpytorch, or the interaction of the two. Thanks for any pointers!

import torch
from gpytorch.kernels import RBFKernel

ls = 0.24
dist = 0.23
kernel = RBFKernel()
kernel.lengthscale = ls

X = torch.tensor([0, dist], requires_grad=True)

Gram = kernel(X, X).evaluate()

print(Gram)
k = Gram[0, 1]
print(k.item())

k.backward()
dkdl = kernel.get_parameter("raw_lengthscale").grad.item()
print(dkdl)

Output:

tensor([[1.0000, 0.6318],
        [0.6318, 1.0000]], grad_fn=<RBFCovarianceBackward>)
0.6317879557609558
0.5158590078353882

Sanity check:

[jacobrgardner]

You said you might be doing something wrong with raw/transformed parameters, and it seems to me like this is definitely the case. In your example Mathematica code, you're differentiating directly with respect to \ell, but in GPyTorch you're differentiating with respect to the inverse softplus of \ell (e.g., the raw lengthscale).

In other words, GPyTorch's version of the RBF kernel is basically exp[-dist^2 / softplus(raw_ls)^2], and what you're computing is the derivative with respect to raw_ls.

[bcolloran]

Dang, I guess I was on the right track, but I somehow convinced myself that the raw/transformed params weren't it. Back on track now thanks to your pointer, sorry for the noise. :-/