[Question] Odd Behavior for simple GP/BO

I’m trying to fit a Guassian Process to a simple polynomial without noise using Ax’s get_botorch function, but I’m seeing some unexpected behavior. In certain cases, the GP fails to fit to the points, even though I am using an ExactGP with a noiseless Expected Improvement acquisition function. I’ve modified the Using a custom botorch model with Ax tutorial to evaluate the function y = x1**2 instead of the Branin function, and to use the (analytic) ExpectedImprovement acquisition function instead of the qNoisyExpectedImprovement acquisition function, since I thought that might be what’s causing the issue. I’ve attached plots of several past outcomes (generated by calling render(plot_slice(...)), and I’d be happy to post my code if necessary.

[keeganharris]

Great, thanks for your help!

[Balandat]

Thanks.

So I ran this a few times in a notebook and was able to reproduce this. If there are only 2-3 data points, the behavior that the model produces a constant fit with high variance is to be expected – there is just not enough information in 3 data points for a non-parametric model with a Matern Kernel to make much sense of it.

The fact that in your example this occurs also for > 3 data points is most likely due to the fact that you do not put a prior on the lengthscales of your Kernel. As a result, a priori any lengthscale is equally likely to the model. However, Ax by default normalizes the inputs to the unit cube [0, 1]^d, so lengthscales > 1 are pretty meaningless.

That’s why the SingleTaskGP BoTorch model Ax uses by default puts a prior on the lengthscales with very little probability mass greater than 1. If you do the same thing, that is, add a lengthscale_prior=GammaPrior(3.0, 6.0) arg to your MaternKernel, I don’t see this behavior anymore for n > 3 data points in your example.

More generally speaking, if you have prior knowledge that your function has a particular structure (e.g. is a polynomial of some degree, monotonic, periodic, …), using a different, more parametric Kernel (e.g. a polynomial Kernel) would be the way to go, and would result in more interpretable fits (assuming that the function indeed satisfies the structural assumptions).

I hope this helps.