romangarnett

Do you not care if your vector space has a basis?

That's a fortunate scenario! If you have good noise estimates available then you can sidestep the need to infer the noise scale and instead simply proceed with "typical" heteroskedastic inference. When the observation noise variances are known, you only need to modify the typical GP inference equations to replace the σ²I term that appears in the homoskedastic case (where σ² is the constant noise scale) with a diagonal matrix N indicating the noise variances associated with each observation along the diagonal.

(One might imagine a slightly more flexible model including a scaling parameter, replacing N with c²N and inferring c from data.)

Indeed -- safety in exploration is not a relevant concern in all (or even in most) scenarios. However, in domains such as robotics where certain configurations may have unforeseen and severe consequences, you may want to slow down exploration and proceed more cautiously. As you suggest, you can optimize more efficiently if you can explore the domain without any safety concerns.

You're correct that the approach taken in the linked paper could be adapted to increase the safety of other sequential design settings (when needed), assuming you have access to a model that can quantify uncertainty.

The book includes an annotated bibliography outlining successes in a wide range of areas spanning science and engineering.

Replying as the author -- I do spend some time discussing hetereoskedastic noise (beginning in §2.2 and intermittedly throughout following chapters), although you're right that I don't discuss this particular modeling approach. Personally I think that inferring hetereoskedastic noise from data alone during Bayesian optimization is likely to be very difficult, as you'll need either a lot of data and/or to be in a very small dimension in order to identify the variable noise scale. (Note that the example in the hetGP writeup is only in one dimension.)

However, when the noise scale is either variable (but known) or can be modeled with a relatively simple (e.g., parametric) model, there may be some benefit to the added model complexity. Here you could include the parameters of the noise model into the model hyperparameters and proceed following the discussion in chapter 4. In doing so, I would be careful to ensure that the data actually support the heteroskedastic noise hypothesis.

Another approach that might be useful in some contexts is a heavy-tailed noise model such as Student-t errors (§§ 2.8, 11.9, 11.11).

Some research has explicitly considered the question of safety during exploration; see for example https://arxiv.org/pdf/1602.04450.pdf which includes theoretical analysis of the resulting algorithm.