romangarnett (u/romangarnett)

romangarnett commented on Mathematicians don't care about foundations (2022) matteocapucci.wordpress.c... · Posted by u/scrivanodev

AnimalMuppet · 3 months ago

The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets.

You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abstraction away from what I'm doing. I pretty much only care about stuff at my layer, and maybe one layer above or below.

romangarnett · 3 months ago

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

romangarnett commented on Bayesian Optimization Book bayesoptbook.com//... · Posted by u/signa11

Djrhfbfnsks · 4 years ago

Thanks for your suggestions. For my use case (tuning parameters of a financial market simulation), I'm essentially able to get good noise estimates for free by re-sampling a set of parameters multiple times.

So for example, rather than simulate an entire month in one shot, I'll simulate a day 30 times and therefore have a decent estimate of the noise for that result and be able to clearly distinguish the noise from the covariance of the Gaussian process.

The noise in these simulations can vary dramatically in parameter space (easily 10-100x), so it seems like it would be important to model.

romangarnett · 4 years ago

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.)

romangarnett commented on Bayesian Optimization Book bayesoptbook.com//... · Posted by u/signa11

janto · 4 years ago

Interesting, although the existence of this research actually suggests that normal Bayesian opt by itself is not safe and needs to be modified to be so.

A relatively unique trait of Bayesian opt is the modelling of unexplored space. The attraction to exploring that space makes it actually less safe than other methods that do not explicitly care.

One could go through similar steps to model and generate safe steps in other methods. It doesn't seem specific to Bayesian opt. It might even be better since it's less likely to be so computationally expensive as the auxiliary opt within Bayesian opt tends to be.

romangarnett · 4 years ago

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.

romangarnett commented on Bayesian Optimization Book bayesoptbook.com//... · Posted by u/signa11

quanto · 4 years ago

Is Bayesian optimization widely used on the field with success? It had much fanfare 5 years ago, but I don't hear much nowadays.

I have some mathematical background in optimization, and I am quite curious how Bayesian optimization compares to other more established methods.

romangarnett · 4 years ago

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

romangarnett commented on Bayesian Optimization Book bayesoptbook.com//... · Posted by u/signa11

Djrhfbfnsks · 4 years ago

For most of the problems for which I've tried to use Bayesian Optimization, I've had poor results because of unknown and heterogeneous noise in the underlying process that I'm trying to optimize.

I believe that modeling the noise directly using a 2nd Gaussian Process [1] could help, but I haven't gotten reliable results. I was hoping this topic would be addressed in the book, but don't see it.

[1] https://rdrr.io/cran/hetGP/

romangarnett · 4 years ago

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).

romangarnett commented on Bayesian Optimization Book bayesoptbook.com//... · Posted by u/signa11

janto · 4 years ago

I'm having a hard time imagining how using Bayesian optimization would improve safety during the search process, especially since in practice it often tends to just give approximately uniform sampling of a space anyway. It really likes the unexplored territory and is likely to try things far away from the optimum that are more risky.

romangarnett · 4 years ago

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.