Eventually, this article will contain some moderately cute mathematics, a new use of “rapidity” (stolen from special relativity), and some trigonometry on a Brownian bridge. Unfortunately, I can’t show you that now — it would ruin the survey I wish to take, with your kind help.
An Exponential Ornstein-Uhlenbeck Process
Let us suppose that at the beginning of time, God created a mountain range.
Let us further assume that she used a stochastic process. It could just as easily have been:
Here if X is a random walk reverting to zero, assume you are looking at the exponential of that random walk: Y =exp(X).
You get to move at most twice from your starting location, in order to try to maximize E[Y]. In this quiz, I’m hinting that it pays to move at least once… but you have to decide on the third point (if you choose to move again at all).
In the management science literature, rugged optimization problems are regarded as “useful metaphors” for a strategy space (proximate example, seminal example). Maybe you are a company deciding whether to change your logo or a golfer deciding to change their swing. You don’t want to do that very often.
I have in mind something I can code, however, and the test I give you I also give to allegedly meta-learning few-shot exploration algorithms that are tasked with this challenge repeatedly. They should eventually perform well in the poll. Their failure to do so is a very simple, clear marker of incomplete learning.
Also… well-performing optimization techniques like DIRECTR, DLIB and others, and theoretical results as well, often rest on an estimate of a Lipchitz coefficient (or assumption one exists). But here the function is not Lipchitz. Optimization techniques using space-filling curves to reduce multi-dimensional problems to univariate tend to lean even more heavily on that notion, so it is interesting to me to consider a univariate task outside that assumption.
And maybe humans, who are allegedly capable meta-learners, actually find this particular limited exploration task quite challenging … at least according to the exit polls thus far. So on to the problem at hand.
Your original location is x=-2 where, as it turns out, the height is Y=1.3 roughly. This gives you some information about where the mountain range might be — though not much. A sample of paths from that Bayesian posterior is shown below.
(ML folks: X is a gaussian process with Ornstein-Uhlenbeck kernel.)
You aren’t satisfied. You move to another location x=2 and discover, to your joy, that the height is Y=1.35, a little better. Now, your belief might be (noisily) represented by a sampling of paths that all pass through the points (-2,1.3) and (2,1.35).
Now we come to the critical decision. You have found two above-average locations. What will be your third and final resting place? Assume you want to maximize the mean of the height you will live at forever.
I don’t want silly answers. There are nine qualitatively different possibilities, but I allow only those in bold.
- Move to negative infinity even though I told you that’s worse on average than either x=2 or x=-2
- Move to some finite x<-2, even though that’s worse than x>2 by symmetry
- Move back to x=-2 even though that’s worse than x=2
- Move to somewhere in between x=-2 and zero even though that’s worse than (0,2) by symmetry
- Move to zero.. call this Option A
- Move to somewhere in between zero and 2 … Option B
- Stay at x=2 … Option C
- Move to some finite x>2… Option D
- Move to positive infinity … the same as negative infinity
So what’s it going to be? I’m interested in your mathematical gut reaction … mostly … but I’d be loathed to ever stop anyone putting pen to paper. I hope you enjoy this little teaser whether or not you try to solve it.
Submit your answer
The poll is here.
The application to time-series models-in-a-hurry is fairly obvious. Stop by our little time-series community some time (slack).