# A Test of Human Intuition for Mean-Reverting Processes

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).

# The game

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).

# Some motivation

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.

# First location

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.)

# Second location

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).

# Third location?

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.

# Into time-series?

The application to time-series models-in-a-hurry is fairly obvious. Stop by our little time-series community some time (slack).