Bayesian Optimization

First of all, Katherine Bailey is one of my favorite smart ladies, her explanation of Gaussian Processes (the flavor of Bayesian optimization I get into there) is totally worth the read. This is a topic that’s worth taking the time to build intuition with, as once you have that, you can start to see the possibilities of applying Bayesian optimization everywhere.

What’s nice about Bayesian optimization is that you don’t need derivatives. What’s challenging is selecting your prior and sampling from your posterior distribution. What makes the optimization “Bayesian” is that we treat the objective function as an unknown, that is, we start with a prior assumption, apply that assumption to our data to generate a posterior distribution, and then use the posterior distribution to create a sampling function that will give us the next point to sample to gain the most information about the objective function.

Gaussian Processes

In Rassmussen’s book two methods are presented - the function approach and the “ ” approach. I prefer the function approach, that is to say I see GPs as being this performing inference on a distribution of functions that can define our observed data. Our goal is not to arrive at a particular function, but to determine the next best point based on the distribution we arrive at.

Covariance Matrix

If I showed you three points on a 2D plane, and told you would be using a line to map the relationship of points, how would you determine which two points were closest to draw a line between? Well, this seems pretty straightforward, but what if we were looking at the relationship between two points in a 10D space, and are searching over all the functions we could use to map their relationship? Well, this is where we can use a covariance matrix to ensure that points close together in our domain will be considered close when observed in our codomain.

Acquisition Functions

Probability of Improvement

Expected Improvement

Upper Confidence Bounds (UCB)

Thompson Sampling

Choose the action that maximizes the expected reward in regards to a randomly selected belief.