Introduction to Bayesian Optimization

Bayesian Optimization (BO) concerns gradient-free, assumption-free optimization. The goal is to solve a general optimization problem with no known structure (e.g. convexity or linearity) to exploit, and where we do not have access to any of the function \(f(\dot)\)’s derivatives.

\[\max_{x \in X} f(x)\]

Many other assumptions are also often applied, including:

• Evaluating the function is expensive. For instance, \(f(\cdot)\) might be the outcome of an economic policy and the input the available economic levers (e.g. tax credits).
• The feasible set \(X\) is easy to assess membership.
• The function \(f\) is continuous. This assumption is necessary to use a common approach in the field (i.e. Gaussian process regression)

All/most approaches to Bayesian Optimization have two components:

1. A statistical model of the objective function, often called the surrogate function
2. A method for deciding where to sample next, often called the acquisition function.

There are a variety of different acquisition functions, but almost always, the surrogate function is Gaussian Process regression.

The psuedo-code is then:

• Place a GP prior on \(f\)
• For as many iterations as you can afford
  • Update the posterior on \(f\) using all available data
  • Choose a point \(x_n\) as the maximizer of some acquisition function
  • Observe \(y_n = f(x_n)\)