Bayesian optimization


Bayesian optimization is a sequential design strategy for global optimization of black-box functions that does not assume any functional forms. It is usually employed to optimize expensive-to-evaluate functions.

History

The term is generally attributed to Jonas Mockus and is coined in his work from a series of publications on global optimization in the 1970s and 1980s.

Strategy

Since the objective function is unknown, the Bayesian strategy is to treat it as a random function and place a prior over it. The prior captures beliefs about the behavior of the function. After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution over the objective function. The posterior distribution, in turn, is used to construct an acquisition function that determines the next query point.
There are several methods used to define the prior/posterior distribution over the objective function. The most common two methods use Gaussian Processes in a method called Kriging. Another less expensive method uses the Parzen-Tree Estimator to construct two distributions for 'high' and 'low' points, and then finds the location that maximizes the expected improvement.

Examples

Examples of acquisition functions include probability of improvement, expected improvement, Bayesian expected losses, upper confidence bounds, Thompson sampling and hybrids of these. They all trade-off exploration and exploitation so as to minimize the number of function queries. As such, Bayesian optimization is well suited for functions that are expensive to evaluate.

Solution methods

The maximum of the acquisition function is typically found by resorting to discretization or by means of an auxiliary optimizer. Acquisition functions are typically well-behaved and are often maximized with implementations of Newton's Method such as Broyden–Fletcher–Goldfarb–Shanno algorithm or the Nelder-Mead method.

Applications

The approach has been applied to solve a wide range of problems, including learning to rank, computer graphics and visual design, robotics, sensor networks, automatic algorithm configuration, automatic machine learning toolboxes, reinforcement learning, planning, visual attention, architecture configuration in deep learning, static program analysis and experimental particle physics.