Components of Bayesian Optimization

Bayesian optimization is a powerful optimization technique that is particularly effective when the objective function is expensive to evaluate or noisy. It combines the concepts of Bayesian inference and optimization to efficiently explore and exploit the search space in order to find the optimal solution. Bayesian optimization maintains a probabilistic model of the objective function and uses it to guide the search toward promising regions. It iteratively suggests new candidate solutions to evaluate based on the information gained from previous evaluations. In this detailed overview, we'll explore the different components of Bayesian optimization and how they work together to solve optimization problems effectively.

The components of Bayesian optimization are shown below:

Surrogate model

At the core of Bayesian optimization is the surrogate model, also known as the surrogate function or the probabilistic model. The surrogate model serves as an approximation of the objective function and offers a way to understand how it behaves throughout the search space with a degree of uncertainty.

GPs are frequently chosen as surrogate models in Bayesian optimization because they are versatile and adept at handling intricate functions. GPs not only capture the average behavior of the objective function but also its variability, which means they can predict what the function might look like at points where we haven’t collected data. This surrogate model provides a probabilistic guess of the actual objective function, which, in turn, facilitates a more efficient way to explore and exploit the search space. GPs find applications in various aspects of machine learning, including:

• Regression: GPs are used for nonlinear regression tasks, allowing us to model complex relationships between input and output variables while quantifying uncertainty in predictions.

• Bayesian optimization: GPs serve as surrogate models in Bayesian optimization, guiding the search for optimal hyperparameters or configurations in a data-efficient manner.

• Active learning: GPs help in selecting informative data points for labeling and reducing the need for a large labeled dataset by choosing samples that maximize the uncertainty in the model’s predictions.

The mathematical explanation of GPs

A GP defines a prior distribution over functions, where any finite set of points in the search space has a joint Gaussian distribution. Let’s consider a set of observed data points:

Here, $x_{}$ ...