Bayesian Optimization with Uncertainty Quantification Explained for Experts

Bayesian optimization is a sequential model-based optimization method that models a probabilistic surrogate function \( f \), usually using Gaussian processes (GP), and uses an acquisition function \( \alpha \) to determine the next evaluation point.

Uncertainty quantification is done via the variance \( \sigma^2(x) \) of the GP prediction at a point \( x \), which expresses the epistemic uncertainty of the model.

Formally, the next evaluation point is chosen as

\[ x^* = \arg\max_x \alpha(x; \mu(x), \sigma(x)) \]

where \( \mu(x) \) and \( \sigma(x) \) are the mean and standard deviation of the prediction.

Examples of acquisition functions are:

• Expected Improvement (EI): Maximizes the expected gain over the current optimum.
• Upper Confidence Bound (UCB): Combines mean and uncertainty with a parameter to control exploration.

Uncertainty quantification thus enables an efficient search in complex, expensive search spaces.

Definition:
“Bayesian optimization with uncertainty quantification is an iterative optimization method that uses a probabilistic model to estimate both predictions and uncertainties and deliberately uses these uncertainties when selecting new evaluation points.”

Source:
Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical Bayesian Optimization of Machine Learning Algorithms. Advances in Neural Information Processing Systems (NeurIPS).