Statistical methods of interest: Bayesian Optimization

Optimization remains a vibrant research area applicable across numerous fields — finance, healthcare, computer science, cybersecurity, transportation, and sports. As computational power increases alongside problem complexity, more sophisticated algorithms emerge.

A particular challenge arises when optimizing “black box functions” — scenarios where only outputs are observable for given inputs, yet the function itself remains unknown. Complicating matters further, evaluation may require significant time or monetary investment.

Understanding Bayesian Optimization

The method is based on the Bayesian paradigm of prior and posterior probabilities. Two core components drive this approach:

1. A probability distribution modeling the unknown function
2. An acquisition function determining optimal evaluation points

The process unfolds iteratively:

• Build a surrogate model using initial function evaluations
• While stopping criteria remain unmet:
  • Apply the acquisition function to identify the next point
  • Evaluate the function at this point
  • Update the surrogate model with the new posterior distribution

Upon completion, the algorithm returns candidate points, allowing selection of the maximum value.

Implementation Strategy

This approach transforms the challenge of handling an unknown function into managing an acquisition function — typically simpler and more tractable. A popular implementation combines:

• Gaussian processes for modeling the black box function
• Expected improvement as the acquisition function, describing potential gains relative to observed maximums

This combination yields closed-form solutions, facilitating computation of optimal evaluation points.

Practical Application

Optimization algorithms offer substantial value within cybersecurity operations. Praxis Security Labs leverages Bayesian optimization among innovative solutions for clients seeking competitive advantages.

Reference: Frazier, P. I. (2018) A tutorial on Bayesian optimization. arXiv preprint arXiv:1807.02811.