Bayesian Optimization: Approximate to the optimum of an expensive Black Box system by using a cheap surrogate model

This code demonstrates the use of Bayesian optimization to approximate the optimal solution of a black box system using surrogate models. Bayesian optimization is an efficient technique for optimizing expensive-to-evaluate functions by iteratively selecting the next point to evaluate based on the surrogate model's predictions and an acquisition function.

Objective

The idea is to find the optimal configuration of parameters for a black box system, where evaluating the black box system is time-consuming. Bayesian Optimization uses a surrogate model, in this case, a Gaussian Process Regressor (but can be Random Forest Regressor, for example), to predict the behavior of the black box system and an acquisition function (a common one is the Upper Confidence Bound (UCB), or the probability of improvement) that guides the search for the optimal configuration. The general idea of the Bayesian Optimization workflow can be seen in the following image:

Considerations

When using surrogate models for optimization, keep the following considerations in mind:

• Surrogate Model Accuracy: The accuracy of the surrogate model is crucial. It's important to validate the model's predictions against actual evaluations to ensure reliable optimization. The surrogate_model_quality function shows two metrics with some interpretations.

• Sampling Strategy: The selection of sampling points and the balance between exploration and exploitation impact the optimization process. Choose a sampling strategy that suits your optimization problem. The get_parameter_space_samples function gets random parameter space samples.

• Model Selection: Experiment with different surrogate models (kernels) such as RBF, Matern, and others to find the best fit for your problem. The find_best_kernel function selects the "best" model. You can play with that to try more kernels and configurations.

• Overfitting: Be cautious of overfitting. Regularly validate the surrogate model's performance and consider adding more data points if necessary.

• Visualization: Visualize the surrogate model's predictions against the black box system to assess its accuracy and behavior. The plot_black_box_and_surrogate provides this plot when the system has one parameter.

Getting Started

1. Install the required Python packages using pip install -r requirements.txt.

2. Configure your black box system and its parameters, as well as other settings in the main program.

3. If you are working with a one-dimensional problem, you can call the optimization with show_plots=True to see, every 5 iterations, the surrogate model plot.

4. Run the program to perform Bayesian optimization with the surrogate model.

Result Interpretation

After running the program, you will obtain the optimal configuration of parameters as predicted by the surrogate model. Compare this result with the actual evaluation of the black box system to determine the accuracy of the surrogate model's predictions.

Disclaimer

While surrogate models are effective in approximating the behavior of the black box system, there may be cases where the surrogate model doesn't precisely capture the system's shape or nuances. The model's predictions rely on existing data and assumptions inherent in the chosen kernel.

Conclusion

Bayesian optimization with surrogate models provides an efficient way to optimize complex and time-consuming processes. By using the surrogate model's predictions, you can make informed decisions about which points to evaluate, saving computational resources and time.

Feel free to experiment with different surrogate models, acquisition functions, sampling strategies, and configurations to find the best optimization approach for your specific problem.