Key Concepts in Bayesian Optimization: Acquisition Functions

Role of Acquisition Functions in Bayesian Optimization

In Bayesian optimization, acquisition functions play a crucial role in guiding the search process to efficiently locate the optimal solution in an unknown objective function space. The core idea behind Bayesian Optimization is to model the objective function using a surrogate model. This probabilistic model provides a mean and a variance prediction for the function value at any given point, giving us both an estimate of the function's behavior and a measure of uncertainty.

An acquisition function leverages this probabilistic model to determine the next point to evaluate, balancing exploration (sampling in regions with high uncertainty) and exploitation (sampling in regions likely to yield high values based on current knowledge). The acquisition function thus provides a heuristic for where the optimizer should sample next, aiming to minimize the number of evaluations needed to find the optimum.

Choosing the next set of input points in Bayesian Optimization amounts to finding the input points that maximize the acquisition function.

Balancing Exploration and Exploitation

The exploration-exploitation trade-off is a crucial concept for acquisition functions.

Exploitation involves sampling points where the model predicts high performance, focusing on regions that seem promising based on current knowledge (data). This strategy seeks to improve the known best result quickly. However, it risks missing out on potentially better regions that haven’t been thoroughly explored.

On the other hand, exploration means sampling points in areas with high uncertainty, where the model has less information. This approach helps discover new areas of the search space that might have better solutions, even if they don’t necessarily look promising based on current knowledge. However, exploring too much can lead to unnecessary evaluations in low-value areas.

Effective acquisition functions balance these two approaches, strategically selecting points that either promise a high result (exploitation) or provide valuable information (exploration).

We will look at one such example, the Expected Improvement (EI) acquisition function.

Expected Improvement as the Acquisition Function

The Expected Improvement (EI) acquisition function guides the search for an optimal solution in Bayesian Optimization by estimating the potential improvement over the current best-known value. 

Roughly speaking, EI balances two strategies by adding two components corresponding to exploitation and exploration.  The exploitation contribution gives a high value where the predicted value is higher than the current best value in data.  The exploration contribution gives a high value where there is high uncertainty in the model.  The two contributions are added to form a balanced trade-off between the two.

This trade-off makes the Bayesian Optimization using EI as the acquisition function to be computationally efficient, requiring fewer evaluations while still exploring strategically.

Limitations and Considerations

While the Expected Improvement (EI) acquisition function is popular for its efficiency and balanced search strategy, it has limitations and considerations that impact its performance.

• Over-exploration in Bad Models: In problems where uncertainty is consistently high across the search space, EI can overly favor exploration, leading to excessive evaluations in less promising areas. This is the case when the surrogate model is not able to accurately model the data. The acquisition function is effective when we have a good model.
• Exploitation Bias due to Premature Convergence: In functions with highly variable landscapes, EI may overly exploit known high-value regions, converging prematurely on local optima. In such cases, the acquisition function can be tuned to adjust the weights which are used in balancing the exploration-exploitation trade-off.
• Sensitivity to the Current Best Solution: EI relies heavily on the best-known result. If this is suboptimal or inaccurately estimated, EI may focus on regions that are less promising. It is therefore important that the best value in the data is measured as accurately as possible. Another way is to use an alternate acquisition function strategy that properly takes into account the presence of noise in the data.

These limitations highlight that while EI is versatile, it may require careful tuning or alternative acquisition functions depending on the problem’s complexity and dimensionality.

Conclusion

Acquisition functions are essential for efficient Bayesian optimization by guiding sampling decisions to minimize evaluations. The Expected Improvement (EI) acquisition function is particularly effective, balancing exploration of uncertain areas with exploitation of high-potential regions to efficiently approach the optimum. A good choice of the acquisition functions in Bayesian Optimization helps optimize complex functions with fewer evaluations and minimal resource use.