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## Bayesian Optimization and its Effectiveness

Bayesian optimization is a powerful optimization strategy for dealing with black-box objective functions. It has been widely used in various real-world applications, such as scientific discovery and experimental design. The strength of Bayesian optimization lies in its ability to efficiently explore and exploit the search space, leading to the discovery of global optima.

Traditionally, the performance of Bayesian optimization algorithms has been evaluated using regret-based metrics. These metrics, including instantaneous, simple, and cumulative regrets, solely rely on function evaluations. While they provide valuable insights into the effectiveness of the algorithms, they fail to consider important geometric relationships between query points and global solutions.

## The Limitations of Regret-Based Metrics

Regret-based metrics do not take into account the geometric properties of query points and global optima. For instance, they cannot differentiate between the discovery of a single global solution and multiple global solutions. Furthermore, these metrics do not assess the ability of Bayesian optimization algorithms to explore and exploit the search space effectively.

## The Introduction of Geometric Metrics

In order to address these limitations, the authors propose four new geometric metrics: precision, recall, average degree, and average distance. These metrics aim to quantify the geometric relationships between query points, global optima, and the search space itself. By considering both the positions of query points and global optima, these metrics offer a more comprehensive evaluation of Bayesian optimization algorithms.

### Precision:

Precision measures the proportion of correctly identified global optima among all identified optima. In other words, it evaluates how well the algorithm can locate global optima and avoid false positives.

### Recall:

Recall measures the proportion of correctly identified global optima compared to the total number of global optima present in the search space. This metric indicates how effectively the algorithm can discover all the true global optima.

### Average Degree:

Average degree quantifies the average number of global optima that a query point is connected to in the search space. It offers insights into the connectivity between query points and global solutions, helping to assess the algorithm’s exploration ability.

### Average Distance:

Average distance evaluates the average distance between query points and their assigned global optima. This metric signifies the efficiency of the algorithm in approaching and converging towards the global solutions.

## Parameter-Free Forms of Geometric Metrics

The proposed geometric metrics come with an additional parameter that needs to be determined carefully. Recognizing the importance of simplicity and ease of use, the authors introduce parameter-free forms of the geometric metrics. These forms remove the need for an additional parameter, making the metrics more accessible and practical for evaluation purposes.