Analyzing Single-objective Black-box Optimization Algorithms Using the Empirical Attainment Function

Core Concepts

The empirical attainment function (EAF) provides a more comprehensive and precise way to analyze the performance of single-objective black-box optimization algorithms compared to the commonly used target-based empirical cumulative distribution function (ECDF).

Abstract

The paper discusses the use of the empirical attainment function (EAF) for analyzing the performance of single-objective black-box optimization algorithms. The key insights are:

The target-based ECDF is an approximation of the EAF, and the EAF-based ECDF converges to the EAF as the number of well-spread targets increases.
The area under the curve (AUC) of the EAF-based ECDF is equivalent to the AUC of the EAF, which is also equivalent to the mean area over the convergence curves (AOCC) of individual runs. This means the AOCC can be used as a simpler-to-calculate measure of anytime performance.
The EAF provides additional insights beyond the ECDF, such as the ability to visualize percentile convergence curves and compute the Vorob'ev expectation as a summary statistic.
The authors have integrated the computation of the EAF into the IOHanalyzer platform to facilitate its use in practice.
Empirical analysis on the BBOB benchmark functions shows that the target-based ECDF can over- or under-estimate the EAF-based ECDF, especially with a small number of targets, and this can lead to differences in algorithm rankings.

Stats

The paper presents the following key figures and statistics:

The EAF of the BFGS and CMA-ES algorithms on the 24 10-dimensional BBOB functions (Figures 3 and 4).
The difference between the target-based ECDF and the EAF-based ECDF for different numbers of targets, averaged over 211 algorithms (Figure 8).
The distribution of differences between the AUC of the EAF-based ECDF and the AUC of the target-based ECDF, for different numbers of targets and problem dimensions (Figure 9).
The rank differences between the AUC-based rankings of 211 algorithms using the EAF-based ECDF and the target-based ECDF with 51 targets (Figure 10).

Quotes

"The EAF has several advantages over the target-based ECDF. In particular, it does not require defining a priori quality targets per function, captures performance differences more precisely, and enables the use of additional summary statistics that enrich the analysis."
"The area under the curve (AUC) of the EAF-based ECDF would be a more precise measure of anytime performance."
"The AUC of the EAF-based ECDF is equivalent to the AUC of the EAF itself, which is also equivalent to the mean area over the convergence curves (AOCC) of the individual runs."

Key Insights Distilled From

Using the Empirical Attainment Function for Analyzing Single-objective Black-box Optimization Algorithms

by Manu... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.02031.pdf

Deeper Inquiries

How can the EAF be extended to handle multi-objective optimization problems with more than two objectives

To extend the Empirical Attainment Function (EAF) to handle multi-objective optimization problems with more than two objectives, we need to consider the concept of Pareto dominance. In multi-objective optimization, a solution is considered better than another if it is not worse in any objective and strictly better in at least one objective.
When dealing with more than two objectives, the EAF can be extended to calculate the probability of attaining a specific region in the objective space within a given budget of function evaluations. This involves considering the Pareto front, which represents the set of non-dominated solutions in the objective space. The EAF can then be used to estimate the probability of finding solutions that lie within certain regions of the Pareto front within a specified runtime.
By analyzing the EAF in the context of multi-objective optimization, we can gain insights into the algorithm's performance in terms of finding diverse and well-distributed solutions across the Pareto front, providing a more comprehensive evaluation of the algorithm's effectiveness in handling multiple conflicting objectives.

What other summary statistics derived from the EAF could be useful for analyzing the performance of single-objective black-box optimization algorithms

Several summary statistics derived from the Empirical Attainment Function (EAF) can be useful for analyzing the performance of single-objective black-box optimization algorithms:

Vorob’ev Expectation: This statistic represents the "mean" convergence curve and can be used to summarize the performance of an algorithm across multiple runs. It provides a synthetic representation of the expected performance based on the EAF quantiles.

Second-Order EAF: This statistic measures the probability of attaining an objective function value z not later than t, given that another objective value z' was attained not later than t'. It helps diagnose convergence issues and changes in algorithm behavior during optimization.

Dispersion Statistics: By analyzing the dispersion of the EAF, we can determine the probability of a single run deviating from the mean convergence curve represented by the Vorob’ev expectation. This can provide insights into the variability of algorithm performance.

Level Sets as Quantile Functions: The level sets of the EAF can be interpreted as quantile-like functions, representing different percentiles of convergence. These can be used to analyze the distribution of algorithm performance across different levels of achievement.

By utilizing these summary statistics derived from the EAF, researchers and practitioners can gain a deeper understanding of algorithm behavior, convergence patterns, and robustness in single-objective optimization scenarios.

How can the EAF-based analysis be integrated into the design and tuning of optimization algorithms to improve their anytime performance

Integrating Empirical Attainment Function (EAF)-based analysis into the design and tuning of optimization algorithms can significantly improve their anytime performance. Here are some ways this integration can be beneficial:

Parameter Tuning: By using the EAF to analyze the performance of optimization algorithms under different parameter settings, researchers can identify optimal configurations that lead to improved convergence and robustness. The EAF can guide the tuning process by providing insights into how changes in parameters affect the algorithm's performance over time.

Algorithm Comparison: EAF-based analysis can be used to compare the performance of different optimization algorithms in terms of their anytime performance. By evaluating the EAF curves and associated statistics, researchers can identify strengths and weaknesses of each algorithm and make informed decisions about which algorithm to use for specific optimization tasks.

Performance Monitoring: Integrating EAF-based analysis into the optimization algorithm runtime can enable real-time monitoring of performance. Algorithms can adapt their behavior based on the EAF feedback, adjusting their strategies to improve convergence rates and overall optimization efficiency.

Benchmarking and Evaluation: EAF-based metrics can serve as standard benchmarks for evaluating optimization algorithms. By incorporating EAF analysis into benchmarking platforms, researchers can establish a common framework for assessing algorithm performance and promoting advancements in optimization research.

Overall, integrating EAF-based analysis into the design and tuning of optimization algorithms provides a systematic and data-driven approach to improving their anytime performance, leading to more effective and efficient optimization processes.