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R2 v2: The Pareto-compliant R2 Indicator for Better Benchmarking in Bi-objective Optimization

Created by
  • Haebom

Author

Lennart Sch apermeier, Pascal Kerschke

Outline

This paper revisits the R2 metric, a widely used metric in multi-objective optimization. Existing R2 metrics typically utilize discretized utility function distributions and suffer from weak Pareto fit. In this study, we analyze the properties of the R2 metric using a Chebyshev utility function with a continuous uniform distribution. We demonstrate that this continuous variant is Pareto-fitting and present an efficient computational procedure. Specifically, (a) it has a computational complexity of $\mathcal O(N \log N)$ for bivariate problems, and (b) the metric can be incrementally updated without recomputing the entire set when solutions are added or removed. This presents an efficient and promising alternative to existing Pareto-fitting unary performance metrics, such as the hypervolume metric.

Takeaways, Limitations

Takeaways:
The continuous transformed R2 metric ensures Pareto fit.
It provides an efficient computational method of $\mathcal O (N \log N)$ for bivariate problems.
Progressive updates allow you to efficiently update metrics as you add or remove solutions.
It presents an efficient alternative to hypervolume metrics.
Limitations:
The specific Limitations is not specified in the paper. (However, judging from the abstract, the potential Limitations may be the increased computational complexity due to the increased problem dimensionality.)
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