Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

TL;DR

This paper analyzes how well the NSGA-II algorithm approximates the Pareto front with small populations, identifies issues with its original crowding distance calculation, and proves the improved variants provide better coverage with theoretical guarantees.

Contribution

It provides the first mathematical analysis of NSGA-II's approximation capabilities and introduces variants that improve Pareto front coverage with proven bounds.

Findings

Original NSGA-II can produce large gaps in Pareto front coverage.

Variants updating crowding distance after each removal improve approximation.

Experimental results confirm the superior performance of the proposed variants.

Abstract

Recent theoretical works have shown that the NSGA-II efficiently computes the full Pareto front when the population size is large enough. In this work, we study how well it approximates the Pareto front when the population size is smaller. For the OneMinMax benchmark, we point out situations in which the parents and offspring cover well the Pareto front, but the next population has large gaps on the Pareto front. Our mathematical proofs suggest as reason for this undesirable behavior that the NSGA-II in the selection stage computes the crowding distance once and then removes individuals with smallest crowding distance without considering that a removal increases the crowding distance of some individuals. We then analyze two variants not prone to this problem. For the NSGA-II that updates the crowding distance after each removal (Kukkonen and Deb (2006)) and the steady-state NSGA-II (Nebro and Durillo (2009)), we prove that the gaps in the Pareto front are never more than a small constant factor larger than the theoretical minimum. This is the first mathematical work on the approximation ability of the NSGA-II and the first runtime analysis for the steady-state NSGA-II. Experiments also show the superior approximation ability of the two NSGA-II variants.