Near-Tight Runtime Guarantees for Many-Objective Evolutionary Algorithms

2024-04-19

Parallel Problem Solving from Nature

AI Summary

The theoretical runtime performance of multi-objective evolutionary algorithms (MOEAs) on discrete optimization problems remains poorly understood. Method: We conduct rigorous mathematical runtime analysis, integrate structural characterizations of Pareto fronts, and leverage recent structural lemmas specific to NSGA-III, establishing near-tight upper bounds on expected optimization time for classic MOEAs—including SEMO, global SEMO, SMS-EMOA, and NSGA-III—on four canonical benchmark problems (e.g., OneMinMax). Contribution/Results: Our analysis breaks the previously known quadratic dependence of runtime bounds on Pareto front size, achieving instead a linear dependence. This is the first such result extended to arbitrary numbers of objectives (high-dimensional objective spaces). The derived bounds involve only low-degree polynomial factors in bit-string length and number of objectives. These findings demonstrate that mainstream MOEAs exhibit significantly stronger scalability under multi-objective extensions than previously recognized, thereby establishing new theoretical benchmarks for algorithm design and analysis.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Despite significant progress in the field of mathematical runtime analysis of multi-objective evolutionary algorithms (MOEAs), the performance of MOEAs on discrete many-objective problems is little understood. In particular, the few existing bounds for the SEMO, global SEMO, and SMS-EMOA algorithms on classic benchmarks are all roughly quadratic in the size of the Pareto front. In this work, we prove near-tight runtime guarantees for these three algorithms on the four most common benchmark problems OneMinMax, CountingOnesCountingZeros, LeadingOnesTrailingZeros, and OneJumpZeroJump, and this for arbitrary numbers of objectives. Our bounds depend only linearly on the Pareto front size, showing that these MOEAs on these benchmarks cope much better with many objectives than what previous works suggested. Our bounds are tight apart from small polynomial factors in the number of objectives and length of bitstrings. This is the first time that such tight bounds are proven for many-objective uses of these MOEAs. While it is known that such results cannot hold for the NSGA-II, we do show that our bounds, via a recent structural result, transfer to the NSGA-III algorithm.

Problem

• Analyzes runtime of MOEAs on many-objective problems
• Proves near-tight bounds for common benchmark problems
• Shows linear dependency on incomparable set size

Innovation

• Proves near-tight runtime guarantees for MOEAs
• Analyzes performance on common benchmark problems
• Bounds depend linearly on incomparable set size