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Overcoming Binary Adversarial Optimisation with Competitive Coevolution

Per Kristian Lehre, Shishen Lin

TL;DR

This work tackles binary test-based adversarial optimisation by introducing the Diagonal problem and analyzing two evolutionary paradigms. It proves that a traditional (1,λ)-EA cannot efficiently reach an $ε$-approximation of the Maximin optimum, while a competitive (1,λ)-CoEA with alternating updates attains polynomial-time convergence under suitable settings, via a two-phase drift analysis that leverages crossing the diagonal and c-tubes. The authors develop a rigorous toolkit (including a Characteristic Lemma, Phase 1 and Phase 2 analyses, and a restart argument) to bound runtime, supported by experiments with $n=λ=1000$ showing the practical viability of large λ and moderate mutation rates (notably around $χ=0.6$). The findings highlight coevolution’s potential for binary adversarial optimisation and provide concrete guidance on when and how to apply CoEAs to such flat, high-contrast landscapes. Overall, the paper makes a significant theoretical and empirical contribution by establishing polynomial-time solvability of a binary Maximin problem via competitive coevolution and outlining future directions for more general, boundary-driven problems.

Abstract

Co-evolutionary algorithms (CoEAs), which pair candidate designs with test cases, are frequently used in adversarial optimisation, particularly for binary test-based problems where designs and tests yield binary outcomes. The effectiveness of designs is determined by their performance against tests, and the value of tests is based on their ability to identify failing designs, often leading to more sophisticated tests and improved designs. However, CoEAs can exhibit complex, sometimes pathological behaviours like disengagement. Through runtime analysis, we aim to rigorously analyse whether CoEAs can efficiently solve test-based adversarial optimisation problems in an expected polynomial runtime. This paper carries out the first rigorous runtime analysis of $(1,λ)$ CoEA for binary test-based adversarial optimisation problems. In particular, we introduce a binary test-based benchmark problem called \Diagonal problem and initiate the first runtime analysis of competitive CoEA on this problem. The mathematical analysis shows that the $(1,λ)$-CoEA can efficiently find an $\varepsilon$ approximation to the optimal solution of the \Diagonal problem, i.e. in expected polynomial runtime assuming sufficiently low mutation rates and large offspring population size. On the other hand, the standard $(1,λ)$-EA fails to find an $\varepsilon$ approximation to the optimal solution of the \Diagonal problem in polynomial runtime. This suggests the promising potential of coevolution for solving binary adversarial optimisation problems.

Overcoming Binary Adversarial Optimisation with Competitive Coevolution

TL;DR

This work tackles binary test-based adversarial optimisation by introducing the Diagonal problem and analyzing two evolutionary paradigms. It proves that a traditional (1,λ)-EA cannot efficiently reach an -approximation of the Maximin optimum, while a competitive (1,λ)-CoEA with alternating updates attains polynomial-time convergence under suitable settings, via a two-phase drift analysis that leverages crossing the diagonal and c-tubes. The authors develop a rigorous toolkit (including a Characteristic Lemma, Phase 1 and Phase 2 analyses, and a restart argument) to bound runtime, supported by experiments with showing the practical viability of large λ and moderate mutation rates (notably around ). The findings highlight coevolution’s potential for binary adversarial optimisation and provide concrete guidance on when and how to apply CoEAs to such flat, high-contrast landscapes. Overall, the paper makes a significant theoretical and empirical contribution by establishing polynomial-time solvability of a binary Maximin problem via competitive coevolution and outlining future directions for more general, boundary-driven problems.

Abstract

Co-evolutionary algorithms (CoEAs), which pair candidate designs with test cases, are frequently used in adversarial optimisation, particularly for binary test-based problems where designs and tests yield binary outcomes. The effectiveness of designs is determined by their performance against tests, and the value of tests is based on their ability to identify failing designs, often leading to more sophisticated tests and improved designs. However, CoEAs can exhibit complex, sometimes pathological behaviours like disengagement. Through runtime analysis, we aim to rigorously analyse whether CoEAs can efficiently solve test-based adversarial optimisation problems in an expected polynomial runtime. This paper carries out the first rigorous runtime analysis of CoEA for binary test-based adversarial optimisation problems. In particular, we introduce a binary test-based benchmark problem called \Diagonal problem and initiate the first runtime analysis of competitive CoEA on this problem. The mathematical analysis shows that the -CoEA can efficiently find an approximation to the optimal solution of the \Diagonal problem, i.e. in expected polynomial runtime assuming sufficiently low mutation rates and large offspring population size. On the other hand, the standard -EA fails to find an approximation to the optimal solution of the \Diagonal problem in polynomial runtime. This suggests the promising potential of coevolution for solving binary adversarial optimisation problems.
Paper Structure (16 sections, 26 theorems, 56 equations, 3 figures, 2 algorithms)

This paper contains 16 sections, 26 theorems, 56 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diagonal Games
  4. Drift Analysis Toolbox
  5. Traditional Evolutionary Algorithm cannot Solve Diagonal efficiently
  6. Competitive Coevolution Solves Diagonal Efficiently
  7. Characteristic Lemma for Alternating Update
  8. Phase 1
  9. Phase 2
  10. Phase 2.1
  11. Phase 2.2
  12. Phase 2.3
  13. Experiments
  14. Settings
  15. Results
  16. ...and 1 more sections

Key Result

theorem thmcountertheorem

For constants $a,b,\delta, \eta, r>0$, with $a<b$, there exist $c>0$, $n_{0}\in \mathbb{N}$ such that the following holds for all $n\geq n_{0}$. Suppose $(X_{t})_{t\geq 0}$ is a sequence of random variables with a finite state space $S\subset \mathbb{R}_{0}^{+}$ and with associated filtration $\math Then, $\Pr[T_{a} \leq e^{cn}] \leq e^{-cn}.$

Figures (3)

  • Figure 1: Example of Diagonal problem. The horizontal axis represents the number of $1$-bits in the designs $x$, and the vertical axis represents the number of $1$-bits in the test cases $y$. The grey area represents search points of payoff $1$, and the rest represents search points with payoff $0$.
  • Figure 2: Runtime of $(1,\lambda)$-CoEA on Diagonal. Fig. \ref{['fig:Diagonal1']} (left): Runtime against different mutation rates under $n=\lambda=1000$. Fig. \ref{['fig:Diagonal2']} (right): Runtime against specific $\chi =0.6$. The red curve is $f(n,\lambda)=6 \lambda n$ where in this case we set $n=\lambda$.
  • Figure 3: Sketch of proof idea for Theorem \ref{['thm:Main0']}. The offspring are sorted according to fitness. We denote the number of offspring with fitness $1$ by $N$.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • theorem thmcountertheorem: Negative Drift Theorem oliveto_erratum_2012rowe_choice_2012doerr2019theory
  • theorem thmcountertheorem: Additive Drift Theorem he_drift_2001doerr2019theory
  • theorem thmcountertheorem
  • Lemma 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 2
  • ...and 33 more