Self-adjusting offspring population sizes outperform fixed parameters on the cliff function

Mario Alejandro Hevia Fajardo, Dirk Sudholt

Abstract: In the discrete domain, self-adjusting parameters of evolutionary algorithms (EAs) have emerged as a fruitful research area with many runtime analyses showing that self-adjusting parameters can outperform the best fixed parameters. Most existing runtime analyses focus on elitist EAs on simple problems, for which moderate performance gains were shown. Here we consider a much more challenging scenario: the multimodal function Cliff, defined as an example where a (1,λ)<math><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo is="true">,</mo><mi is="true">λ</mi><mo stretchy="false" is="true">)</mo></math> EA is effective, and for which the best known upper runtime bound for standard EAs is O(n25)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">25</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math>.We prove that a (1,λ)<math><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo is="true">,</mo><mi is="true">λ</mi><mo stretchy="false" is="true">)</mo></math> EA self-adjusting the offspring population size λ using success-based rules optimises Cliff in O(n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> expected generations and O(nlog⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> expected evaluations. Along the way, we prove tight upper and lower bounds on the runtime for fixed λ (up to a logarithmic factor) and identify the runtime for the best fixed λ as nη<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">η</mi></mrow></msup></math> for η≈3.97677<math><mi is="true">η</mi><mo is="true">≈</mo><mn is="true">3.97677</mn></math> (up to sub-polynomial factors). Hence, the self-adjusting (1,λ)<math><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo is="true">,</mo><mi is="true">λ</mi><mo stretchy="false" is="true">)</mo></math> EA outperforms the best fixed parameter by a factor of at least n2.9767<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2.9767</mn></mrow></msup></math>.