Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lower Bounds from Fitness Levels Made Easy

Benjamin Doerr, Timo Kötzing

TL;DR

The paper develops a simple, intuitive variant of the fitness level method for lower bounds by incorporating visit probabilities of levels. By coupling leaving-probabilities $p_i$ with visit probabilities $v_i$, it derives bounds of the form $\mathbb{E}[T] \ge \sum_i \frac{v_i}{p_i}$ and demonstrates practical estimations for both quantities. Applying this to the (1+1) EA on LeadingOnes, OneMax, jump functions, and long $k$-paths yields tight or near-tight results, often matching or improving known bounds with clearer, more accessible arguments. The work offers a versatile tool for elitist EAs and lays groundwork for extensions to non-elitist settings, with potential broad impact on runtime analyses in evolutionary computation.

Abstract

One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the algorithm in increasing order, possibly skipping levels. An easy, but often strong upper bound for the run time can then be derived by adding the reciprocals of the probabilities to leave the levels (or upper bounds for these). Unfortunately, a similarly effective method for proving lower bounds has not yet been established. The strongest such method, proposed by Sudholt (2013), requires a careful choice of the viscosity parameters $γ_{i,j}$, $0 \le i < j \le n$. In this paper we present two new variants of the method, one for upper and one for lower bounds. Besides the level leaving probabilities, they only rely on the probabilities that levels are visited at all. We show that these can be computed or estimated without greater difficulties and apply our method to reprove the following known results in an easy and natural way. (i) The precise run time of the (1+1) EA on \textsc{LeadingOnes}. (ii) A lower bound for the run time of the (1+1) EA on \textsc{OneMax}, tight apart from an $O(n)$ term. (iii) A lower bound for the run time of the (1+1) EA on long $k$-paths. We also prove a tighter lower bound for the run time of the (1+1) EA on jump functions by showing that, regardless of the jump size, only with probability $O(2^{-n})$ the algorithm can avoid to jump over the valley of low fitness.

Lower Bounds from Fitness Levels Made Easy

TL;DR

The paper develops a simple, intuitive variant of the fitness level method for lower bounds by incorporating visit probabilities of levels. By coupling leaving-probabilities with visit probabilities , it derives bounds of the form and demonstrates practical estimations for both quantities. Applying this to the (1+1) EA on LeadingOnes, OneMax, jump functions, and long -paths yields tight or near-tight results, often matching or improving known bounds with clearer, more accessible arguments. The work offers a versatile tool for elitist EAs and lays groundwork for extensions to non-elitist settings, with potential broad impact on runtime analyses in evolutionary computation.

Abstract

One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the algorithm in increasing order, possibly skipping levels. An easy, but often strong upper bound for the run time can then be derived by adding the reciprocals of the probabilities to leave the levels (or upper bounds for these). Unfortunately, a similarly effective method for proving lower bounds has not yet been established. The strongest such method, proposed by Sudholt (2013), requires a careful choice of the viscosity parameters , . In this paper we present two new variants of the method, one for upper and one for lower bounds. Besides the level leaving probabilities, they only rely on the probabilities that levels are visited at all. We show that these can be computed or estimated without greater difficulties and apply our method to reprove the following known results in an easy and natural way. (i) The precise run time of the (1+1) EA on \textsc{LeadingOnes}. (ii) A lower bound for the run time of the (1+1) EA on \textsc{OneMax}, tight apart from an term. (iii) A lower bound for the run time of the (1+1) EA on long -paths. We also prove a tighter lower bound for the run time of the (1+1) EA on jump functions by showing that, regardless of the jump size, only with probability the algorithm can avoid to jump over the valley of low fitness.

Paper Structure

This paper contains 15 sections, 15 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The $(1 + 1)$ EA
  3. The Fitness Level Methods
  4. Level Processes
  5. Original Fitness Level Method
  6. Fitness Level Method with Viscosity
  7. Fitness Level Method with Visit Probabilities
  8. The Precise Run Time for Leading-Ones
  9. A Tight Lower Bound for OneMax
  10. Upper and Lower Bounds Via Fitness Levels
  11. Estimating the Fitness Level Estimate $\tilde{T}_{k,\ell}$
  12. Comparison with the Literature
  13. Jump Functions
  14. A Bound for Long $k$-Paths
  15. Conclusion

Key Result

Lemma 3

Let $(X_t)_t$ be a non-decreasing level process with underlying Markov chain $(Y_t)_t$ and level function $\ell$. Assume $X_t$ starts on some particular level. Let $p \in (0,1]$ be a lower bound on the probability for level process to leave this level regardless of the state of the underlying Markov

Theorems & Definitions (27)

  • Definition 1: Fitness-Based Partition Wegener02
  • Definition 2: Non-decreasing Level Process
  • Lemma 3
  • proof
  • Theorem 4: Fitness Level Method, upper bound Wegener02
  • Theorem 5: Fitness Level Method, lower bound Wegener02
  • Theorem 6: Fitness Level Method with Viscosity, lower bound Sudholt13
  • Theorem 7: Fitness Level Method with Viscosity, upper bound Sudholt13
  • Theorem 8: Fitness Level Method with visit probabilities, lower bound
  • proof
  • ...and 17 more