Fast Genetic Algorithms (1703.03334v2)

Abstract: For genetic algorithms using a bit-string representation of length~$n$, the general recommendation is to take $1/n$ as mutation rate. In this work, we discuss whether this is really justified for multimodal functions. Taking jump functions and the $(1+1)$ evolutionary algorithm as the simplest example, we observe that larger mutation rates give significantly better runtimes. For the $\jump_{m,n}$ function, any mutation rate between $2/n$ and $m/n$ leads to a speed-up at least exponential in $m$ compared to the standard choice. The asymptotically best runtime, obtained from using the mutation rate $m/n$ and leading to a speed-up super-exponential in $m$, is very sensitive to small changes of the mutation rate. Any deviation by a small $(1 \pm \eps)$ factor leads to a slow-down exponential in $m$. Consequently, any fixed mutation rate gives strongly sub-optimal results for most jump functions. Building on this observation, we propose to use a random mutation rate $\alpha/n$, where $\alpha$ is chosen from a power-law distribution. We prove that the $(1+1)$ EA with this heavy-tailed mutation rate optimizes any $\jump_{m,n}$ function in a time that is only a small polynomial (in~$m$) factor above the one stemming from the optimal rate for this $m$. Our heavy-tailed mutation operator yields similar speed-ups (over the best known performance guarantees) for the vertex cover problem in bipartite graphs and the matching problem in general graphs. Following the example of fast simulated annealing, fast evolution strategies, and fast evolutionary programming, we propose to call genetic algorithms using a heavy-tailed mutation operator \emph{fast genetic algorithms}.

• The paper challenges the traditional 1/n mutation rate in genetic algorithms for multimodal optimization, demonstrating exponential speed-ups with larger rates (2/n to m/n) on jump functions.
• Research reveals that optimal mutation rates are highly sensitive to problem instances, rendering any single fixed rate sub-optimal and leading to exponential slowdowns if deviated.
• Authors propose using a heavy-tailed, random mutation rate derived from a power-law distribution, significantly enhancing performance on problems like vertex cover and matching by providing near-universal optimality.

Analysis of Mutation Dynamics in Genetic Algorithms for Multimodal Optimization

The paper "Fast Genetic Algorithms" explores an innovative approach to mutation in genetic algorithms (GAs), particularly when dealing with multimodal functions. The common practice in GAs utilizing bit-string representations has been to use a mutation rate of $1/n$, where $n$ is the length of the bit-string. This rate is traditionally considered optimal for problems like the OneMax function. However, the authors challenge this conventional wisdom by demonstrating that large mutation rates can significantly improve runtimes for multimodal functions, using the $(1+1)$ evolutionary algorithm (EA) and jump functions as a paradigm.

Key Findings:

1. Mutation Rate Effectiveness: The analysis reveals that for the $Jump_{m,n}$ function, mutation rates between $2/n$ and $m/n$ provide exponential speed-ups over the standard $1/n$ rate. This finding emphasizes the inefficiency of fixed mutation rates for complex multimodal optimization problems.
2. Sensitivity to Mutation Rate: The research outlines that the best asymptotic runtime using the mutation rate $m/n$ offers a super-exponential speed-up in $m$. However, this optimal rate is highly sensitive to deviation. Even slight variations lead to exponential slowdowns, rendering any fixed mutation rate sub-optimal for most jump functions.
3. Proposing Heavy-Tailed Mutation Rates: Building on these insights, the authors propose using a random mutation rate derived from a power-law distribution, facilitating both diverse and large mutations when required. They introduce a probabilistic mutation rate $\alpha/n$, with $\alpha$ randomly selected according to this distribution.
4. Performance on Various Tasks: The newly proposed heavy-tailed mutation operator significantly enhances performance on the vertex cover problem in bipartite graphs and on matching problems in general graphs. The runtime obtained using this operator is only slightly above the optimal for each instance size $m$ by a polynomial factor, thus providing a near-universal optimality across varying problem instances.

Theoretical and Practical Implications:

The research highlights the importance of adaptive mutation strategies that align more closely with the problem's landscape rather than adhering to generic rules. This shift in strategy is crucial for effectively tackling complex multimodal problems, where the optimization landscape can trap algorithms at local optima.

Theoretically, this opens avenues for further analysis of dynamic evolutionary strategies that can adapt based on problem characteristics rather than static parameters. Practically, adopting heavy-tailed mutation operators in real-world applications can potentially elevate the performance of GAs, especially in scenarios where conventional mutation rates fail to offer significant gains.

Future Directions:

This work serves as a foundation for extending heavy-tailed mutations to other types of representations beyond bit-strings and to more diverse problem sets. Exploring the broad spectrum of mutation distributions in conjunction with other GA parameters could create more robust, all-encompassing algorithms suitable for diverse optimization challenges.

Overall, this research makes a substantial contribution to the understanding and optimization of mutation strategies in genetic algorithms, presenting a notable shift towards more flexible and adaptive methods that consider the problem's intricacies, particularly in the context of multimodal landscapes.