Rigorous Runtime Analysis of Permutation-based Evolutionary Algorithms on Classic Benchmarks

Core Concepts

The author proposes a general way to translate classic pseudo-Boolean optimization benchmarks into permutation-based problems, and conducts a rigorous runtime analysis of the permutation-based (1+1) EA on the analogues of the LeadingOnes and Jump benchmarks. The analysis reveals new challenges and insights compared to the bit-string case, leading to the suggestion of using scramble mutations over swap mutations, and heavy-tailed mutation strengths.

Abstract

The author first proposes a general method to transform classic pseudo-Boolean optimization benchmarks, such as OneMax, LeadingOnes, and Jump, into permutation-based problems. This allows for a systematic study of permutation-based evolutionary algorithms (EAs) using well-established benchmark problems. The author then conducts a detailed runtime analysis of the permutation-based (1+1) EA on the permutation versions of the LeadingOnes and Jump benchmarks: LeadingOnes Benchmark: The author proves a Θ(n^3) runtime for the permutation-based (1+1) EA on the permutation-LeadingOnes problem. This is higher by a factor of Θ(n) compared to the bit-string case, due to the lower probability of fitness improvements in the permutation setting. Jump Benchmark: The analysis reveals that the probability of jumping from a local optimum to the global optimum depends critically on the cycle structure of the local optimum, ranging from Θ(n^(-2(m-1))) to Θ(n^(-2⌈m/2⌉)), where m is the jump size. By analyzing the random walk on the plateau of local optima, the author proves an upper bound of Θ(n^2⌈m/2⌉) on the runtime. The author also considers a variant of the scramble mutation operator, which leads to simpler proofs and better runtime bounds, especially for jump functions with odd jump size. Furthermore, a heavy-tailed version of the scramble operator is shown to provide a speed-up of order mΘ(m) on jump functions with jump size m. Finally, a short empirical analysis confirms the theoretical findings and reveals that small implementation details, such as the rate of void mutations, can have a significant impact on the performance.

Stats

The probability of a fitness improvement in each iteration of the permutation-based (1+1) EA on the PLeadingOnes benchmark is at most 6/(n-1)^2. The probability that the permutation-based (1+1) EA increases the fitness by at least 3 in a single iteration is at most 44/(n-1)^3. The probability that one iteration of the (1+1) EA changes the number of cycles in the cycle decomposition of a local optimum is at most 3(m/(n-1))^2.

Quotes

"Different from the bit-string case, also in the optimization of a permutation-based jump function, the most difficult step is to mutate a local optimum and say that for a jump into the global optimum, which is the only improving solution here. This requires flipping m particular bits in the bit-string case and permuting m particular elements in the permutation-case, where m is the jump size parameter of the jump function." "From our results on Jump functions, we would rather suggest to use scramble mutations than swap mutations, and rather with a heavy-tailed mutation strength than with a Poisson distributed one."

Key Insights Distilled From

Runtime Analysis for Permutation-based Evolutionary Algorithms

by Benjamin Doe... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2207.04045.pdf

Deeper Inquiries

What other types of permutation-based problems (beyond the assignment type considered here) could benefit from a systematic translation of classic pseudo-Boolean benchmarks

Beyond the assignment type problems discussed in the context, other types of permutation-based problems that could benefit from a systematic translation of classic pseudo-Boolean benchmarks include order-based and adjacency-based problems. In order-based problems, the optimization involves respecting precedence relations or profitable orderings, such as in production scheduling where jobs need to be placed in an optimal order on a machine. For adjacency-based problems, the focus is on ensuring certain items are placed next to each other, which is common in scenarios like the traveling salesman problem where the order of visiting locations is crucial. By translating classic benchmarks into these problem types, researchers can establish a diverse set of benchmarks that cover various aspects of permutation-based optimization problems.

How can the insights from the analysis of the permutation-based (1+1) EA be leveraged to design more sophisticated permutation-based evolutionary algorithms with improved performance

The insights gained from the analysis of the permutation-based (1+1) EA can be instrumental in designing more sophisticated permutation-based evolutionary algorithms with enhanced performance. Some ways to leverage these insights include: Mutation Operator Design: The analysis highlights the importance of the mutation operator in navigating permutation spaces efficiently. By experimenting with different mutation strategies, such as swap operations and scramble mutations, algorithm designers can tailor mutation operators to suit the specific characteristics of permutation-based problems. Local Optima Handling: Understanding the behavior around local optima, as seen in the analysis, can guide the development of strategies to escape local optima effectively. Techniques like diversification mechanisms or intelligent restart strategies can be incorporated to improve the algorithm's exploration capabilities. Adaptive Parameter Tuning: The runtime analysis provides insights into the impact of parameters on algorithm performance. Utilizing adaptive parameter tuning mechanisms based on the problem instance characteristics can help optimize the algorithm's behavior dynamically during the optimization process. By incorporating these insights, researchers can design permutation-based evolutionary algorithms that are more robust, efficient, and effective in solving a wide range of permutation-based optimization problems.

Are there natural permutation-based problems that cannot be easily obtained from the translation of bit-string benchmarks, and what new challenges would they pose for the theoretical analysis of permutation-based EAs

There are natural permutation-based problems that may not lend themselves easily to translation from bit-string benchmarks. One such example could be problems involving complex constraints or interdependencies among elements in the permutation. These problems may require specific problem structures or specialized operators that are not directly translatable from bit-string representations. Challenges posed by these types of problems for the theoretical analysis of permutation-based evolutionary algorithms include: Complexity of Interactions: Problems with intricate interdependencies among elements may lead to a combinatorially rich search space, making it challenging to analyze the impact of mutations or crossovers on the solution quality. Non-linear Fitness Landscapes: The presence of non-linear fitness landscapes in certain permutation-based problems can introduce difficulties in predicting algorithm behavior and convergence properties. Constraint Handling: Dealing with constraints in permutation-based problems, especially those that cannot be easily modeled using traditional pseudo-Boolean benchmarks, may require novel approaches for constraint handling and feasibility enforcement. Addressing these challenges would require innovative algorithmic techniques, tailored problem representations, and advanced analytical methods to understand the behavior and performance of permutation-based evolutionary algorithms on such complex problems.