[Paper Review] A Theoretical Framework of Approximation Error Analysis of Evolutionary Algorithms

Jun He, Yu Chen+1 more|arXiv (Cornell University)|Oct 26, 2018

Metaheuristic Optimization Algorithms Research22 references4 citations

TL;DR

This paper proposes a theoretical framework for analyzing approximation error in evolutionary algorithms (EAs) for discrete optimization by modeling EAs as homogeneous Markov chains. It provides exact expressions using Jordan form and Schur’s triangularization, and derives upper bounds via convergence rate analysis and auxiliary matrix iteration, enabling rigorous performance evaluation of EAs beyond empirical fitness values.

ABSTRACT

In the empirical study of evolutionary algorithms, the solution quality is evaluated by either the fitness value or approximation error. The latter measures the fitness difference between an approximation solution and the optimal solution. Since the approximation error analysis is more convenient than the direct estimation of the fitness value, this paper focuses on approximation error analysis. However, it is straightforward to extend all related results from the approximation error to the fitness value. Although the evaluation of solution quality plays an essential role in practice, few rigorous analyses have been conducted on this topic. This paper aims at establishing a novel theoretical framework of approximation error analysis of evolutionary algorithms for discrete optimization. This framework is divided into two parts. The first part is about exact expressions of the approximation error. Two methods, Jordan form and Schur's triangularization, are presented to obtain an exact expression. The second part is about upper bounds on approximation error. Two methods, convergence rate and auxiliary matrix iteration, are proposed to estimate the upper bound. The applicability of this framework is demonstrated through several examples.

Motivation & Objective

• To establish a rigorous theoretical foundation for analyzing approximation error in evolutionary algorithms, addressing the gap between empirical practice and theoretical analysis.
• To model evolutionary algorithms as homogeneous Markov chains to enable mathematical analysis of solution quality over time.
• To derive exact expressions for the expected approximation error $ e^{[t]} $ at generation $ t $, particularly for elitist EAs.
• To develop upper bound estimation techniques for $ e^{[t]} $ when exact expressions are infeasible, applicable to general and elitist EAs.
• To demonstrate the framework’s applicability through concrete examples, including the (1+1) EA and a random search process.

Proposed method

• Model EAs as homogeneous Markov chains with transition matrix $ \mathbf{P} $, where states represent solution quality levels.
• Use Jordan form and Schur’s triangularization to derive exact expressions for $ e^{[t]} $, expressed as $ e^{[t]} = \sum_{i=1}^{k} \sum_{m=1}^{L_i} c_{i_m} \binom{t}{L_i - m + 1} \lambda_i^{t - m + 1} $.
• Apply convergence rate analysis to bound $ e^{[t]} \leq c(\lambda_{\text{max}} + \epsilon)^t $, where $ \lambda_{\text{max}} $ is the spectral radius of the transition matrix.
• Introduce the auxiliary matrix iteration method combined with power factors to derive tighter bounds for upper triangular transition matrices: $ e^{[t]} \leq \sum_{i=1}^{L} c_i(\lambda_i + \epsilon_i)^t $.
• Define the approximation error as $ e^{[t]} = \mathbb{E}[|f(X^{[t]}) - f_{\text{opt}}|] $, and use matrix decomposition techniques to analyze its time evolution.
• Utilize eigenvalue decomposition and matrix power analysis to express error dynamics in terms of eigenvalues and coefficients of the transition matrix.

Experimental results

Research questions

• RQ1 What is the exact analytical expression for the expected approximation error $ e^{[t]} $ of an evolutionary algorithm at generation $ t $?
• RQ2 How can the convergence rate of the approximation error be theoretically bounded for general EAs?
• RQ3 Can tighter upper bounds on $ e^{[t]} $ be derived for elitist EAs using matrix decomposition and iterative methods?
• RQ4 How do matrix properties such as diagonalizability or triangularization affect the structure of the error expression?
• RQ5 What is the relationship between the spectral radius of the transition matrix and the long-term decay rate of the approximation error?

Key findings

• The exact expression for $ e^{[t]} $ is derived as $ e^{[t]} = \sum_{i=1}^{k} \sum_{m=1}^{L_i} c_{i_m} \binom{t}{L_i - m + 1} \lambda_i^{t - m + 1} $, where $ \lambda_i $ are eigenvalues of the reduced transition matrix $ \mathbf{R} $.
• For diagonalizable transition matrices, the expression simplifies to $ e^{[t]} = \sum_{i=1}^{L} c_i \lambda_i^t $, enabling direct eigenvalue-based analysis.
• The convergence rate method yields an upper bound $ e^{[t]} \leq c(\lambda_{\text{max}} + \epsilon)^t $, valid for all EAs, with $ \lambda_{\text{max}} $ being the spectral radius of $ \mathbf{R} $.
• For upper triangular transition matrices, the auxiliary matrix iteration method provides a tighter bound: $ e^{[t]} \leq \sum_{i=1}^{L} c_i(\lambda_i + \epsilon_i)^t $, where $ \epsilon_i $ can be made arbitrarily small.
• In the (1+1) EA example, the bound $ e^{[t]} \leq [f(n) - f(n-1)](1 - \frac{1}{ne})^t $ is derived when starting from a single-bit-flip solution.
• The framework is demonstrated on a random search process, showing that $ e^{[t]} \leq 2[f(n)-f(n-1)](1 - \frac{1}{ne})^t + [f(n)-f(n-2)](1 - \frac{2}{ne})^t $ under a two-individual initial distribution.