Abstract
In this paper we show how to use drift analysis in the case of two random variables \(X_1, X_2\), when the drift is approximatively given by \(A\cdot (X_1,X_2)^T\) for a matrix A. The non-trivial case is that \(X_1\) and \(X_2\) impede each other’s progress, and we give a full characterization of this case. As application, we develop and analyze a minimal example TwoLinof a dynamic environment that can be hard. The environment consists of two linear function \(f_1\) and \(f_2\) with positive weights 1 and n, and in each generation selection is based on one of them at random. They only differ in the set of positions that have weight 1 and n. We show that the \((1 + 1)\)-EAwith mutation rate \(\chi /n\) is efficient for small constant \(\chi \) on TwoLin, but does not find the shared optimum in polynomial time for large constant \(\chi \).
Extended Abstract. All proofs and further details are available on arxiv [11].
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Notes
- 1.
- 2.
We follow the convention that we always call drift towards the optimum positive, and drift away from the optimum negative. Since the optimum in our case is at 0, this means that we consider the difference \(X^{t}-X^{t+1}\) for the drift, not vice versa.
- 3.
For direct comparison it is important to note that [28] works with the matrix \(I-A\) instead of A, where I is the identity matrix.
- 4.
This is because for any matrix M of norm 1 we can write \(\lambda _1(A+\beta M) = \lambda _1(A) + \beta D{\lambda _1}(A)\cdot M + O(\beta ^2)\), where the total differential \(D\lambda _1(A)\) has bounded norm, and analogously for the other eigenvalues and eigenvectors.
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Janett, D., Lengler, J. (2022). Two-Dimensional Drift Analysis:. In: Rudolph, G., Kononova, A.V., Aguirre, H., Kerschke, P., Ochoa, G., Tušar, T. (eds) Parallel Problem Solving from Nature – PPSN XVII. PPSN 2022. Lecture Notes in Computer Science, vol 13399. Springer, Cham. https://doi.org/10.1007/978-3-031-14721-0_43
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