Abstract
It is known that step size adaptive evolution strategies (ES) do not converge (prematurely) to regular points of continuously differentiable objective functions. Among critical points, convergence to minima is desired, and convergence to maxima is easy to exclude. However, surprisingly little is known on whether ES can get stuck at a saddle point. In this work we establish that even the simple (1+1)-ES reliably overcomes most saddle points under quite mild regularity conditions. Our analysis is based on drift with tail bounds. It is non-standard in that we do not even aim to estimate hitting times based on drift. Rather, in our case it suffices to show that the relevant time is finite with full probability.
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Notes
- 1.
It should be noted that a few interesting cases exist for zero eigenvalues (which should be improbable in practice), like the “Monkey saddle” \(f(x) = x_1^3 - 3 x_1 x_2^2\). We believe that this case can be analyzed with the same techniques as developed below, but it is outside the scope of this paper.
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Glasmachers, T. (2022). The (1+1)-ES Reliably Overcomes Saddle Points. 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_22
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DOI: https://doi.org/10.1007/978-3-031-14721-0_22
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