Global Convergence of the (1+1) Evolution Strategy
This provides a theoretical foundation for understanding convergence in evolutionary algorithms, which is incremental but addresses a known bottleneck in optimization theory.
The paper tackles the problem of proving global convergence for the (1+1) evolution strategy, showing that it converges to a critical point from any initial state with full probability in many cases, as illustrated across smooth, saddle, ridge, discontinuous, and rugged functions.
We establish global convergence of the (1+1) evolution strategy, i.e., convergence to a critical point independent of the initial state. More precisely, we show the existence of a critical limit point, using a suitable extension of the notion of a critical point to measurable functions. At its core, the analysis is based on a novel progress guarantee for elitist, rank-based evolutionary algorithms. By applying it to the (1+1) evolution strategy we are able to provide an accurate characterization of whether global convergence is guaranteed with full probability, or whether premature convergence is possible. We illustrate our results on a number of example applications ranging from smooth (non-convex) cases over different types of saddle points and ridge functions to discontinuous and extremely rugged problems.