Single-Funnel and Multi-funnel Landscapes and Subthreshold-Seeking Behavior

Algorithms for parameter optimization display subthreshold-seeking behavior when the majority of the points that the algorithm samples have an evaluation less than some target threshold. Subthreshold-seeking algorithms avoid the curse of the general and Sharpened No Free Lunch theorems in the sense that they are better than random enumeration on a specific (but general) family of functions. In order for subthreshold-seeking search to be possible, most of the solutions that are below threshold must be localized in one or more regions of the search space. Functions with search landscapes that can be characterized as single-funnel or multi-funnel landscapes have this localized property. We first analyze a simple “Subthreshold-Seeker” algorithm. Further theoretical analysis details conditions that would allow a Hamming neighborhood local search algorithm using a Gray or binary representation to display subthreshold-seeking behavior. A very simple modification to local search is proposed that improves its subthreshold-seeking behavior.

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