Emergence of L\'evy walks from second order stochastic optimization

TL;DR

This paper demonstrates that Le9vy walks and flights naturally emerge from second-order stochastic optimization processes with noisy observations, unifying directed and random search behaviors without explicit switching mechanisms.

Contribution

It introduces a novel mechanism where Le9vy search patterns arise from second-order gradient-based optimization, linking biological foraging behaviors to mathematical optimization.

Findings

Le9vy tail index b1=1 in various scenarios

Directed and random search behaviors emerge from the same process

No explicit switching mechanism needed between search modes

Abstract

In natural foraging, many organisms seem to perform two different types of motile search: directed search (taxis) and random search. The former is observed when the environment provides cues to guide motion towards a target. The latter involves no apparent memory or information processing and can be mathematically modeled by random walks. We show that both types of search can be generated by a common mechanism in which L\'evy flights or L\'evy walks emerge from a second-order gradient-based search with noisy observations. No explicit switching mechanism is required -- instead, continuous transitions between the directed and random motions emerge depending on the Hessian matrix of the cost function. For a wide range of scenarios the L\'evy tail index is $\alpha=1$, consistent with previous observations in foraging organisms. These results suggest that adopting a second-order optimization method can be a useful strategy to combine efficient features of directed and random search.