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

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.