Instantons for rare events in heavy-tailed distributions

TL;DR

This paper develops a method to compute instantons for rare events in heavy-tailed distributions by convexifying the rate function, enabling analysis of systems with super-exponential or algebraic tail decay.

Contribution

It introduces a nonlinear reparametrization technique to overcome convergence issues in instanton calculations for heavy-tailed stochastic systems.

Findings

Successfully computes instantons in systems with heavy tails.

Applies method to extreme power spikes in fiber optics.

Addresses divergence of the scaled CGF in heavy-tailed contexts.

Abstract

Large deviation theory and instanton calculus for stochastic systems are widely used to gain insight into the evolution and probability of rare events. At its core lies the realization that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by "convexifying" the rate function through nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation.