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 fact 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 a 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.
Highlights
In many situations of physical relevance, rare events are tremendously important despite their infrequent occurrence: heat waves, stock market crashes, or earth quakes all occur with small probability but devastating consequences
To demonstrate the applicability of our approach, we show several examples of instantons for heavy-tailed distributions in section 4: toy models with super-exponential and powerlaw tails, and a banana-shaped potential, and high-amplitude events in fiber-optics described by the focusing nonlinear Schrödinger (NLS) equation
Estimating the probability of tail events can efficiently be done via large deviation theory (LDT) and instanton calculus, which transforms an inefficient sampling problem into a deterministic optimization problem
Summary
In many situations of physical relevance, rare events are tremendously important despite their infrequent occurrence: heat waves, stock market crashes, or earth quakes all occur with small probability but devastating consequences. Rare event algorithms [1] are typically based on one of the two following ideas: either to increase the rate of occurrence of the rare event by biasing the underling system (importance sampling), or to substitute all possible ways of observing a rare event by its most common realization (large deviations/instanton theory) Under the hood both are connected to the exponentially tilted measure and the cumulant generating function (CGF). Though, if the probability measure Pε is super-exponential, or has even ε→0 heavier tails, the expectation in equation (3) diverges and the CGF is no longer defined This does not mean that the corresponding rare events are special in any way, but merely that.
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