First-Exit Time Analysis for Truncated Heavy-Tailed Dynamical Systems

TL;DR

This paper analyzes the first-exit times of truncated heavy-tailed stochastic systems, revealing phase transitions and metastable behaviors, and extends classical Freidlin-Wentzell theory to heavy-tailed contexts.

Contribution

It develops a framework connecting large deviations with metastability for truncated heavy-tailed systems, providing a detailed characterization of exit times and locations.

Findings

Identifies a hierarchy of phase transitions in exit times as truncation varies.

Unveils the catastrophe principle where key events are driven by few components.

Extends Freidlin-Wentzell theory to heavy-tailed stochastic systems.

Abstract

In this paper, we study the first-exit time of stochastic difference equation $X^\eta_{j+1}(x) = X^\eta_{j}(x) + \eta a\big( X^\eta_{j}(x)\big) + \eta \sigma\big( X^\eta_{j}(x)\big)Z_{j+1}$ and its truncated variant $X^{\eta|b}_{j+1}(x) = X^{\eta|b}_{j}( x) + \varphi_b\big(\eta a\big( X^{\eta|b}_{j}( x)\big) + \eta \sigma\big( X^{\eta|b}_{j}( x)\big) Z_{j+1}\big)$, where $\varphi_b(x) = (x/|x|)\min\{|x|, b\}$ and the law of the noise $Z_t$ is multivariate regularly varying. The truncation operator $\varphi_b(\cdot)$ is often introduced as a modulation mechanism in heavy-tailed systems, such as stochastic gradient descent algorithms in deep learning. By developing a framework that connects large deviations with metastability, we leverage the locally uniform sample-path large deviations for both processes in Wang and Rhee (2024) to obtain precise characterizations of the joint distributions of the first exit times and exit locations. The resulting limit theorem unveils a discrete hierarchy of phase transitions (i.e., exit times) as the truncation threshold $b$ varies, and manifests the catastrophe principle, whereby key events or metastable behaviors in heavy-tailed systems are driven by catastrophic behavior in a few components while the rest of the system behaves nominally. These developments lead to a comprehensive heavy-tailed counterpart of the classical Freidlin-Wentzell theory.