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

Xingyu Wang, Chang-Han Rhee

Abstract

In this paper, we study the first-exit time of stochastic difference equation $X^η_{j+1}(x) = X^η_{j}(x) + ηa\big( X^η_{j}(x)\big) + ησ\big( X^η_{j}(x)\big)Z_{j+1}$ and its truncated variant $X^{η|b}_{j+1}(x) = X^{η|b}_{j}( x) + \varphi_b\big(ηa\big( X^{η|b}_{j}( x)\big) + ησ\big( X^{η|b}_{j}( x)\big) Z_{j+1}\big)$, 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.

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

Abstract

In this paper, we study the first-exit time of stochastic difference equation and its truncated variant , where and the law of the noise is multivariate regularly varying. The truncation operator 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 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.
Paper Structure (16 sections, 14 theorems, 164 equations, 1 figure)

This paper contains 16 sections, 14 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Heavy-Tailed Large Deviations
  5. Main Results
  6. Metastability Analysis
  7. First Exit Times and Locations
  8. General Framework: Asymptotic Atoms
  9. Numerical Examples
  10. Proof of Theorem \ref{['thm: exit time analysis framework']}
  11. Proof of Theorem \ref{['theorem: first exit time, unclipped']}
  12. Technical Lemmas for $\widecheck{ \mathbf C }^{(k)|b}$
  13. Proof of Theorem \ref{['theorem: first exit time, unclipped']} via Asymptotic Atoms
  14. Proof of Corollary \ref{['corollary: first exit time, untruncated case']}
  15. Results under General Scaling
  16. ...and 1 more sections

Key Result

Theorem 1

Let Assumptions assumption gradient noise heavy-tailed and assumption: lipschitz continuity of drift and diffusion coefficients hold. Let $k \in \mathbb N$, $b,T,\epsilon \in (0,\infty)$, and $A\subset \mathbb{R}^m$ be compact. Given $B \in \mathscr{S}_{\mathbb{D}[0,T]}$ that is bounded away from ${

Figures (1)

  • Figure 1.1: Numerical examples of the metastability analysis. (i) The univariate potential $U(\cdot)$ defined in \ref{['aeq: potential U, first exit time']}. (ii) First exit times $\tau^{\eta|b}(m)$ from $I$ under different truncation thresholds $b$ and scale parameters $\eta$. Dashed lines are predictions from our results in Section \ref{['sec: first exit time simple version']}, whereas the dots are the exit times estimated using 20 samples. Non-solid dot represents an underestimation due to early termination after long runtime ($5 \times 10^7$ steps).

Theorems & Definitions (28)

  • Theorem 1: Theorem 2.5 of wang2024largedeviationsmetastabilityanalysis
  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 2
  • Definition 3
  • Theorem 4
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['thm: exit time analysis framework']}
  • ...and 18 more