Deep Learning’s ‘catastrophe Principle’ Mapped with New Exit Time Analysis

Scientists investigate the first-exit time of stochastic difference equations and their truncated variants, a problem crucial for understanding heavy-tailed dynamical systems. Xingyu Wang from the University of Amsterdam and Chang-Han Rhee from Northwestern University, along with their colleagues, demonstrate how truncation acts as a modulation mechanism in such systems, with relevance to areas like stochastic gradient descent in deep learning. Their research establishes a link between large deviations and metastability, providing precise characterisations of exit times and locations and revealing a discrete hierarchy of phase transitions as truncation varies. This work extends the classical Freidlin-Wentzell theory to encompass heavy-tailed systems, illuminating the catastrophic behaviour driving key events within them.

Heavy-tailed dynamical systems and truncation-induced phase transitions

Scientists have developed a new theoretical framework for analysing the first-exit times of stochastic dynamical systems exhibiting heavy-tailed behaviour. This work introduces a comprehensive understanding of how these systems transition between states, revealing a discrete hierarchy of phase transitions governed by the truncation threshold.

Researchers connected large deviation theory with metastability to precisely characterise the joint distributions of both exit times and locations within these complex systems. The resulting limit theorem demonstrates that variations in truncation levels induce distinct phase transitions, manifesting a catastrophe principle where system behaviour is driven by significant events in a few components.

The study focuses on stochastic difference equations and their truncated variants, where the noise follows a multivariate regularly varying law. A key innovation lies in the development of a truncation operator, frequently employed as a modulation mechanism in heavy-tailed systems such as stochastic gradient descent algorithms used in deep learning.

By leveraging locally uniform sample-path large deviations, the research provides precise characterisations of the system’s dynamics. This allows for a detailed understanding of how the system escapes from metastable states and the pathways it takes during these transitions. These developments establish a heavy-tailed analogue of the classical Freidlin-Wentzell theory, extending its applicability to a broader range of stochastic processes.

The findings illuminate the catastrophe principle, whereby critical events in heavy-tailed systems are not necessarily driven by a single large fluctuation, but rather by the combined effect of multiple significant changes. This contrasts with light-tailed systems, where exponential scaling and smooth tilting dominate.

The research has implications for fields beyond theoretical probability, offering insights into the behaviour of complex systems in machine learning, finance, and operations research. Specifically, the work provides a mathematical foundation for understanding the impact of techniques like gradient clipping in deep learning, which prevent drastic changes in model weights and promote exploration of flat local minima. This advancement promises to refine the analysis of metastability and improve the design of algorithms operating in heavy-tailed environments.

Characterising first-exit time distributions via large deviations and truncation analysis

Researchers investigated the first-exit time of a stochastic difference equation and its truncated variant, focusing on systems with multivariate regularly varying noise. Central to the methodology was the analysis of how truncation, a modulation mechanism, impacts heavy-tailed systems, such as those found in stochastic gradient descent algorithms used in deep learning.

The research revealed a discrete hierarchy of phase transitions, or exit times, as the truncation parameter varied. This demonstrated the catastrophe principle, whereby events in heavy-tailed systems are driven by catastrophic behaviour in a few components, while the remainder of the system behaves predictably.

The work extends the classical Freidlin-Wentzell theory to encompass heavy-tailed systems, providing a comprehensive counterpart for analysing metastability. Asymptotic atoms were developed as a key tool, enabling the precise characterisation of the scaling behaviour of exit times. Numerical examples were generated to validate the theoretical findings and illustrate the impact of truncation on the system’s dynamics. These examples confirmed the polynomial scaling of exit times observed in heavy-tailed systems and the role of multiple large jumps in driving key events.

Heavy-tailed stochastic dynamics exhibit catastrophe-driven metastability and discrete phase transitions

Research into stochastic difference equations and their truncated variants reveals precise characterizations of first exit times and locations when the noise follows a multivariate regularly varying law. The study establishes a framework linking large deviations with metastability, utilising locally uniform sample-path large deviations to analyse both processes.

Results demonstrate a discrete hierarchy of phase transitions, or exit times, as the truncation parameter varies, highlighting how changes in truncation levels induce distinct system behaviours. This work unveils the catastrophe principle in heavy-tailed systems, whereby key events and metastable behaviours are driven by catastrophic changes in a few components, while the remainder of the system maintains nominal behaviour.

The research provides a comprehensive heavy-tailed analogue to the classical Freidlin-Wentzell theory, extending its applicability to systems exhibiting heavy-tailed characteristics. Analysis of the joint distributions of first exit times and exit locations provides detailed insights into the system’s dynamics.

The developed framework allows for precise characterisation of the asymptotic behaviour of these systems, moving beyond merely logarithmic asymptotics of exit times. This advancement enables the determination of the full scale of exit times, including prefactor identification. The study builds upon existing metastability analysis, extending the scope to encompass non-reversible Markov processes and, crucially, heavy-tailed systems which exhibit fundamentally different behaviours. These findings have implications for understanding complex systems across diverse fields, including machine learning and finance, where heavy-tailed phenomena are prevalent.

Truncated heavy-tailed dynamics and scaling to Markov jump processes

Scientists have established a comprehensive understanding of the first-exit times and locations for stochastic difference equations with regularly varying noise and truncation. This work connects large deviations with metastability, providing precise characterizations of the joint distributions of these exit events.

The resulting analysis reveals a discrete hierarchy of phase transitions in exit times as the truncation level changes, demonstrating how catastrophic behaviour in a few components can drive events within heavy-tailed systems. These findings represent a heavy-tailed extension of the classical Freidlin-Wentzell theory, offering insights into the global dynamics of such processes.

The research demonstrates that truncated heavy-tailed dynamics scale to a Markov jump process visiting only the widest minima of a potential function, unlike untruncated cases which explore all local minima. This understanding is particularly relevant to stochastic gradient descent algorithms used in deep learning, explaining why these algorithms can effectively train neural networks despite their stochastic nature.

The authors acknowledge that their analysis focuses on systems with regularly varying perturbations and does not cover all possible tail behaviours. Future research will extend this framework to systems not governed by a single large jump, further characterising their global behaviour and scaling limits at the process level.