Threshold Crossings as Tail Events for Catastrophic AI Risk

TL;DR

The work investigates the existence and nontriviality of $T$-periodic solutions for convex Hamiltonian systems with subquadratic growth at infinity. It develops a dual-action variational framework for the boundary-value problem $\dot{x}=JH'(t,x)$, under $(A_{\infty},B_{\infty})$-subquadratic assumptions, and identifies spectral-gap conditions involving $\gamma$ and $\lambda$ (and $\delta$) that guarantee nontrivial periodic orbits. The results unify and extend classical subharmonic existence results (Rabinowitz; Clarke–Ekeland; Michalek–Tarantello), provide explicit nontriviality criteria, and clarify the role of subquadratic growth in generating periodic solutions. The framework is positioned as a foundation for further analysis of periodic orbits in convex Hamiltonian dynamics and related subquadratic variational problems.

Abstract

We analyse circumstances in which bifurcation-driven jumps in AI systems are associated with emergent heavy-tailed outcome distributions. By analysing how a control parameter's random fluctuations near a catastrophic threshold generate extreme outcomes, we demonstrate in what circumstances the probability of a sudden, large-scale, transition aligns closely with the tail probability of the resulting damage distribution. Our results contribute to research in monitoring, mitigation and control of AI systems when seeking to manage potentially catastrophic AI risk.

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Theorems & Definitions (16)

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