HiPPO-Prophecy: State-Space Models can Provably Learn Dynamical Systems in Context

2024-07-12

arXiv.org

AI Summary

This work addresses the fundamental question of how state space models (SSMs) can perform zero-shot in-context learning to predict the next state of arbitrary dynamical systems without parameter fine-tuning. Method: We propose HiPPO-Prophecy, a novel weight construction method for SSMs grounded in the HiPPO framework, enabling both continuous- and discrete-time modeling. Our approach theoretically establishes that continuous SSMs can asymptotically approximate the derivative of any input signal, and we derive provable next-state prediction guarantees for discrete SSMs. Through rigorous signal-theoretic analysis and asymptotic error bound derivation, we obtain an explicit upper bound on the derivative approximation error. Results: Experiments demonstrate high-accuracy zero-shot state prediction across diverse dynamical systems. This work provides the first theoretical characterization of SSMs’ capacity for modeling dynamical system evolution in zero-shot settings, significantly enhancing both their theoretical interpretability and practical applicability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Intelligent Robots: State Estimation

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.

Problem

• Explores in-context learning of State Space Models
• Introduces weight construction for predicting dynamical systems
• Extends HiPPO framework to approximate signal derivatives

Innovation

• Novel weight construction for SSMs
• Extends HiPPO framework for signal approximation
• Provides asymptotic error bound on derivatives