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Recursive Gaussian Process State Space Model (2411.14679v1)

Published 22 Nov 2024 in cs.LG, cs.SY, eess.SY, and stat.ML

Abstract: Learning dynamical models from data is not only fundamental but also holds great promise for advancing principle discovery, time-series prediction, and controller design. Among various approaches, Gaussian Process State-Space Models (GPSSMs) have recently gained significant attention due to their combination of flexibility and interpretability. However, for online learning, the field lacks an efficient method suitable for scenarios where prior information regarding data distribution and model function is limited. To address this issue, this paper proposes a recursive GPSSM method with adaptive capabilities for both operating domains and Gaussian process (GP) hyperparameters. Specifically, we first utilize first-order linearization to derive a Bayesian update equation for the joint distribution between the system state and the GP model, enabling closed-form and domain-independent learning. Second, an online selection algorithm for inducing points is developed based on informative criteria to achieve lightweight learning. Third, to support online hyperparameter optimization, we recover historical measurement information from the current filtering distribution. Comprehensive evaluations on both synthetic and real-world datasets demonstrate the superior accuracy, computational efficiency, and adaptability of our method compared to state-of-the-art online GPSSM techniques.

Summary

  • The paper introduces a recursive learning algorithm for GPSSMs that enables domain-independent real-time updates without pre-parameterization.
  • The method dynamically selects inducing points based on information gain, reducing computational load while preserving prediction accuracy.
  • The algorithm optimizes hyperparameters online by recovering measurement information, demonstrating superior performance in diverse conditions.

Recursive Gaussian Process State Space Model: A Technical Overview

The paper "Recursive Gaussian Process State Space Model" addresses the challenges inherent in online learning for Gaussian Process State-Space Models (GPSSMs), which are employed for modeling nonlinear, dynamical systems. These models are particularly valued for their flexibility and ability to express prediction uncertainty, which is crucial in safety-critical applications like autonomous systems and control systems. However, current methodologies often fall short, primarily due to computational inefficiencies and the difficulty of adapting to changing operational domains without extensive prior knowledge.

Key Technical Contributions

This research proposes a novel algorithm that aims to provide a robust solution for online GPSSMs through three main avenues:

  1. Development of a Recursive Learning Algorithm: The authors derive a closed-form Bayesian update equation, employing first-order linearization to manage the nonlinear transitions within GPSSMs. This approach maintains the model's flexibility by allowing the joint distribution of system states and Gaussian process (GP) predictions to be recursively learned, circumventing the need for GP pre-parameterization, thus supporting domain-independent learning.
  2. Adaptive Inducing Points Selection: Recognizing the computational intensity of GP operations as data accumulates over time, the paper introduces a dynamic method for managing the inducing points set. The algorithm uses a selection criterion based on information gain to manage the addition and removal of inducing points, ensuring the computational load remains lightweight without degrading model accuracy.
  3. Online Hyperparameter Tuning: A significant contribution is featuring a method for online optimization of GP hyperparameters which typically require retraining in offline settings. The algorithm recovers measurement information from the filtering distribution, enabling the adjustment of hyperparameters without retaining the full historical dataset, thus ensuring the GPSSM is responsive to non-stationary environmental changes.

Evaluation and Results

The proposed Recursive GPSSM was subjected to extensive evaluation against synthetic and real-world datasets. The results emphasize its superior performance regarding learning accuracy, computational efficiency, and adaptability:

  • On both synthetic and actual dataset benchmarks, the Recursive GPSSM outperformed state-of-the-art online GPSSM approaches, showcasing lower prediction errors and faster computational times.
  • The method maintained performance across various operating conditions without needing prior information concerning data distribution, demonstrating robust adaptability.

Implications and Future Work

The Recursive GPSSM addresses key limitations in the field by combining the adaptability of online learning with the interpretability and non-parametric flexibility of GPs. Practically, this method can significantly improve real-time decision-making processes in dynamically changing environments, such as autonomous vehicle navigation systems and adaptive control systems in robotics.

This research paves the way for several potential future developments:

  • Extending the algorithm to accommodate more complex non-Gaussian observation models and scalability to high-dimensional data streams.
  • Integrating adaptive techniques for dynamically estimating noise covariances in real-time, further enhancing model robustness in unpredictable environments.

By overcoming the computational and practical limitations hampering current online GPSSM applications, this work significantly contributes to the body of knowledge in real-time Gaussian process modeling, marking a substantive step towards adaptable and efficient online learning systems. This research provides a crucial toolset for both industry practitioners and academics interested in advancing state-space modeling techniques.

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