[Paper Review] Structured Variational Inference in Unstable Gaussian Process State Space Models

Silvan Melchior, Sebastian Curi+2 more|arXiv (Cornell University)|Jul 16, 2019

Gaussian Processes and Bayesian Inference24 references4 citations

TL;DR

This paper proposes CBF-SSM, a structured variational inference algorithm for Gaussian Process State-Space Models (GPSSMs) that enables learning in unstable and partially observable systems. By combining a backward smoother with soft conditioning via a tunable Kalman gain parameter, CBF-SSM achieves superior test performance on both stable and unstable systems, outperforming prior methods like PR-SSM and VCDT in scenarios with hidden states and long sequences.

ABSTRACT

We propose a new variational inference algorithm for learning in Gaussian Process State-Space Models (GPSSMs). Our algorithm enables learning of unstable and partially observable systems, where previous algorithms fail. Our main algorithmic contribution is a novel approximate posterior that can be calculated efficiently using a single forward and backward pass along the training trajectories. The forward-backward pass is inspired on Kalman smoothing for linear dynamical systems but generalizes to GPSSMs. Our second contribution is a modification of the conditioning step that effectively lowers the Kalman gain. This modification is crucial to attaining good test performance where no measurements are available. Finally, we show experimentally that our learning algorithm performs well in stable and unstable real systems with hidden states.

Motivation & Objective

• To address the failure of existing variational inference methods in learning unstable and partially observable dynamical systems using GPSSMs.
• To improve posterior approximation in GPSSMs by introducing a backward pass that better captures temporal correlations than prior methods.
• To enhance test-time generalization in systems with no measurements by modifying the conditioning step to reduce Kalman gain.
• To enable training on longer sequences in unstable systems where previous methods fail due to divergence or overfitting.
• To combine physical models with data-driven GP corrections for accurate, uncertainty-aware dynamics prediction in real-world robotics.

Proposed method

• Introduces a backward pass inspired by Kalman smoothing, applied to GPSSMs to improve posterior approximation of hidden states.
• Proposes a smoothed estimate-based conditioning step that replaces direct observation conditioning, improving robustness in partially observed systems.
• Introduces a tunable parameter $k$ to control the Kalman gain, enabling interpolation between full conditioning (VCDT-like) and no conditioning (PR-SSM-like).
• Uses a structured posterior that preserves temporal correlations through forward and backward passes along training trajectories.
• Employs doubly stochastic variational inference with a recognition model to approximate the posterior over latent states.
• Combines the GP correction term with a known physical model (e.g., ODE-based rigid body dynamics) to improve generalization and reduce data requirements.

Experimental results

Research questions

• RQ1 Can variational inference in GPSSMs be made robust to instability and partial observability, where prior methods fail?
• RQ2 Does a backward smoothing pass improve posterior approximation in GPSSMs compared to filtering-based methods?
• RQ3 Can soft conditioning via a tunable Kalman gain parameter improve test-time generalization in systems without measurements?
• RQ4 Can CBF-SSM train on longer sequences in unstable systems without divergence or overfitting?
• RQ5 Does combining CBF-SSM with a physical model yield more accurate and reliable uncertainty estimates in real-world robotics tasks?

Key findings

• CBF-SSM achieves lower test error than PR-SSM and VCDT on a simulated unstable Dubin’s car model, especially as training sequence length increases.
• On the VoliroX drone dataset, CBF-SSM outperforms PR-SSM and VCDT in predictive accuracy and uncertainty calibration, even with large process noise and hidden states.
• CBF-SSM can train on longer sequences (up to 300 steps) in unstable systems, while PR-SSM and VCDT fail due to divergence or overfitting.
• The CBF-SSM with $k=50$ achieves the lowest test error across all datasets, demonstrating the benefit of soft conditioning.
• CBF-SSM with $k=1$ (closer to VCDT) performs well when full state information is available, showing consistency with prior methods in favorable settings.
• The algorithm is 3.7× slower at training due to the extra backward pass, but prediction time remains identical across all methods.