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Variational Gaussian Process State-Space Models
Roger Frigola, Yutian Chen, Carl E. Rasmussen
TL;DR
Flexible nonlinear state-space models are difficult to learn when principled parametric dynamics are unavailable, while posterior inference and prediction can be computationally expensive for long time series. The paper combines sparse-GP variational inference with sequential Monte Carlo to obtain a tractable posterior, linear-time learning, and prediction complexity independent of series length, alongside stochastic and online variants. The resulting approach supports expressive uncertainty-aware dynamical models while avoiding overfitting within the stated approximations and scope.
Problem
Learning flexible nonlinear state-space models requires tractable posterior inference and prediction, but sampling length-T smoothing trajectories is expensive.
Method
The paper uses sparse-GP variational Bayesian inference with sequential Monte Carlo for auxiliary-system smoothing, plus stochastic variational inference and online learning.
Results
The method yields a tractable posterior over nonlinear dynamical systems, linear learning complexity in time-series length, and prediction computation independent of time-series length.
Takeaways & Limitations
The approach supports expressive nonlinear dynamical models with probabilistic future-trajectory predictions and a user-controlled trade-off between model capacity and computation.
Takeaways & Limitations
Future work is needed to characterize GP-SSM prior properties such as stability, equilibria, and limit cycles; predictions also rely on inducing-variable approximations.
Abstract
from arXiv · showhide
State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient variational Bayesian learning of nonlinear state-space models based on sparse Gaussian processes. The result of learning is a tractable posterior over nonlinear dynamical systems. In comparison to conventional parametric models, we offer the possibility to straightforwardly trade off model capacity and computational cost whilst avoiding overfitting. Our main algorithm uses a hybrid inference approach combining variational Bayes and sequential Monte Carlo. We also present stochastic variational inference and online learning approaches for fast learning with long time series.
1 Introduction
The paper develops Bayesian learning for flexible nonlinear state-space models using sparse Gaussian processes, enabling a trade-off between model capacity and computation. Its variational approach yields tractable posteriors and supports efficient prediction, stochastic inference, and online learning.
- Sparse Gaussian processes encode general assumptions such as continuity or smoothness while supporting flexible nonparametric nonlinear dynamics.
- The approach lets users trade off model capacity and computation time, while variational training allows complex models to be learned without risk of overfitting.
- The variational formulation produces a tractable approximate posterior over nonlinear dynamical systems and fast probabilistic predictions of future trajectories.
- Learning has computational complexity linear in the time-series length through variational sparse Gaussian processes and smoothing in an auxiliary dynamical system.
- Stochastic variational inference accelerates learning for long time series, and the paper also presents an online learning scheme.
- Uncertainty-aware explicit dynamics models are useful when future uncertainty matters, parametric dynamics are difficult to specify, or future simulation is required.
2 Gaussian Process State-Space Models
GP state-space models place a Gaussian-process prior over nonlinear transition functions, producing a rich distribution over dynamical systems and trajectories. The paper uses this prior to represent varied behaviors while retaining a sequential generative interpretation.
- The model uses latent states with deterministic transition and observation functions plus independent process and observation noise.
- Gaussian processes provide flexible functions for dynamics when parametric models cannot combine complex-function modeling with resistance to overfitting.
- The GP transition induces sequential predictive distributions over latent function values using kernel matrices and GP marginals.
- A GP-SSM is doubly stochastic: one can sample a transition function and then sample state trajectories and observations conditioned on that function.
- The GP prior generates qualitatively different dynamical behaviors under fixed hyperparameters, including nearly linear non-oscillatory behavior and nonlinear limit cycles.
3 Variational Inference in GP-SSMs
The method uses variational sparse-GP inference to obtain a tractable posterior over nonlinear dynamical systems and efficient predictions. Inducing variables, auxiliary-state smoothing, and flexible smoothing methods support a capacity–computation trade-off.
- 3 Variational Inference in GP-SSMs: Variational sparse-GP techniques replace costly smoothing-based posterior construction with a tractable distribution over the state transition function.The approach is designed to make the posterior representation independent of time-series length.
- 3 Variational Inference in GP-SSMs: The variational distribution factorizes the model so difficult transition-function terms cancel from the objective, yielding a tractable evidence lower bound.The bound includes a Kullback–Leibler divergence and entropy terms, while the integral over transition-function values is solved analytically.
- 3 Variational Inference in GP-SSMs: Increasing the number of inducing variables tightens the ELBO and provides a direct trade-off between model capacity and computation cost without increasing overfitting risk.The variational distribution precisely learns the latent function at inducing-input locations and uses the prior conditioned on inducing variables elsewhere.
- 3 Variational Inference in GP-SSMs: The optimal q*(x) is the smoothing distribution of a simpler auxiliary parametric state-space model with Markovian latent-state structure.Its transition density is Gaussian, and standard smoothers can be selected according to the system’s characteristics.
- 3 Variational Inference in GP-SSMs: The learning algorithm alternates smoothing-distribution sampling with updates to hyperparameters and q*(u), using sequential Monte Carlo as a hybrid inference component.Other smoothing methods may be more appropriate for particular dynamical systems, and vanilla sequential Monte Carlo can produce biased estimators.
- 3 Variational Inference in GP-SSMs: After pre-computation, prediction costs O(M) for the mean and O(M^2) for the variance, while smoothing-sample approaches cost O(TL) and O(T^2L).The variational method condenses latent-function learning onto inducing points and does not explicitly require p(x|y) for predictions.
4 Stochastic Variational Inference
Stochastic variational inference estimates trajectory-wide sufficient statistics from one or more segments, enabling unbiased optimization for long observed time series. This reduces the cost of computing q*(u) or hyperparameter gradients without requiring the full trajectory at every update.
- 4 Stochastic Variational Inference: For long time series, stochastic variational inference estimates sufficient statistics from one or more trajectory segments to reduce optimization cost.Both q*(u) and the hyperparameter gradient depend linearly on q(x) through sums over the state trajectory.
5 Online Learning
The variational GP-SSM approach supports online Bayesian updating as new sequential data arrives, while stochastic variational inference can improve performance for long time series.
- Online Bayesian Updating: The variational approach naturally yields an online learning implementation for dynamical systems with sequentially arriving data.This setting includes robots learning object dynamics through interaction.
- Online Bayesian Updating: Online learning applies Bayes’ rule sequentially, using the previous approximate posterior as the next prior.The update replaces p(u) = N(u|0, K_u,u) with N(u|μ, Σ) and updates natural parameters using a new mini-batch.
6 Experiments
Experiments evaluate variational GP-SSMs on a difficult one-dimensional nonlinear system and neural spike recordings, testing nonlinear dynamics, latent dimensionality, and non-Gaussian observations.
- Experimental Goals: The experiments target learning nonlinear dynamics, handling latent states higher-dimensional than observations, and using non-Gaussian likelihoods.These goals organize the evaluation across the two experimental settings.
- 1D Nonlinear System: The one-dimensional system has a pronounced kink, and the study learns its transition-function posterior with a Matérn-covariance GP-SSM.Inference uses a bootstrap particle fixed-lag smoother with 1000 particles and lag 10.
- 1D Nonlinear System: PMCMC gives the best test performance with 500 training points but requires substantial training and test time.With T = 10^4, stochastic variational inference improves test performance and has test time independent of training-set size.
- Neural Spike Train Recordings: The neural experiment models hippocampal spike counts as an autonomous nonlinear dynamical system using a second-order GP-SSM structure and Poisson observations.The two states are arranged so one represents the derivative of the other, with spike counts binned at 0.01 seconds.
- Neural Spike Train Recordings: Posterior simulations capture the roughly 6 Hz spike frequency and the nonlinear limit cycle visible in the learned dynamics.Figure 2 presents observed counts, a smoothing sample, predictive spike-count distributions, and corresponding latent states; Figure 3 shows transition-function contours.
7 Discussion and Future Work
The paper concludes that variational GP-SSMs provide tractable posteriors and efficient predictions for expressive nonlinear dynamical models, while identifying structured variational inference and theoretical analysis as future work.
- Discussion: The variational formulation learns expressive GP-SSMs without risk of overfitting and supports latent spaces larger than the observation space.The latter capability is demonstrated as a distinction from previous approaches.
- Discussion: The resulting tractable posterior enables prediction times that do not depend on the time-series length, unlike approaches based on smoothing-distribution sampling.The method is intended for settings lacking a principled parametric dynamics model.
- Future Work: Future work includes structured variational distributions that could remove auxiliary-system smoothing and theoretical characterization of stability, equilibria, and limit cycles.Eliminating smoothing would require optimizing more variational parameters.
Supplementary Material
The supplementary material interprets GP-SSM terms through Gaussian-process regression and summarizes a Gaussian representation of their product.
- GP Regression Interpretation: The right-hand side is equivalent to Gaussian process regression with x_0:T−1 as inputs and x_1:T as outputs.
- GP Regression Interpretation: The left-hand side is a product of terms equivalent to Gaussian-process prediction with noiseless observations.
- Gaussian Representation: This product can be represented succinctly by a Gaussian distribution.
B.1 Markovian Model with Heteroscedastic Noise
This model makes state transitions conditionally independent given inducing variables and interprets them through a deterministic transition function with heteroscedastic process noise. Its variational evidence lower bound matches equation (9).
- Given the inducing variables, state transitions are independent of one another.
- A_{t−1}u acts as the deterministic transition function, while B_{t−1} describes heteroscedastic process noise.
- The variational evidence lower bound becomes identical to the lower bound in equation (9).
B.2 Bayesian RBF Model
The Bayesian RBF model combines GP-based transition and observation functions with inducing variables and variational inference. Its transition dynamics are deterministic and parameterized by inducing quantities, with a known uncertainty limitation away from inducing inputs.
- The transition dynamics are fully deterministic and parameterized by u and z, analogous to a Subset of Regressors sparse GP.
- The predictive variance shrinks to zero away from inducing inputs, even though greater uncertainty would be desirable there.
- The model places Gaussian process priors over both the state transition and emission functions.
- Inducing variables u and v augment the transition and emission GPs, yielding a joint distribution over observations, states, functions, and inducing variables.
- The variational formulation introduces a distribution over the latent variables and inducing quantities.
- The optimal variational distributions for the inducing variables are Gaussian, including the optimal distribution for the emission inducing variables.
- The state trajectory has an optimal variational distribution, while hyperparameters and inducing inputs are optimized by gradient ascent.
D Plots from Experiments Section
Figure 4 depicts the posterior distribution over the latent state transition function by comparing the ground truth, posterior mean, and posterior uncertainty.
- Green marks the ground truth, blue marks the posterior mean, and red marks the mean ±1 standard deviation.