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Differentiable Interacting Multiple Model Particle Filtering

John-Joseph Brady, Yuhui Luo, Wenwu Wang, Víctor Elvira, Yunpeng Li

TL;DR

The paper tackles learning in regime-switching state-space models where switching dynamics are unknown. It develops the Differentiable Interacting Multiple Model Particle Filter (DIMMPF), a differentiable extension of the IMMPF that jointly learns per-regime models and the switching mechanism via gradient descent. The authors prove consistency of their estimators and gradient signals, and show through simulations that DIMMPF achieves state-of-the-art filtering accuracy across Markov, Polya-urn, and Erlang-like switching patterns. While computationally intensive during training, the DIMMPF yields competitive inference speed and offers a principled framework for learning high-dimensional switching dynamics in sequential data.

Abstract

We propose a sequential Monte Carlo algorithm for parameter learning when the studied model exhibits random discontinuous jumps in behaviour. To facilitate the learning of high dimensional parameter sets, such as those associated to neural networks, we adopt the emerging framework of differentiable particle filtering, wherein parameters are trained by gradient descent. We design a new differentiable interacting multiple model particle filter to be capable of learning the individual behavioural regimes and the model which controls the jumping simultaneously. In contrast to previous approaches, our algorithm allows control of the computational effort assigned per regime whilst using the probability of being in a given regime to guide sampling. Furthermore, we develop a new gradient estimator that has a lower variance than established approaches and remains fast to compute, for which we prove consistency. We establish new theoretical results of the presented algorithms and demonstrate superior numerical performance compared to the previous state-of-the-art algorithms.

Differentiable Interacting Multiple Model Particle Filtering

TL;DR

The paper tackles learning in regime-switching state-space models where switching dynamics are unknown. It develops the Differentiable Interacting Multiple Model Particle Filter (DIMMPF), a differentiable extension of the IMMPF that jointly learns per-regime models and the switching mechanism via gradient descent. The authors prove consistency of their estimators and gradient signals, and show through simulations that DIMMPF achieves state-of-the-art filtering accuracy across Markov, Polya-urn, and Erlang-like switching patterns. While computationally intensive during training, the DIMMPF yields competitive inference speed and offers a principled framework for learning high-dimensional switching dynamics in sequential data.

Abstract

We propose a sequential Monte Carlo algorithm for parameter learning when the studied model exhibits random discontinuous jumps in behaviour. To facilitate the learning of high dimensional parameter sets, such as those associated to neural networks, we adopt the emerging framework of differentiable particle filtering, wherein parameters are trained by gradient descent. We design a new differentiable interacting multiple model particle filter to be capable of learning the individual behavioural regimes and the model which controls the jumping simultaneously. In contrast to previous approaches, our algorithm allows control of the computational effort assigned per regime whilst using the probability of being in a given regime to guide sampling. Furthermore, we develop a new gradient estimator that has a lower variance than established approaches and remains fast to compute, for which we prove consistency. We establish new theoretical results of the presented algorithms and demonstrate superior numerical performance compared to the previous state-of-the-art algorithms.
Paper Structure (20 sections, 6 theorems, 42 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 20 sections, 6 theorems, 42 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Background
  4. The interacting multiple model particle filter
  5. Differentiable particle filtering
  6. The Differentiable Interacting Multiple Model Particle Filter
  7. Parameterising the model
  8. Estimating the gradient of the DIMMPF
  9. Training the DIMMPF
  10. The full algorithm and practical considerations
  11. Numerical Experiments
  12. Simulated environments
  13. Experiment details
  14. Results
  15. Conclusions
  16. ...and 5 more sections

Key Result

Theorem 3.1

Defining $\mathcal{F}_{t}\left({\psi}\right) \mathrel{\stackrel{\hbox{\normalfont\tiny def}}{=}} \sum^{N}_{n=1}\bar{w}^{n}_{t}\psi\left({\hat{x}^{n}_{t}}\right)$, and $\mathbb{P}_{t}\left({\psi}\right)$ to be true posterior mean of some test function $\psi: \mathcal{X} \to \mathbb{R}$, respectively. as $N \to \infty$, implying that $\mathcal{F}_{t}\left({\psi}\right)$ is a weakly consistent estima

Figures (2)

  • Figure 1: Bayesian network representation of the considered regime switching model.
  • Figure 2: Graphical representation of our proposed switching dynamic. Blue nodes are input/outputs. Purple nodes are fully connected network layers with the specified activation. Yellow nodes are non-learned functions. The switching probability mass, $K^{\theta}\left({k_{t+1}\mid r_{t}}\right)$, is the value at the $k_{t+1}^{\text{th}}$ index of the model output $K^{\theta}$.

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Corollary Appendix B.1
  • Corollary Appendix C.1
  • proof
  • Corollary Appendix C.2
  • ...and 1 more