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Differentiable Particle Filtering using Optimal Placement Resampling

Domonkos Csuzdi, Olivér Törő, Tamás Bécsi

TL;DR

This work tackles the nondifferentiability of resampling in particle filters by introducing optimal placement resampling (OPR), a deterministic, gradient-friendly resampling scheme built from an empirically constructed CDF of the weighted particle set. By deterministically relocating particles to optimally spaced positions, OPR preserves particle diversity while concentrating mass in high-probability regions, enabling backpropagation through time for both state and parameter learning. Empirical results on a 1D linear Gaussian model, proposal learning, and a stochastic volatility model show that PF-OPR matches or surpasses traditional multinomial resampling in ELBO quality and gradient signal, with a manageable computational cost dominated by particle sorting. The approach demonstrates clear potential for differentiable PFs in real-world time-series applications, though it currently remains limited to one dimension, motivating future multidimensional extensions.

Abstract

Particle filters are a frequent choice for inference tasks in nonlinear and non-Gaussian state-space models. They can either be used for state inference by approximating the filtering distribution or for parameter inference by approximating the marginal data (observation) likelihood. A good proposal distribution and a good resampling scheme are crucial to obtain low variance estimates. However, traditional methods like multinomial resampling introduce nondifferentiability in PF-based loss functions for parameter estimation, prohibiting gradient-based learning tasks. This work proposes a differentiable resampling scheme by deterministic sampling from an empirical cumulative distribution function. We evaluate our method on parameter inference tasks and proposal learning.

Differentiable Particle Filtering using Optimal Placement Resampling

TL;DR

This work tackles the nondifferentiability of resampling in particle filters by introducing optimal placement resampling (OPR), a deterministic, gradient-friendly resampling scheme built from an empirically constructed CDF of the weighted particle set. By deterministically relocating particles to optimally spaced positions, OPR preserves particle diversity while concentrating mass in high-probability regions, enabling backpropagation through time for both state and parameter learning. Empirical results on a 1D linear Gaussian model, proposal learning, and a stochastic volatility model show that PF-OPR matches or surpasses traditional multinomial resampling in ELBO quality and gradient signal, with a manageable computational cost dominated by particle sorting. The approach demonstrates clear potential for differentiable PFs in real-world time-series applications, though it currently remains limited to one dimension, motivating future multidimensional extensions.

Abstract

Particle filters are a frequent choice for inference tasks in nonlinear and non-Gaussian state-space models. They can either be used for state inference by approximating the filtering distribution or for parameter inference by approximating the marginal data (observation) likelihood. A good proposal distribution and a good resampling scheme are crucial to obtain low variance estimates. However, traditional methods like multinomial resampling introduce nondifferentiability in PF-based loss functions for parameter estimation, prohibiting gradient-based learning tasks. This work proposes a differentiable resampling scheme by deterministic sampling from an empirical cumulative distribution function. We evaluate our method on parameter inference tasks and proposal learning.
Paper Structure (12 sections, 25 equations, 6 figures)

This paper contains 12 sections, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. State-Space Models
  3. Particle Filters
  4. Maximum Likelihood Estimation with Particle Filters
  5. Related Work and Contributions
  6. Deterministic Sampling
  7. Optimal Placement Resampling
  8. Empirical Study
  9. Linear Gaussian State-Space Model
  10. Proposal Distribution Learning
  11. Stochastic Volatility Model
  12. Conclusions and Future Work

Figures (6)

  • Figure 1: Directed probabilistic graphical model of an SSM. Shaded nodes are observed random variables.
  • Figure 2: Illustration of the marginal data log-likelihood estimation of a standard PF with multinomial resampling (left), and a differentiable PF with optimal placement resampling (right) for a 1-dimensional linear Gaussian SSM, described in Section \ref{['sec:lgssm']}.
  • Figure 3: Illustration of optimal placement resampling. The goal is to perform resampling, i.e., create an unweighted particle set from a weighted particle set \ref{['eq:DiracSum']} by moving particles to positions with high probability mass. The true pdf (black) and true cdf (dashed black) do not change during resampling, only their representation by particles. The two rows show two consecutive timesteps. Left: weighted particle set before resampling, with stem length proportional to particle weight. Middle: resampled particle set by optimal placement resampling. Right: empirical cdfs. In the second row, the resampled particles in the previous timestep are weighted according to \ref{['eq:weights']}, and the resampling is performed again. Due to the particles' finite representational power, their histograms slightly differ before and after resampling.
  • Figure 4: Illustration of empirical pdf (left) and cdf (right) based on the optimal particle locations. Black line: analytical pdf and cdf. Blue line: empirical pdf and cdf. Blue triangles: particle locations. Red line: traditional step-wise empirical cdf, which we replaced by the blue line approximation given in \ref{['eq:empCDF']}.
  • Figure 5: Proposal distribution learning for an LGSSM using PF-MR (orange) and PF-OPR (blue). Higher ELBO is better.
  • ...and 1 more figures