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Learning to Bias: Machine Learning-Enhanced Particle Filters

Apoorv Srivastava, Eric Darve

arXiv:2609.30498v1cs.LGstat.ML

TL;DR

Particle Filters can be inefficient and scale poorly when their proposals place particles away from observation-consistent regions. NOPFs learn amortized proposals from offline simulated conditioning tuples and use standard importance weighting in PF updates. Across benchmarks, they match Kalman-based and Gaussian baselines in approximately unimodal settings and outperform standard PF baselines in multimodal problems.

  • Problem

    PF sample efficiency depends on proposal quality, and the number of particles required can grow sharply with state dimension.

    Source: §1 · p. 1

  • Method

    NOPFs learn an amortized proposal offline and use it as a replacement in standard PF updates, with samples corrected by standard importance weights.

    Source: §1 · p. 1

  • Results

    NOPF matches Kalman-based and Gaussian baselines in approximately unimodal settings and outperforms standard PF baselines in multimodal problems.

    Source: §5 · p. 9

  • Takeaways & Limitations

    Learned proposals can improve PF practicality without sacrificing principled Bayesian weighting.

    Source: §5 · p. 10

    Offline training can cause train-test mismatch when inference drifts outside the training distribution or the simulator poorly represents the true system, potentially degrading proposal quality.

    Source: §5 · p. 9

Abstract

from arXiv · show

Sequential inference estimates latent states from noisy and incomplete observations. Particle Filters (PFs), a class of Monte Carlo methods based on importance sampling, provide a flexible framework for this task, but often suffer from poor sample efficiency and unfavorable scaling with dimension, partly due to suboptimal proposal distributions. We address these challenges by integrating learned proposals into the PF framework. We introduce Neural Optimal Particle Filters (NOPFs), which learn an amortized approximation to the optimal proposal from offline simulated one-step conditioning tuples. The learned proposal is used as a drop-in replacement in standard PF updates, with samples corrected by standard importance weights so that the method asymptotically targets the same filtering distribution under standard support and density-evaluation assumptions. Across stochastic nonlinear benchmarks of varying inference complexity, NOPFs improve sample efficiency and distributional accuracy over standard PF baselines with modest computational overhead. The approach integrates data-driven proposal learning into classical inference without altering the underlying filtering objective.

1 Introduction

Particle Filters can scale poorly because sample efficiency depends on proposal quality, while naive proposals may send particles to regions inconsistent with observations. Neural Optimal Particle Filters learn proposals for standard PF updates, with contributions including a residual Gaussian design and evaluation across nonlinear localization regimes.

1 Introduction

  • PF sample efficiency depends critically on proposal choice, and particle counts can grow sharply with state dimension as naive proposals yield negligible weights.Effective proposals use the current observation to guide particles toward plausible posterior regions while retaining proper importance weighting.Source: §1 · p. 1
  • NOPFs learn proposal distributions as plug-and-play replacements in standard PF updates, correcting samples with standard importance weights.The update preserves the filtering objective while improving particle placement.Source: §1 · p. 1
  • The contributions include an offline amortized approximation to the locally optimal one-step proposal and a residual Gaussian around the deterministic transition skeleton.The evaluation spans nonlinear localization regimes that are approximately Gaussian, multimodal, or data-association based, against Kalman-based and proposal-enhanced PF baselines.Source: §1 · pp. 1–2

2 Background

Particle Filters approximate filtering distributions with weighted particles, but proposal choices can cause weight degeneracy that resampling cannot fully repair. The background motivates learning the locally optimal one-step proposal and distinguishes NOPF from other SMC and learned-proposal approaches.

2 Background

  • PFs approximate the filtering distribution using weighted particles, with SIS updates whose normalized weights determine each particle’s contribution.When proposals place particles in low-likelihood regions, a few particles dominate and the effective number of useful particles falls.Source: §2 · p. 2
  • Resampling duplicates high-weight particles and resets weights, but cannot restore diversity when few particles survive poor proposals.This can cause sample impoverishment, leaving proposal design central to PF efficiency.Source: §2 · p. 2
  • The locally optimal proposal is a natural target because it gives the least-degenerate benchmark in recursive one-step SIS updates.It approximates p(xt | xt−1, yt), biasing particles toward dynamically plausible, observation-consistent states while retaining importance weighting.Source: §2 · p. 2
  • Prior approaches include predictive-likelihood ancestor selection, local Gaussian proposals, particle optimization, likelihood nudging, and learned score-based filters.Other SMC work addresses parameter learning, differentiable model training, or online adaptation; NASMC instead optimizes a trajectory-level objective using SMC-derived gradients.Source: §2 · pp. 2–3
  • NOPF assumes known transition and observation models and learns an amortized one-step Markov proposal offline from simulated conditioning tuples.It uses analytic model densities and a residual Gaussian, then applies standard PF importance weights without online proposal optimization or weighted SMC training trajectories.Source: §2 · p. 3

3 Method

NOPF learns a conditional proposal for each ancestor particle and current observation, then inserts it into the standard PF update. Its Gaussian proposal uses the transition’s deterministic skeleton; offline ELBO training and complementary evaluation metrics complete the method.

3 Method

  • NOPF learns a conditional density approximating the locally optimal one-step proposal for each particle under the Markov structure.The proposal is inserted into the standard PF update, where particles are weighted and resampled as usual.Source: §3 · p. 3

3.1 Learned Proposal Density

  • The proposal is Gaussian, with neural networks producing its mean and covariance from the deterministic transition prediction, observation, and time.Using the transition skeleton lets the model learn an observation-informed correction rather than relearning the deterministic dynamics; sampling and density evaluation require a network pass and Gaussian evaluation.Source: §3.1 · pp. 3–4

3.2 Training and Inference

  • NOPF trains a family of conditional proposals offline by minimizing expected KL divergence over conditioning tuples from simulated trajectories.The ELBO objective rewards states that explain the observation and remain likely under the transition while penalizing proposal-density mismatch.Source: §3.2 · p. 4
  • Training uses simulated trajectories rather than PF rollouts, and inference samples from the learned proposal before applying standard weighting, normalization, and resampling.The offline approach does not require differentiating through resampling or using weighted particle trajectories as training targets.Source: §3.2 · p. 4

3.3 Evaluation Metrics

  • Evaluation reports NMSE, PNLL, and energy score for accuracy and distributional quality, ESS for proposal efficiency, and wall time for computational cost.The metrics are time-averaged; computations, including proposal-network passes, are performed on CPU for consistency across filters.Source: §3.3 · p. 5

4 Experiments

Across increasingly difficult localization benchmarks, NOPF improves proposal efficiency and posterior tracking while keeping computational costs moderate. The experiments compare it with Kalman-based and particle-filter baselines across unimodal, multimodal, and data-association settings.

4 Experiments

  • The benchmarks span increasing dimensionality and inference complexity, use trajectories simulated from the true stochastic model, and report mean ± standard deviation across 10 random seeds.Comparisons include applicable Kalman-based baselines and standard or proposal-enhanced particle filters.Source: §4 · p. 5

4.1 Linear System with Biased Sensors

  • In the linear-Gaussian sensor-bias benchmark, NOPF closely matches OPF and ISPF in estimation accuracy and ESS, with wall time comparable to simple PF baselines.All methods achieve NMSE and PNLL comparable to the exact Kalman Filter reference.Source: §4.1 · p. 6, Figure 1 · p. 6

4.2 2D Robot Localization with Range and Bearing Observations

  • In approximately unimodal nonlinear localization, NOPF achieves NMSE, PNLL, and ES comparable to EnKF while yielding substantially higher ESS than other PF baselines.ISPF also improves over standard PF proposals but is considerably more expensive because it uses inner Monte Carlo sampling.Source: §4.2 · pp. 6–7, Figure 2 · p. 7

4.3 2D Robot Localization with Biased Range-Only Observations

  • NOPF preserves and resolves multimodal posterior structure more reliably than competing filters, improving PNLL and ES while maintaining high ESS; ISPF is substantially more expensive.The EnKF Gaussian approximation spreads probability mass across separated modes and produces an overly diffuse approximation.Source: §4.3 · p. 7, Figure 3 · p. 8

4.4 3D Robot Localization with Unordered Range-Only Observations

  • With 50,000 particles, NOPF improves PNLL and ES over standard PF baselines while maintaining high ESS and achieving similar performance to ISPF at substantially lower wall time.ISPF attains comparable distributional accuracy and higher ESS, suggesting the local proposal structure is more complex than a single Gaussian.Source: §4.4 · p. 8

5 Discussion and Limitations

NOPF combines learned proposals with standard particle-filter weighting, with favorable accuracy, sample-efficiency, and cost tradeoffs across the benchmarks. The discussion highlights limits from offline training, differentiability requirements, and Gaussian proposals in strongly multimodal local settings.

5 Discussion and Limitations

  • Within recursive one-step updates that leave previous particle locations fixed, the locally optimal proposal gives the least-degenerate benchmark, while importance weighting preserves the target filtering distribution.NOPF’s added cost is primarily a neural-network forward pass and Gaussian density evaluation.Source: §5 · pp. 8–9
  • Across benchmarks, NOPF matches Kalman-based and Gaussian baselines in approximately unimodal settings and outperforms standard PF baselines in multimodal problems.Gains grow with observational ambiguity and posterior multimodality; inference requires no online optimization or sigma-point construction beyond a neural-network forward pass.Source: §5 · p. 9
  • Offline training on forward-model trajectories can cause train-test mismatch when inference drifts beyond the training distribution or the simulator poorly represents the true system.With adequate proposal support, poor proposals primarily increase weight variance rather than asymptotic bias.Source: §5 · p. 9
  • ELBO training uses reparameterized proposal samples and requires differentiable transition-density and likelihood evaluations with respect to the proposed state.Non-differentiable simulators or observation operators require alternative gradient estimators or differentiable surrogate likelihood models.Source: §5 · p. 9
  • A Gaussian conditional proposal may be restrictive when one ancestor’s locally optimal proposal is strongly multimodal, as under severe data-association ambiguity.A unimodal proposal can average across modes and increase importance-weight variance; mixtures or normalizing flows may help in more strongly multimodal local settings.Source: §5 · pp. 9–10

A Additional Results

  • The supplementary results section examines metric sensitivity to particle count and includes additional diagnostics not shown in the main paper.The stated sensitivity analysis covers all benchmarks and averages results over 10 random seeds.Source: A Additional Results · p. 12

A.1 Metrics vs Number of Particles

The section examines how reported metrics vary with particle count across benchmarks. Across the tested range, NOPF maintains higher effective sample sizes and competitive or improved accuracy.

A.1 Metrics vs Number of Particles

  • Metrics are reported as the number of particles N varies across all benchmarks.The results are averaged over 10 random seeds.Source: §A.1 · p. 12
  • Across the tested particle counts, NOPF maintains consistently higher effective sample sizes.Source: §A.1 · p. 12
  • Across the tested particle counts, NOPF has competitive or improved accuracy.Source: §A.1 · p. 12

A.2 Tabulated Results

This appendix reports complete per-filter results for four benchmarks, with seven metrics per filter aggregated across 10 independent seeds. Figures 5–8 show metrics as particle count varies.

A.2 Tabulated Results

  • Each benchmark table reports seven metrics per filter: NMSE, PNLL, ES, ESS, WT, MADE, and Joint PICP.The tables indicate each metric’s preferred direction; the Joint PICP target is 0.9.Source: §A.2 · p. 12
  • Figures 5–8 plot metrics against particle count for the linear sensor-bias system and the three robot-localization benchmarks.The figure captions identify the benchmark for each plot but do not provide specific metric values or comparisons.Source: Figure 5 · p. 13, Figure 6 · p. 13, Figure 7 · p. 14
  • Tables 1–4 cover the linear system with biased sensors and three robot-localization benchmarks, reporting mean ± standard deviation across 10 seeds.The robot tasks use range and bearing, biased range-only, or unordered range-only observations.Source: §A.2 · p. 12, Table 1 · p. 15, Table 2 · p. 15

B Neural Proposal: Architecture and Training

The proposal network learns an observation-informed correction around deterministic state propagation, using a conditional-Gaussian parameterization. It is trained on simulated one-step transitions by minimizing a negative-ELBO objective, then used in online filtering.

B.1 Parameterization of Mean and Covariance Matrices for the Learned Proposal

  • A conditional-Gaussian proposal predicts a residual mean and Cholesky-factor covariance, with positive definiteness enforced through the Cholesky parameterization.The covariance head predicts the lower-triangular factor’s entries; the default backbone is a residual multilayer perceptron.Source: §B.1 · p. 16
  • The proposal network uses the deterministic noise-free prediction as input so it learns an observation-informed correction rather than relearning the forward model.The proposal is parameterized in whitened residual coordinates, scaled to the forward uncertainty to improve numerical conditioning.Source: §B.1 · p. 16

B.2 Loss Function

  • Training minimizes the negative ELBO, which is equivalent to minimizing the KL divergence from the learned proposal to the locally optimal one-step proposal.The objective’s expectation is estimated by Monte Carlo samples and averaged over minibatches of simulated one-step transitions.Source: §B.2 · p. 17

B.3 Training and Inference Algorithms

  • Training data are generated by rolling out the forward model and forming sliding-window state-observation pairs; proposal training then uses these simulated tuples.The algorithm descriptions specify the trajectory rollout and proposal-network training procedure.Source: §B.3 · pp. 17–18
  • At inference, the trained proposal supplies particles within the filtering algorithm, which returns a weighted particle approximation at each step.The inference algorithm takes the trained proposal, forward model, observation operator, observations, initial prior, and particle count as inputs.Source: §B.3 · p. 18

C Evaluation Metrics

  • Evaluation metrics are time-averaged over an index set that excludes the initial transient and, where needed, the final undefined step.The evaluation passage also defines state dimension and the number of Monte Carlo replicates used by sampling-based estimators.Source: C Evaluation Metrics · pp. 18–19

C.1 Normalized Mean Squared Error

This section defines complementary measures of state error, predictive quality, particle degeneracy, runtime, support coverage, and calibration. It also states where these measures have interpretive limits.

C.1 Normalized Mean Squared Error

  • NMSE compares the time-averaged squared error of the weighted posterior mean with the time-and-coordinate-averaged squared truth.Because it reduces the particle cloud to a point estimate, NMSE can be uninformative when that mean falls in a low-posterior-mass region of a multimodal distribution.Source: §C.1 · p. 19

C.2 Predictive Negative Log Likelihood

  • PNLL evaluates the time-averaged negative log predictive density assigned to the true next observation by the weighted particle mixture.It remains defined for symmetric or label-ambiguous posteriors because states producing the same observation distribution receive identical predictive scores.Source: §C.2 · p. 19

C.3 Energy Score

  • The energy score evaluates the weighted empirical distribution using wrapped angular distances, with Monte Carlo pairs replacing an O(N^2) term.The stated number of Monte Carlo pairs is M = 2048.Source: §C.3 · p. 19

C.4 Effective Sample Size

  • ESS is the time-averaged inverse sum of squared normalized weights, computed before resampling to diagnose degeneracy in the importance-sampling update.ESS ranges from 1 under full degeneracy to N under uniform weights, but does not by itself certify filter accuracy.Source: §C.4 · p. 20

C.5 Wall Clock Time

  • Wall-clock time measures end-to-end CPU runtime for one full trajectory, including proposal evaluation and filtering operations but excluding offline training and data simulation.The experiments run inference on CPU and use the same CPU node for filters within a benchmark to limit hardware confounding.Source: §C.5 · p. 20, D Computing Environment · p. 22

C.6.1 Mean Absolute Deviation Error

  • MADE records the minimum wrapped distance from any particle to the truth, regardless of that particle’s weight.It is a support-coverage diagnostic: a near-truth particle can keep MADE small even if it has negligible weight.Source: §C.6.1 · p. 20

C.6.2 Joint Particle In Credible-region Probability

  • Joint PICP checks whether the truth falls within a Gaussian-surrogate Mahalanobis ellipsoid, with 0.9 as the target coverage rate.Its calibration interpretation depends on the Gaussian residual assumption; multimodal posteriors can yield artificially high coverage, so it should be read alongside shape-aware scores.Source: §C.6.2 · pp. 20–21

E Particle Filter Implementations

The section describes standard and proposal-enhanced particle-filter implementations, including learned and sampling-based local proposals. It also specifies resampling and summarizes computational-cost assumptions.

E.3 Oracle Particle Filter

  • The Oracle Particle Filter provides a closed-form locally optimal proposal reference for the linear-Gaussian sensor-bias benchmark.It is used to assess how closely the learned proposal approaches that reference in the setting where the optimal proposal is available.Source: §E.3 · pp. 22–23

E.4 Unscented particle filter

  • The Unscented Particle Filter constructs a local Gaussian proposal for each particle using an Unscented Kalman Filter and sigma points.Its implementation adds diagonal jitter during sigma-point construction if the estimated covariance is not positive definite.Source: §E.4 · p. 23

E.5 Inner-Sampling Proposal Filter

  • ISPF approximates the locally optimal proposal by drawing among inner candidates with likelihood-proportional probabilities, using 200 inner samples per outer particle.The selected candidate likelihood is not applied again in the outer weight update because the inner draw already samples candidates according to likelihood.Source: §E.5 · p. 24

E.6 Auxiliary particle filter

  • The auxiliary particle filter selects ancestors using weights combining prior particle weights with likelihoods at deterministic predictive means, then corrects the auxiliary draw in a second-stage weight.The second-stage ratio uses the propagated particle’s true likelihood and the predictive likelihood used for ancestor selection.Source: §E.6 · pp. 24–25

E.7 Ensemble Kalman filter

  • The Ensemble Kalman Filter estimates update covariances from an ensemble and assigns uniform weights to the updated members.It recovers the Kalman filtering update in the large-ensemble limit when state and observation are jointly Gaussian.Source: §E.7 · p. 25

E.8 Neural Optimal Particle filter

  • NOPF replaces the importance density with a learned conditional proposal, conditioned on the deterministic prediction, observation, and time.It uses standard SIS weights evaluated from the observation likelihood, transition density, and proposal density; the network adds one evaluation per particle.Source: §E.8 · p. 25

E.9 Systematic resampling

  • Systematic resampling selects each resampled particle using a cumulative-weight threshold and resets all weights to 1/N.The threshold locations are evenly spaced after a single initial offset.Source: §E.9 · p. 26

E.10 Computational Complexity of PFs

  • The per-step complexity summary absorbs systematic resampling, Gaussian density evaluations, and other operations linear in the particle count into listed terms.The stated cost assumptions include additive process noise, allowing transition-density evaluation from the cached deterministic forward map without an additional forward-model call.Source: §E.10 · p. 26, Table 6 · p. 26

F Benchmark Specifications

The four benchmarks use additive process noise, while their observation models vary. The 3D unordered-range benchmark applies independent noise before sorting and uses a permutation-marginalized range likelihood.

F Benchmark Specifications

  • All four benchmarks use additive Gaussian process noise with diagonal covariance; the first three also use additive observation noise with diagonal covariance.Source: F Benchmark Specifications · p. 26
  • The 3D unordered-range benchmark adds independent noise to labeled raw ranges before sorting, then uses a permutation-marginalized likelihood.Table 7 summarizes key parameters for the four benchmarks.Source: F Benchmark Specifications · p. 26, Table 7 · p. 26

F.1 Benchmark 1: Linear System with Biased Sensors

The four benchmark specifications range from a linear sensor-bias system to 2D and 3D robot-localization settings. They vary in state components, observation structure, and prior, with the 3D case treating sorted range observations as unordered.

F.1 Benchmark 1: Linear System with Biased Sensors

  • The linear-Gaussian sensor-bias benchmark tracks position, velocity, and additive sensor bias on a 60-step grid, with specified diagonal process and observation noise and a Gaussian prior.The time grid is tk = 0.1(k + 1), k = 0, . . . , 59.Source: §F.1 · pp. 26–27

F.2 Benchmark 2: 2D Robot Localization with Range and Bearing Observations

  • Known-map 2D localization estimates robot pose, motion states, and odometry biases using range-bearing observations from four fixed landmarks and biased odometry channels.The bias states evolve as Gaussian random walks through additive process noise.Source: §F.2 · p. 27

F.3 Benchmark 3: 2D Robot Localization with Biased Range-Only Observations

  • The 2D range-only benchmark uses an anchor at (0, 0), and its motion and bias states evolve as Gaussian random walks under additive process noise.Its filtering prior is a mirrored two-component Gaussian mixture.Source: §F.3 · p. 28

F.4 Benchmark 4: 3D Unordered-Range Localization

  • The 3D benchmark stores four unordered range returns and three labeled proprioceptive measurements; because sorting destroys anchor-to-range correspondence, its range likelihood is permutation invariant across all 24 assignments.Source: §F.4 · p. 29
  • The state contains position, yaw and pitch, forward speed, angular rates, proprioceptive biases, per-anchor range biases, and slowly varying position-drift disturbances; its prior is a two-component positive-z Gaussian mixture.Source: §F.4 · pp. 28–29
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