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Error analysis of a compositional score-based algorithm for simulation-based inference

Camille Touron, Gabriel V. Cardoso, Julyan Arbel, Pedro L. C. Rodrigues

TL;DR

This work addresses how errors accumulate when combining multiple observations in compositional score-based SBI using the GAUSS framework. It derives an upper bound on the mean-squared error of the compositional score in terms of per-observation denoising score matching errors $\epsilon_{\mathrm{DSM}}^2$, precision-matrix estimation errors, and the number of observations $n$, with explicit results under Gaussian priors and posteriors. The analysis links the precision-error to a Wasserstein control $\mathcal{W}_2(p,\tilde{p})\le \eta$ and provides a detailed bound that illuminates how the diffusion hyperparameters and the aggregation via $\Lambda^{-1}$ temper error growth. A Gaussian 2D numerical example validates the bounds and highlights that the bound mirrors empirical behavior while revealing a nuanced interaction between $n$ and diffusion time. The results offer guidance for tuning diffusion parameters and precision-matrix estimation to achieve accurate sampling from the multi-observation posterior $p_t(\bm{\theta}|\bm{x}_{1:n})$ and outline avenues for extending the theory to alternative estimators beyond GAUSS.

Abstract

Simulation-based inference (SBI) has become a widely used framework in applied sciences for estimating the parameters of stochastic models that best explain experimental observations. A central question in this setting is how to effectively combine multiple observations in order to improve parameter inference and obtain sharper posterior distributions. Recent advances in score-based diffusion methods address this problem by constructing a compositional score, obtained by aggregating individual posterior scores within the diffusion process. While it is natural to suspect that the accumulation of individual errors may significantly degrade sampling quality as the number of observations grows, this important theoretical issue has so far remained unexplored. In this paper, we study the compositional score produced by the GAUSS algorithm of Linhart et al. (2024) and establish an upper bound on its mean squared error in terms of both the individual score errors and the number of observations. We illustrate our theoretical findings on a Gaussian example, where all analytical expressions can be derived in a closed form.

Error analysis of a compositional score-based algorithm for simulation-based inference

TL;DR

This work addresses how errors accumulate when combining multiple observations in compositional score-based SBI using the GAUSS framework. It derives an upper bound on the mean-squared error of the compositional score in terms of per-observation denoising score matching errors , precision-matrix estimation errors, and the number of observations , with explicit results under Gaussian priors and posteriors. The analysis links the precision-error to a Wasserstein control and provides a detailed bound that illuminates how the diffusion hyperparameters and the aggregation via temper error growth. A Gaussian 2D numerical example validates the bounds and highlights that the bound mirrors empirical behavior while revealing a nuanced interaction between and diffusion time. The results offer guidance for tuning diffusion parameters and precision-matrix estimation to achieve accurate sampling from the multi-observation posterior and outline avenues for extending the theory to alternative estimators beyond GAUSS.

Abstract

Simulation-based inference (SBI) has become a widely used framework in applied sciences for estimating the parameters of stochastic models that best explain experimental observations. A central question in this setting is how to effectively combine multiple observations in order to improve parameter inference and obtain sharper posterior distributions. Recent advances in score-based diffusion methods address this problem by constructing a compositional score, obtained by aggregating individual posterior scores within the diffusion process. While it is natural to suspect that the accumulation of individual errors may significantly degrade sampling quality as the number of observations grows, this important theoretical issue has so far remained unexplored. In this paper, we study the compositional score produced by the GAUSS algorithm of Linhart et al. (2024) and establish an upper bound on its mean squared error in terms of both the individual score errors and the number of observations. We illustrate our theoretical findings on a Gaussian example, where all analytical expressions can be derived in a closed form.
Paper Structure (16 sections, 9 theorems, 54 equations, 1 figure)

This paper contains 16 sections, 9 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on compositional score estimation with GAUSS
  3. Contributions
  4. Technical Appendices and Supplementary Material
  5. Conditional score-based modelling
  6. Gaussian test case
  7. Theoretical setting
  8. Numerical illustration
  9. Proof of Proposition \ref{['prop:diffusion_discretization_1']}
  10. Intermediate results
  11. Proof of Proposition \ref{['prop:diffusion_discretization_1']}
  12. Time-independence of precision errors
  13. Proof of Proposition \ref{['prop:diffusion_discretization_2']} - detailed version
  14. Proof of intermediate results on compositional score error
  15. Proof of detailed version of Proposition \ref{['prop:diffusion_discretization_2']}
  16. ...and 1 more sections

Key Result

Proposition 1

Choosing $0 < \eta < \min(\sqrt{\frac{\|\Sigma\|}{2}},f(\|\Sigma\|, \|\Sigma^{-1}\|))$ we get where and function $f:\mathbb{R}_+^2\to\mathbb{R}_+$ is defined in Appendix proof_prop_1.

Figures (1)

  • Figure 1: Solid lines stand for the evolution of the empirical MSE of the compositional score estimate (\ref{['eq:score-estimate']}) computed with the algorithm GAUSS at different times $t\in[0,1]$ of a diffusion process for the 2D Gaussian example (see Appendix \ref{['gaussian_ex']}). Dashed lines represent the evolution of the theoretical bound of the aforementioned compositional score error derived in Proposition \ref{['prop:diffusion_discretization_2']}. The empirical and theoretical evolutions are represented with respect to (A) the precision estimation error $\epsilon$ assuming exact individual scores are known, (B) the number of conditional observations $n$ and (C) the individual score error $\epsilon_{\mathrm{DSM}}^2$ that contributes both directly to the compositional score error and indirectly through the precision estimation. The choice of the fixed parameters, especially $\epsilon$ in panel (B) and (C), is discussed in detail in Appendix \ref{['numerical_illustration']}.

Theorems & Definitions (17)

  • Proposition 1: Precision matrix error
  • Proposition 2: Compositional score error
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3: prec_bound
  • proof
  • Lemma 2.1
  • proof
  • ...and 7 more