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Distributional Computational Graphs: Error Bounds

Olof Hallqvist Elias, Michael Selby, Phillip Stanley-Marbell

TL;DR

Distributional computational graphs (DCGs) propagate known input distributions through deterministic operations and study how finite input representations affect the output distribution. The authors derive non-asymptotic Wasserstein-1 bounds that quantify error propagation along all source-to-output paths, with the amplification determined by path Lipschitz constants and the graph structure. They provide a general bound for arbitrary DCGs and a specialized bound for the Euler–Maruyama scheme, including a sharp Gaussian quantization rate $W_1(\mu,\mu^{(n)})=O(2^{-n})$ and an EM error bound of the form $W_1(\mu_k,\mu_k^{(n),c})\le c e^{c' k\sqrt{n\Delta t}}/2^{n}$. This work offers a deterministic alternative to Monte Carlo with explicit error control for propagating distributions in numerical schemes and probabilistic reasoning, highlighting trade-offs between graph complexity, approximation level, and accuracy.

Abstract

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated using finite approximations of continuous probability distributions. Such an approximation might be the result of representing a continuous real-valued distribution using a discrete representation or from constructing an empirical distribution from samples (or might be the output of another distributional computational graph). We establish non-asymptotic error bounds in terms of the Wasserstein-1 distance, without imposing structural assumptions on the computational graph.

Distributional Computational Graphs: Error Bounds

TL;DR

Distributional computational graphs (DCGs) propagate known input distributions through deterministic operations and study how finite input representations affect the output distribution. The authors derive non-asymptotic Wasserstein-1 bounds that quantify error propagation along all source-to-output paths, with the amplification determined by path Lipschitz constants and the graph structure. They provide a general bound for arbitrary DCGs and a specialized bound for the Euler–Maruyama scheme, including a sharp Gaussian quantization rate and an EM error bound of the form . This work offers a deterministic alternative to Monte Carlo with explicit error control for propagating distributions in numerical schemes and probabilistic reasoning, highlighting trade-offs between graph complexity, approximation level, and accuracy.

Abstract

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated using finite approximations of continuous probability distributions. Such an approximation might be the result of representing a continuous real-valued distribution using a discrete representation or from constructing an empirical distribution from samples (or might be the output of another distributional computational graph). We establish non-asymptotic error bounds in terms of the Wasserstein-1 distance, without imposing structural assumptions on the computational graph.
Paper Structure (16 sections, 15 theorems, 137 equations, 5 figures, 3 algorithms)

This paper contains 16 sections, 15 theorems, 137 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main result
  3. Notation and Definitions
  4. Directed acyclic graphs
  5. Wasserstein spaces
  6. Quantization and compression algorithms
  7. Distributional computational graphs
  8. Examples
  9. The Euler-Maruyama approximation scheme
  10. Order statistics and bubble sort
  11. Upper Bound for General Computational Graphs
  12. Approximation errors
  13. Error Bounds for the Euler-Maruyama Scheme
  14. Quantization error for Gaussian distribution
  15. Error bound for the Euler-Maruyama scheme
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(G,\ {\mathcal{F}})$ be a computational graph and let $\mu_s, s \in {\mathsf{S}}$ be a collection of finite-mean probability measures. Let $\mu = \bigotimes_{s\in {\mathsf{S}}} \mu_s$, and let $\mu^{(n)} = \bigotimes_{s\in {\mathsf{S}}} \mu_s^{(n)}$ denote the quantized input measures. Furtherm

Figures (5)

  • Figure 1: Section of the computational graph for Euler-Maruyama. Dots indicate the inductive continuation of the graph.
  • Figure 2: Section of the computational graph for Bubble sort showing iteration $i$ for $j = 0,\dots,n-i-2$.
  • Figure 3: $\log W_1(Y_N, Y_N^{(n),c})$ as a function of $N$ (left) and $n$ (right).
  • Figure :
  • Figure :

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1: Lemma 3.1 in bilgin2025quantizationprobabilitydistributionsdivideandconquer
  • Lemma 2.2: Lemma 4.3 in bilgin2025quantizationprobabilitydistributionsdivideandconquer
  • Lemma 2.3
  • Remark 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 25 more