Distributional Computational Graphs: Error Bounds

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.

💡 Research Summary

This paper introduces a rigorous theoretical framework for analyzing “distributional computational graphs” (DCGs), which are directed acyclic graphs whose inputs are probability distributions rather than deterministic scalars. The authors focus on the error that arises when the continuous input distributions are replaced by finite, discrete approximations – a process they call quantization – and when the resulting discrete measures are subsequently compressed to keep the representation size manageable.

The main contributions are twofold. First, they derive a non‑asymptotic upper bound on the Wasserstein‑1 distance between the true output distribution of a DCG and the output obtained after quantizing all inputs and applying a generic compression scheme. The bound (Theorem 1.1) is expressed as a sum over all source vertices and all directed paths from each source to the unique terminal vertex Δ. For each path γ, the error contributed by that path is the product of the Lipschitz constants of the functions attached to the vertices along γ, multiplied by two terms: (i) the quantization error of the source distribution measured in Wasserstein‑1, and (ii) a compression term proportional to the path length |γ|, the inverse of the quantization depth n, and the diameter of the support of the quantized measure. In a simplified form (Equation 2) the bound can be read as

 error ≤ (graph size) × (max path distortion) × (quantization error + compression error).

Thus four factors govern the overall error: (1) the structural size of the graph (depth times number of source‑to‑terminal paths), (2) the worst‑case Lipschitz distortion along any path, (3) the quality of the quantization of each input distribution, and (4) the loss incurred by compressing the discrete measures. The compression term is shown to be essentially optimal without additional assumptions on the inputs.

Second, the authors apply this general result to a concrete numerical method: the Euler‑Maruyama scheme for stochastic differential equations. By interpreting each time‑step update as a node in a DCG, they obtain Theorem 1.2, which bounds the Wasserstein‑1 distance between the exact solution at step k and the compressed‑quantized approximation. The bound scales as

 W₁(μ_k, μ_k^{(n),c}) ≤ c e^{c′k} √n Δt² n,

where Δt = T/N is the time step, n is the quantization depth, and the factor √n originates from the compression error. Numerical experiments suggest that the term k √Δt captures the dominant behavior, while the √n factor’s tightness remains an open question.

The paper situates its contributions within the broader context of Monte‑Carlo methods, which achieve an expected Wasserstein‑1 error of order N^{-1/2} but suffer from slow convergence and dimensionality issues. In contrast, the deterministic DCG approach avoids sampling variability and provides explicit error guarantees that depend only on graph topology, Lipschitz properties, and quantization/compression parameters. However, the authors acknowledge that the compression bound may be improvable and that extending the analysis to vector‑valued outputs or more complex stochastic operators is a direction for future work.

Overall, the work fills a theoretical gap by delivering concrete, non‑asymptotic error bounds for deterministic propagation of probability measures through arbitrary computational graphs. It offers a solid foundation for designing deterministic, quantization‑based algorithms in areas such as option pricing, optimal stopping, and stochastic control, where controlling approximation error without resorting to large Monte‑Carlo simulations is highly desirable.