ScoreMatchingRiesz: Score Matching for Debiased Machine Learning and Policy Path Estimation

We propose ScoreMatchingRiesz, a family of Riesz representer estimators based on score matching. The Riesz representer is a key nuisance component in debiased machine learning, enabling $\sqrt{n}$-consistent and asymptotically efficient estimation of causal and structural targets via Neyman-orthogonal scores. We formulate Riesz representer estimation as a score estimation problem. This perspective stabilizes representer estimation by allowing us to leverage denoising score matching and telescoping density ratio estimation. We also introduce the policy path, a parameter that captures how policy effects evolve under continuous treatments. We show that the policy path can be estimated via score matching by smoothly connecting average marginal effect (AME) and average policy effect (APE) estimation, which improves the interpretability of policy effects.

💡 Research Summary

ScoreMatchingRiesz introduces a unified framework for estimating the Riesz representer—an essential nuisance component in debiased machine learning—by casting the problem as a score‑matching task. The Riesz representer α₀ appears in Neyman‑orthogonal scores ψ(W; η, θ) and must be estimated accurately to achieve √n‑consistent, asymptotically efficient inference for causal and structural parameters. Traditional approaches (Riesz regression, nearest‑neighbor matching, covariate‑balancing) struggle with high‑dimensional over‑fitting or numerical instability.

The authors propose two complementary notions of “score.”

1. Data score s_data(x)=∇ₓ log p₀(x), the gradient of the log‑density of the observed data. It can be estimated by Hyvärinen score matching (HSM) or, more practically in high dimensions, by Denoising Score Matching (DSM). DSM adds Gaussian noise to the data, trains a neural network to predict the noisy‑sample score, and then recovers the clean‑data score as the limit σ→0. This is exactly the loss used in modern diffusion models and avoids costly Jacobian‑trace calculations.
2. Time score s_timeₜ(x)=∂ₜ log pₜ(x), where {pₜ}_{t∈