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This paper presents a unified theoretical framework for Learn-then-Differentiate (LTD), establishing rigorous accuracy guarantees for gradient estimation via fitted models. It demonstrates that LTD effectively differentiates learned representations of probability measures, recovering established results across kernel regression and neural networks while approaching standard Monte Carlo rates under smoothness conditions. These findings provide a robust mathematical foundation for analyzing gradient estimation techniques in simulation-based inference and probabilistic machine learning.
Learn-Then-Differentiate Gradient Estimation
arXiv:2609.38842v1 Announce Type: cross Abstract: Learn-then-differentiate (LTD) estimates gradients by fitting a model to simulation outputs and differentiating it. We develop a unified framework explaining what LTD differentiates and how accurately it estimates gradients. For models with a weighted representation, LTD differentiates a learned representation of the underlying probability measure. We then show how accuracy guarantees for fitted models translate into guarantees for gradients and higher-order derivatives, with rates approaching the standard Monte Carlo rate under suitable smoothness conditions. The framework recovers established results for kernel regression, local polynomial regression, and kernel ridge regression, and yields further guarantees for multiple kernel learning and smooth neural networks. These results provide a common foundation for understanding and analyzing LTD across learning methods.
Abstract:Learn-then-differentiate (LTD) estimates gradients by fitting a model to simulation outputs and differentiating it. We develop a unified framework explaining what LTD differentiates and how accurately it estimates gradients. For models with a weighted representation, LTD differentiates a learned representation of the underlying probability measure. We then show how accuracy guarantees for fitted models translate into guarantees for gradients and higher-order derivatives, with rates approaching the standard Monte Carlo rate under suitable smoothness conditions. The framework recovers established results for kernel regression, local polynomial regression, and kernel ridge regression, and yields further guarantees for multiple kernel learning and smooth neural networks. These results provide a common foundation for understanding and analyzing LTD across learning methods.
Computation (stat.CO); Machine Learning (cs.LG)
arXiv:2609.38842 [stat.CO]
(or arXiv:2609.38842v1 [stat.CO] for this version)
https://doi.org/10.48550/arXiv.2609.38842 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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