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Nonparametric autoregressive estimation of probability distributions with parallel gradient boosting

Rémy Chapelle · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.31524 · cited by 0 works (OpenAlex) · open-access PDF

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Tal desafío también surge con la estimación de la distribución condicional y ha llevado a algoritmos competitivos flexibles que buscan abordar el "curso de la dimensión" mediante la explotación de estructuras predictivas de una dimensión inferior a la del espacio ambiente. En este documento, estudiamos el uso de tales estimadores de distribución condicional para estimar las distribuciones conjuntas factorizando la última de una manera "autoregresiva". Mostramos que para las funciones optimizadas de pérdida que satisfagan una propiedad de subadditividad, los límites de la tasa de convergencia de un estimador autoregresivo pueden derivarse de los límites de la tasa de convergencia de los estimadores de distribución condicional. Estos límites relacionan la dificultad de estimar la distribución conjunta flexible con la de estimar el factor más desafiante en el

Machine translation (Meta M2M100); the English original follows.

Abstract

Nonparametric estimation of joint probability distributions is a core problem in statistics, which is notoriously challenging with high-dimensional data. Such challenges also arise with conditional distribution estimation and have led to flexible algorithms that aim to tackle the ''curse of dimensionality'' by exploiting predictive structures of lower dimension than the ambient space. In this paper, we study the use of such conditional distribution estimators to estimate joint distributions by factorizing the latter in an ''autoregressive'' way. We show that for loss functions satisfying a subadditivity property, bounds on the convergence rate of an autoregressive estimator can be derived from bounds on the convergence rate of the conditional distribution estimators. These bounds relate the difficulty of estimating the joint distribution to that of estimating the most challenging factor in the autoregressive factorization. When the conditional estimators adapt to sparsity in their conditioning sets, we characterize this difficulty through variable elimination in an undirected dependence graph. This allows bounding the rates of convergence by quantities related to the sparsity of the graph, and provides an additional motivation for the autoregressive approach. We then evaluate the practical performance of an autoregressive estimator based on parallel gradient boosting, a flexible nonparametric method for estimating conditional distributions. The estimator demonstrates competitive estimation performance on various simulated and real datasets, even without optimizing the factorization order. Together, these results support nonparametric autoregressive estimation as a practical approach to flexible joint distribution estimation.

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Topics: Autoregressive model, Estimator, Mathematics, Conditional probability distribution, Joint probability distribution, Nonparametric statistics

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