arXiv subjects

Jérémy Bettinger

Publications and source records attributed to Jérémy Bettinger.

4 recordsLinked to original sources

arXiv2026

Pointwise convergence of purely random partition estimators: from random trees to prototype rules

We study pointwise convergence rates of purely random partition estimators in nonparametric regression, where the partition -- into hyper-rectangles by purely random trees, or into Voronoi cells by prototype rules -- is built independently of the responses. Our analysis rests on a single geometric criterion, shape regularity, relating the diameter of a cell to its volume, which is shown by Bettinger, Portier and Saumard (2026) to be necessary and sufficient, up to logarithmic factors, for achieving the minimax rate $n^{-1/(d+2)}$. We show that centered and uniform trees are not shape-regular -- their cells' aspect ratio grows exponentially with the number of splits with probability bounded away from zero -- explaining the super-logarithmic corrections in their error bounds, whereas Mondrian trees, whose splits adapt to the current cell geometry, are shape-regular in probability and attain the minimax rate. The same analysis applied to Voronoi partitions yields the first pointwise concentration bounds for Proto-NN, resolving an open problem of Györfi and Weiss (2021), and shows that OptiNet achieves the minimax rate with markedly better success probability -- even almost surely, for a suitable choice of parameters -- thanks to its $η$-net construction.

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arXiv2026

Revisiting local regression: shape regularity, uniform rates, and the limits of random splits

Considering pointwise and sup-norm estimation, we analyze the non-asymptotic behavior of local averaging estimators for Lipschitz regression functions. Building on a general deviation bound for estimators based on a VC family of localizing sets, we introduce the notion of shape-regular local maps, where averaging is performed over sets with an almost isotropic geometry. Our main message is a characterization: shape regularity is both necessary and sufficient to attain optimal rates, up to logarithmic factors. Necessity is established non-asymptotically through an explicit anisotropic example, sharpening a phenomenon previously understood only heuristically in asymptotic theory. We then draw two consequences. First, the simple $k$-nearest neighbor rule is shape-regular by construction and attains the optimal rate, even on unbounded supports. Second, and perhaps surprisingly, the popular random-split condition for trees -- known to ensure consistency and vanishing cell diameters -- does not guarantee optimal rates: for blind tree constructions, the cell aspect ratio diverges exponentially with depth, so that shape regularity fails with positive probability. This identifies the absence of a geometric correction mechanism, rather than a slowly shrinking diameter, as the obstruction to optimality. Motivated by this gap, we propose a tree construction that enforces shape regularity through a simple constraint on admissible splits, and prove a uniform deviation inequality showing that it restores the optimal rate for Lipschitz functions.

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arXiv2025

On the pointwise and sup-norm errors for local regression estimators

In this paper, we analyze the behavior of various non-parametric local regression estimators, i.e. estimators that are based on local averaging, for estimating a Lipschitz regression function at a fixed point, or in sup-norm. We first prove some deviation bounds for local estimators that can be indexed by a VC class of sets in the covariates space. We then introduce the general concept of shape-regular local maps, corresponding to the situation where the local averaging is done on sets which, in some sense, have ``almost isotropic'' shapes. On the one hand, we prove that, in general, shape-regularity is necessary to achieve the minimax rates of convergence. On the other hand, we prove that it is sufficient to ensure the optimal rates, up to some logarithmic factors. Next, we prove some deviation bounds for specific estimators, that are based on data-dependent local maps, such as nearest neighbors, their recent prototype variants, as well as a new algorithm, which is a modified and generalized version of CART, and that is minimax rate optimal in sup-norm. In particular, the latter algorithm is based on a random tree construction that depends on both the covariates and the response data. For each of the estimators, we provide insights on the shape-regularity of their respective local maps. Finally, we conclude the paper by establishing some probability bounds for local estimators based on purely random trees, such as centered, uniform or Mondrian trees. Again, we discuss the relations between the rates of the estimators and the shape-regularity of their local maps.

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arXiv2025

A theory of shape regularity for local regression maps

We introduce the concept of shape-regular regression maps as a framework to derive optimal rates of convergence for various non-parametric local regression estimators. Using Vapnik-Chervonenkis theory, we establish upper and lower bounds on the pointwise and the sup-norm estimation error, even when the localization procedure depends on the full data sample, and under mild conditions on the regression model. Our results demonstrate that the shape regularity of regression maps is not only sufficient but also necessary to achieve an optimal rate of convergence for Lipschitz regression functions. To illustrate the theory, we establish new concentration bounds for many popular local regression methods such as nearest neighbors algorithm, CART-like regression trees and several purely random trees including Mondrian trees.

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