arXiv subjects

François Portier

Publications and source records attributed to François Portier.

At least 19 recordsLinked to original sources

arXiv2026

Minimax optimality for sequential gradient-free minimization of smooth functions and their derivatives

We consider the problem of noisy gradient-free minimization of the k-th order partial derivative of a $β$-H{ö}lder function supported on a d-dimensional cube. We show that T ^{($β$+d+k)/(2$β$+d)} log(T )^{(β-k)/(2β+d)} is a non-asymptotic minimax rate of the T step cumulative regret for all $β$ \ge 0. In the special case k = 0, our results cover the problem of noisy gradient-free minimization of $β$-H{ö}lder functions, closing the existing gap between the known upper and lower bounds. We show that a minimizer of a suitably chosen local polynomial estimator is rate-optimal. The minimax optimal upper bound is achieved under the passive design, that is, when the query points are i.i.d. Thus, there is no advantage in considering sequential designs when it is only known that f is a $β$-H{ö}lder function with no additional property. We propose an algorithm feasible in polynomial time that constructs a proxy of the minimizer of the local polynomial estimator. The procedure requires computing the estimator on auxiliary random points. The resulting polynomial time algorithm matches the lower bound.

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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

A uniform relative deviation inequality for VC-subgraph classes

We establish a new Bernstein-type deviation inequality for classes of functions whose complexity is characterized through subgraphs. The inequality is non-asymptotic, involves explicit constants, and features a relative normalization by the probability level. Applied to kernel density estimation, it produces a location and bandwidth-adaptive error bound between the estimator and the smoothed density, holding simultaneously over all points on the real line and all positive bandwidths. The proof is elementary, combining a new symmetrization principle, which incorporates the relative normalization, with the maximal sub-Gaussian inequality, and requires neither concentration nor entropy-integral arguments. When specialized to classes of sets, our technique improves the constants in the classical Vapnik-Chervonenkis inequality with relative deviation of Anthony and Shawe-Taylor (1993), reducing the factor 4 S A (2n) to S A (2n) in the right-tail inequality and to 3 S A (2n) in the left-tail inequality.

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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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arXiv2026

Logistic lasso regression with nearest neighbors for gradient-based dimension reduction

This paper investigates a new approach to estimate the gradient of the conditional probability given the covariates in the binary classification framework. The proposed approach consists of fitting a localized nearest-neighbor logistic model with $\ell_1$-penalty in order to cope with possibly high-dimensional covariates. Our theoretical analysis shows that the pointwise convergence rate of the gradient estimator is optimal under very mild assumptions. Moreover, using an outer product of such gradient estimates at several points in the covariate space, we provide a new method for estimating the central subspace, a well-known object allowing to carry out dimension reduction within the covariate space. Our implementation uses cross-validation on the misclassification rate to estimate the dimension of this subspace. We find that the proposed approach outperforms existing competitors in synthetic and real data applications.

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arXiv2026

Riemannian Stochastic Optimization for Sufficient Dimension Reduction

Sufficient dimension reduction (SDR) makes high-dimensional regression tractable by projecting the covariates onto a low-dimensional subspace that preserves the conditional mean of the response. Existing gradient-based estimators either operate in the ambient space and suffer from the curse of dimensionality, or localize in the reduced space at a per-outer-iteration cost at least quadratic in the sample size. We show that minimizers of the population Minimum Average Variance Estimation (MAVE) risk approximate the same Grassmannian target as the Outer Product of Gradients (OPG), and recast the empirical criterion as a smooth maximization on the Stiefel manifold with closed-form Riemannian gradient. The resulting algorithm, SMAVE, combines sparse projected-space nearest-neighbor localization with Riemannian stochastic gradient ascent. A simplified version comes with almost-sure convergence and a non-asymptotic rate matching the standard non-convex stochastic first-order scaling. Empirically, SMAVE matches or improves on RMAVE's synthetic subspace recovery at moderate-to-high ambient dimension, and on four real datasets it uniformly improves over OPG and is competitive with or outperforms RMAVE at orders of magnitude lower runtime.

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arXiv2026

Importance Sampling Optimization with Laplace Principle

Grid search and random search are widely used techniques for hyperparameter tuning in machine learning, especially when gradient information is unavailable. In these methods, a finite set of candidate configurations is evaluated, and the best-performing one is selected. We propose a simple and computationally inexpensive refinement of this paradigm: instead of selecting a single best point, we form a weighted average of the evaluated configurations, where the weights are chosen using an importance sampling scheme inspired by the Laplace principle. This scheme can be implemented as a post-processing step on top of a random search, with no additional function evaluations. We also propose an iterative variant, where the sampling distributions are chosen adaptively to generate new candidate points around the previous estimate, in the spirit of Evolution Strategy (ES) methods. In a general non-convex setting, we show that, after n evaluations, the error of the proposed methods is of smaller order than n -2/(d+2) . This compares favorably to random search or grid search rates of n -1/d as soon as d > 2. We illustrate the practical benefits of this averaging strategy on several examples.

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arXiv2025

Concentration and excess risk bounds for imbalanced classification with synthetic oversampling

Synthetic oversampling of minority examples using SMOTE and its variants is a leading strategy for addressing imbalanced classification problems. Despite the success of this approach in practice, its theoretical foundations remain underexplored. We develop a theoretical framework to analyze the behavior of SMOTE and related methods when classifiers are trained on synthetic data. We first derive a uniform concentration bound on the discrepancy between the empirical risk over synthetic minority samples and the population risk on the true minority distribution. We then provide a nonparametric excess risk guarantee for kernel-based classifiers trained using such synthetic data. These results lead to practical guidelines for better parameter tuning of both SMOTE and the downstream learning algorithm. Numerical experiments are provided to illustrate and support the theoretical findings

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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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arXiv2024

Stochastic mirror descent for nonparametric adaptive importance sampling

This paper addresses the problem of approximating an unknown probability distribution with density $f$ -- which can only be evaluated up to an unknown scaling factor -- with the help of a sequential algorithm that produces at each iteration $n\geq 1$ an estimated density $q_n$.The proposed method optimizes the Kullback-Leibler divergence using a mirror descent (MD) algorithm directly on the space of density functions, while a stochastic approximation technique helps to manage between algorithm complexity and variability. One of the key innovations of this work is the theoretical guarantee that is provided for an algorithm with a fixed MD learning rate $η\in (0,1 )$. The main result is that the sequence $q_n$ converges almost surely to the target density $f$ uniformly on compact sets. Through numerical experiments, we show that fixing the learning rate $η\in (0,1 )$ significantly improves the algorithm's performance, particularly in the context of multi-modal target distributions where a small value of $η$ allows to increase the chance of finding all modes. Additionally, we propose a particle subsampling method to enhance computational efficiency and compare our method against other approaches through numerical experiments.

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arXiv2024

Cross-validation on Extreme Regions

We conduct a non asymptotic study of the Cross Validation (CV) estimate of the generalization risk for learning algorithms dedicated to extreme regions of the covariates space. In this Extreme Value Analysis context, the risk function measures the algorithm's error given that the norm of the input exceeds a high quantile. The main challenge within this framework is the negligible size of the extreme training sample with respect to the full sample size and the necessity to re-scale the risk function by a probability tending to zero. We open the road to a finite sample understanding of CV for extreme values by establishing two new results: an exponential probability bound on the \Kfold CV error and a polynomial probability bound on the leave-\textrm{p}-out CV. Our bounds are sharp in the sense that they match state-of-the-art guarantees for standard CV estimates while extending them to encompass a conditioning event of small probability. We illustrate the significance of our results regarding high dimensional classification in extreme regions via a Lasso-type logistic regression algorithm. The tightness of our bounds is investigated in numerical experiments.

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arXiv2024

Nearest Neighbor Sampling for Covariate Shift Adaptation

Many existing covariate shift adaptation methods estimate sample weights given to loss values to mitigate the gap between the source and the target distribution. However, estimating the optimal weights typically involves computationally expensive matrix inversion and hyper-parameter tuning. In this paper, we propose a new covariate shift adaptation method which avoids estimating the weights. The basic idea is to directly work on unlabeled target data, labeled according to the $k$-nearest neighbors in the source dataset. Our analysis reveals that setting $k = 1$ is an optimal choice. This property removes the necessity of tuning the only hyper-parameter $k$ and leads to a running time quasi-linear in the sample size. Our results include sharp rates of convergence for our estimator, with a tight control of the mean square error and explicit constants. In particular, the variance of our estimators has the same rate of convergence as for standard parametric estimation despite their non-parametric nature. The proposed estimator shares similarities with some matching-based treatment effect estimators used, e.g., in biostatistics, econometrics, and epidemiology. Our experiments show that it achieves drastic reduction in the running time with remarkable accuracy.

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arXiv2024

Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates

The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.

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arXiv2024

Sharp error bounds for imbalanced classification: how many examples in the minority class?

When dealing with imbalanced classification data, reweighting the loss function is a standard procedure allowing to equilibrate between the true positive and true negative rates within the risk measure. Despite significant theoretical work in this area, existing results do not adequately address a main challenge within the imbalanced classification framework, which is the negligible size of one class in relation to the full sample size and the need to rescale the risk function by a probability tending to zero. To address this gap, we present two novel contributions in the setting where the rare class probability approaches zero: (1) a non asymptotic fast rate probability bound for constrained balanced empirical risk minimization, and (2) a consistent upper bound for balanced nearest neighbors estimates. Our findings provide a clearer understanding of the benefits of class-weighting in realistic settings, opening new avenues for further research in this field.

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arXiv2024

Nearest neighbor empirical processes

In the regression framework, the empirical measure based on the responses resulting from the nearest neighbors, among the covariates, to a given point $x$ is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a local bracketing entropy condition on the underlying class of functions reflecting the localizing nature of the nearest neighbor algorithm. Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is simply equal to the conditional covariance operator given the covariate value. This suggests the possibility of using standard formulas to estimate the variance by using only the nearest neighbors instead of the full data. This is illustrated on two problems: the estimation of the conditional cumulative distribution function and local linear regression.

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arXiv2024

Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence

A novel linear integration rule called $\textit{control neighbors}$ is proposed in which nearest neighbor estimates act as control variates to speed up the convergence rate of the Monte Carlo procedure on metric spaces. The main result is the $\mathcal{O}(n^{-1/2} n^{-s/d})$ convergence rate -- where $n$ stands for the number of evaluations of the integrand and $d$ for the dimension of the domain -- of this estimate for Hölder functions with regularity $s \in (0,1]$, a rate which, in some sense, is optimal. Several numerical experiments validate the complexity bound and highlight the good performance of the proposed estimator.

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arXiv2023

Asymptotic Analysis of Conditioned Stochastic Gradient Descent

In this paper, we investigate a general class of stochastic gradient descent (SGD) algorithms, called Conditioned SGD, based on a preconditioning of the gradient direction. Using a discrete-time approach with martingale tools, we establish under mild assumptions the weak convergence of the rescaled sequence of iterates for a broad class of conditioning matrices including stochastic first-order and second-order methods. Almost sure convergence results, which may be of independent interest, are also presented. Interestingly, the asymptotic normality result consists in a stochastic equicontinuity property so when the conditioning matrix is an estimate of the inverse Hessian, the algorithm is asymptotically optimal.

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