arXiv subjects

Arnak S. Dalalyan

Publications and source records attributed to Arnak S. Dalalyan.

At least 19 recordsLinked to original sources

arXiv2026

Universal Local Error and Realized Amplification for the First-Order EDM Predictor

We analyze the first-order deterministic diffusion sampler of Karras et al. (2022), termed EDM, in 2-Wasserstein distance by separating two sources of error: local discretization error and its amplification by subsequent learned steps. We prove that local error admits a universal bound: for any data distribution with finite second moment, the one-step discretization error is quadratic in the step size, with an explicit constant that does not depend on the data distribution. Error propagation, in contrast, depends on the learned network. At high noise levels, we exploit the network parametrization of EDM to derive an explicit contraction criterion. At low noise levels, we measure propagation through the amplification realized on the distributions transported by the sampler; this realized amplification can be arbitrarily smaller than the worst-case Lipschitz constant. This analysis yields an $O(e^{Λ_K}/K)$ global discretization error for $K$ sampling steps, where $Λ_K$ is the low-noise log-amplification. Experiments on a one-dimensional Gaussian mixture show how measured amplification accounts for slower error decay on finite sampling grids. Diagnostics on a pretrained CIFAR-10 model illustrate related stability mechanisms without certifying the global assumptions.

stat.ML↗

arXiv2026

Boundary-layer asymptotics for Gaussian-smoothed singular measures

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

math.PR↗

arXiv2026

Improved Guarantees for Langevin Monte Carlo with Average Smoothness

We establish improved nonasymptotic bounds for Langevin Monte Carlo in the strongly log-concave setting, when the error is measured by the Wasserstein distance. The main result shows that the discretization error is governed by an average coordinate-wise smoothness constant, rather than by the usual global smoothness constant. The proof is short and probabilistic, and relies on a refined use of the synchronous coupling. We further show that the same ideas lead to improved bounds for variable step sizes, for potentials whose Laplacian is Lipschitz-continuous, and for finite-sum problems sampled by stochastic-gradient Langevin dynamics with fixed point control variates. In the Laplacian-smooth case, the usual Hessian-Lipschitz contribution is replaced by a weaker trace-type third-order smoothness quantity. In the finite-sum setting, the resulting SGLD bound improves the dependence on the root mean square smoothness of the component functions. Applications to generalized linear models with Gaussian design show that these refinements can yield substantial, dimension-dependent improvements over previously known bounds, especially for correlated covariates.

math.ST↗

arXiv2024

User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

math.ST↗

arXiv2023

Graphon Estimation in bipartite graphs with observable edge labels and unobservable node labels

Many real-world data sets can be presented in the form of a matrix whose entries correspond to the interaction between two entities of different natures (number of times a web user visits a web page, a student's grade in a subject, a patient's rating of a doctor, etc.). We assume in this paper that the mentioned interaction is determined by unobservable latent variables describing each entity. Our objective is to estimate the conditional expectation of the data matrix given the unobservable variables. This is presented as a problem of estimation of a bivariate function referred to as graphon. We study the cases of piecewise constant and Hölder-continuous graphons. We establish finite sample risk bounds for the least squares estimator and the exponentially weighted aggregate. These bounds highlight the dependence of the estimation error on the size of the data set, the maximum intensity of the interactions, and the level of noise. As the analyzed least-squares estimator is intractable, we propose an adaptation of Lloyd's alternating minimization algorithm to compute an approximation of the least-squares estimator. Finally, we present numerical experiments in order to illustrate the empirical performance of the graphon estimator on synthetic data sets.

math.ST↗

arXiv2022

Simple proof of the risk bound for denoising by exponential weights for asymmetric noise distributions

In this note, we consider the problem of aggregation of estimators in order to denoise a signal. The main contribution is a short proof of the fact that the exponentially weighted aggregate satisfies a sharp oracle inequality. While this result was already known for a wide class of symmetric noise distributions, the extension to asymmetric distributions presented in this note is new.

math.ST↗

arXiv2022

Nearly minimax robust estimator of the mean vector by iterative spectral dimension reduction

We study the problem of robust estimation of the mean vector of a sub-Gaussian distribution. We introduce an estimator based on spectral dimension reduction (SDR) and establish a finite sample upper bound on its error that is minimax-optimal up to a logarithmic factor. Furthermore, we prove that the breakdown point of the SDR estimator is equal to $1/2$, the highest possible value of the breakdown point. In addition, the SDR estimator is equivariant by similarity transforms and has low computational complexity. More precisely, in the case of $n$ vectors of dimension $p$ -- at most $\varepsilon n$ out of which are adversarially corrupted -- the SDR estimator has a squared error of order $\big(\frac{r_Σ}{n} + \varepsilon^2\log(1/\varepsilon)\big){\log p}$ and a running time of order $p^3 + n p^2$. Here, $r_Σ\le p$ is the effective rank of the covariance matrix of the reference distribution. Another advantage of the SDR estimator is that it does not require knowledge of the contamination rate and does not involve sample splitting. We also investigate extensions of the proposed algorithm and of the obtained results in the case of (partially) unknown covariance matrix.

math.ST↗

arXiv2022

Risk bounds for aggregated shallow neural networks using Gaussian prior

Analysing statistical properties of neural networks is a central topic in statistics and machine learning. However, most results in the literature focus on the properties of the neural network minimizing the training error. The goal of this paper is to consider aggregated neural networks using a Gaussian prior. The departure point of our approach is an arbitrary aggregate satisfying the PAC-Bayesian inequality. The main contribution is a precise nonasymptotic assessment of the estimation error appearing in the PAC-Bayes bound. We also review available bounds on the error of approximating a function by a neural network. Combining bounds on estimation and approximation errors, we establish risk bounds that are sharp enough to lead to minimax rates of estimation over Sobolev smoothness classes.

math.ST↗

arXiv2021

Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets

In this paper, we provide non-asymptotic upper bounds on the error of sampling from a target density using three schemes of discretized Langevin diffusions. The first scheme is the Langevin Monte Carlo (LMC) algorithm, the Euler discretization of the Langevin diffusion. The second and the third schemes are, respectively, the kinetic Langevin Monte Carlo (KLMC) for differentiable potentials and the kinetic Langevin Monte Carlo for twice-differentiable potentials (KLMC2). The main focus is on the target densities that are smooth and log-concave on $\mathbb R^p$, but not necessarily strongly log-concave. Bounds on the computational complexity are obtained under two types of smoothness assumption: the potential has a Lipschitz-continuous gradient and the potential has a Lipschitz-continuous Hessian matrix. The error of sampling is measured by Wasserstein-$q$ distances. We advocate for the use of a new dimension-adapted scaling in the definition of the computational complexity, when Wasserstein-$q$ distances are considered. The obtained results show that the number of iterations to achieve a scaled-error smaller than a prescribed value depends only polynomially in the dimension.

math.ST↗

arXiv2021

All-In-One Robust Estimator of the Gaussian Mean

The goal of this paper is to show that a single robust estimator of the mean of a multivariate Gaussian distribution can enjoy five desirable properties. First, it is computationally tractable in the sense that it can be computed in a time which is at most polynomial in dimension, sample size and the logarithm of the inverse of the contamination rate. Second, it is equivariant by translations, uniform scaling and orthogonal transformations. Third, it has a high breakdown point equal to $0.5$, and a nearly-minimax-rate-breakdown point approximately equal to $0.28$. Fourth, it is minimax rate optimal, up to a logarithmic factor, when data consists of independent observations corrupted by adversarially chosen outliers. Fifth, it is asymptotically efficient when the rate of contamination tends to zero. The estimator is obtained by an iterative reweighting approach. Each sample point is assigned a weight that is iteratively updated by solving a convex optimization problem. We also establish a dimension-free non-asymptotic risk bound for the expected error of the proposed estimator. It is the first result of this kind in the literature and involves only the effective rank of the covariance matrix. Finally, we show that the obtained results can be extended to sub-Gaussian distributions, as well as to the cases of unknown rate of contamination or unknown covariance matrix.

math.ST↗

arXiv2020

Penalized Langevin dynamics with vanishing penalty for smooth and log-concave targets

We study the problem of sampling from a probability distribution on $\mathbb R^p$ defined via a convex and smooth potential function. We consider a continuous-time diffusion-type process, termed Penalized Langevin dynamics (PLD), the drift of which is the negative gradient of the potential plus a linear penalty that vanishes when time goes to infinity. An upper bound on the Wasserstein-2 distance between the distribution of the PLD at time $t$ and the target is established. This upper bound highlights the influence of the speed of decay of the penalty on the accuracy of the approximation. As a consequence, considering the low-temperature limit we infer a new nonasymptotic guarantee of convergence of the penalized gradient flow for the optimization problem.

math.ST↗

arXiv2020

Confidence regions and minimax rates in outlier-robust estimation on the probability simplex

We consider the problem of estimating the mean of a distribution supported by the $k$-dimensional probability simplex in the setting where an $\varepsilon$ fraction of observations are subject to adversarial corruption. A simple particular example is the problem of estimating the distribution of a discrete random variable. Assuming that the discrete variable takes $k$ values, the unknown parameter $\boldsymbol θ$ is a $k$-dimensional vector belonging to the probability simplex. We first describe various settings of contamination and discuss the relation between these settings. We then establish minimax rates when the quality of estimation is measured by the total-variation distance, the Hellinger distance, or the $\mathbb L^2$-distance between two probability measures. We also provide confidence regions for the unknown mean that shrink at the minimax rate. Our analysis reveals that the minimax rates associated to these three distances are all different, but they are all attained by the sample average. Furthermore, we show that the latter is adaptive to the possible sparsity of the unknown vector. Some numerical experiments illustrating our theoretical findings are reported.

math.ST↗

arXiv2019

Outlier-robust estimation of a sparse linear model using $\ell_1$-penalized Huber's $M$-estimator

We study the problem of estimating a $p$-dimensional $s$-sparse vector in a linear model with Gaussian design and additive noise. In the case where the labels are contaminated by at most $o$ adversarial outliers, we prove that the $\ell_1$-penalized Huber's $M$-estimator based on $n$ samples attains the optimal rate of convergence $(s/n)^{1/2} + (o/n)$, up to a logarithmic factor. For more general design matrices, our results highlight the importance of two properties: the transfer principle and the incoherence property. These properties with suitable constants are shown to yield the optimal rates, up to log-factors, of robust estimation with adversarial contamination.

math.ST↗

arXiv2019

A nonasymptotic law of iterated logarithm for general M-estimators

M-estimators are ubiquitous in machine learning and statistical learning theory. They are used both for defining prediction strategies and for evaluating their precision. In this paper, we propose the first non-asymptotic "any-time" deviation bounds for general M-estimators, where "any-time" means that the bound holds with a prescribed probability for every sample size. These bounds are nonasymptotic versions of the law of iterated logarithm. They are established under general assumptions such as Lipschitz continuity of the loss function and (local) curvature of the population risk. These conditions are satisfied for most examples used in machine learning, including those ensuring robustness to outliers and to heavy tailed distributions. As an example of application, we consider the problem of best arm identification in a parametric stochastic multi-arm bandit setting. We show that the established bound can be converted into a new algorithm, with provably optimal theoretical guarantees. Numerical experiments illustrating the validity of the algorithm are reported.

math.ST↗

arXiv2018

On sampling from a log-concave density using kinetic Langevin diffusions

Langevin diffusion processes and their discretizations are often used for sampling from a target density. The most convenient framework for assessing the quality of such a sampling scheme corresponds to smooth and strongly log-concave densities defined on $\mathbb R^p$. The present work focuses on this framework and studies the behavior of Monte Carlo algorithms based on discretizations of the kinetic Langevin diffusion. We first prove the geometric mixing property of the kinetic Langevin diffusion with a mixing rate that is, in the overdamped regime, optimal in terms of its dependence on the condition number. We then use this result for obtaining improved guarantees of sampling using the kinetic Langevin Monte Carlo method, when the quality of sampling is measured by the Wasserstein distance. We also consider the situation where the Hessian of the log-density of the target distribution is Lipschitz-continuous. In this case, we introduce a new discretization of the kinetic Langevin diffusion and prove that this leads to a substantial improvement of the upper bound on the sampling error measured in Wasserstein distance.

math.PR↗

arXiv2018

Restricted eigenvalue property for corrupted Gaussian designs

Motivated by the construction of tractable robust estimators via convex relaxations, we present conditions on the sample size which guarantee an augmented notion of Restricted Eigenvalue-type condition for Gaussian designs. Such a notion is suitable for high-dimensional robust inference in a Gaussian linear model and a multivariate Gaussian model when samples are corrupted by outliers either in the response variable or in the design matrix. Our proof technique relies on simultaneous lower and upper bounds of two random bilinear forms with very different behaviors. Such simultaneous bounds are used for balancing the interaction between the parameter vector and the estimated corruption vector as well as for controlling the presence of corruption in the design. Our technique has the advantage of not relying on known bounds of the extreme singular values of the associated Gaussian ensemble nor on the use of mutual incoherence arguments. A relevant consequence of our analysis, compared to prior work, is that a significantly sharper restricted eigenvalue constant can be obtained under weaker assumptions. In particular, the sparsity of the unknown parameter and the number of outliers are allowed to be completely independent of each other.

math.ST↗

arXiv2018

Minimax estimation of a p-dimensional linear functional in sparse Gaussian models and robust estimation of the mean

We consider two problems of estimation in high-dimensional Gaussian models. The first problem is that of estimating a linear functional of the means of $n$ independent $p$-dimensional Gaussian vectors, under the assumption that most of these means are equal to zero. We show that, up to a logarithmic factor, the minimax rate of estimation in squared Euclidean norm is between $(s^2\wedge n) +sp$ and $(s^2\wedge np)+sp$. The estimator that attains the upper bound being computationally demanding, we investigate suitable versions of group thresholding estimators that are efficiently computable even when the dimension and the sample size are very large. An interesting new phenomenon revealed by this investigation is that the group thresholding leads to a substantial improvement in the rate as compared to the element-wise thresholding. Thus, the rate of the group thresholding is $s^2\sqrt{p}+sp$, while the element-wise thresholding has an error of order $s^2p+sp$. To the best of our knowledge, this is the first known setting in which leveraging the group structure leads to a polynomial improvement in the rate. The second problem studied in this work is the estimation of the common $p$-dimensional mean of the inliers among $n$ independent Gaussian vectors. We show that there is a strong analogy between this problem and the first one. Exploiting it, we propose new strategies of robust estimation that are computationally tractable and have better rates of convergence than the other computationally tractable robust (with respect to the presence of the outliers in the data) estimators studied in the literature. However, this tractability comes with a loss of the minimax-rate-optimality in some regimes.

math.ST↗

arXiv2018

Exponential weights in multivariate regression and a low-rankness favoring prior

We establish theoretical guarantees for the expected prediction error of the exponential weighting aggregate in the case of multivariate regression that is when the label vector is multidimensional. We consider the regression model with fixed design and bounded noise. The first new feature uncovered by our guarantees is that it is not necessary to require independence of the observations: a symmetry condition on the noise distribution alone suffices to get a sharp risk bound. This result needs the regression vectors to be bounded. A second curious finding concerns the case of unbounded regression vectors but independent noise. It turns out that applying exponential weights to the label vectors perturbed by a uniform noise leads to an estimator satisfying a sharp oracle inequality. The last contribution is the instantiation of the proposed oracle inequalities to problems in which the unknown parameter is a matrix. We propose a low-rankness favoring prior and show that it leads to an estimator that is optimal under weak assumptions.

math.ST↗