arXiv subjects

Marco Avella Medina

Publications and source records attributed to Marco Avella Medina.

12 recordsLinked to original sources

arXiv2026

The Influence Function of Transport-based Quantiles

Transport-based quantiles extend univariate quantiles to multivariate distributions via optimal transport. We study the influence function of the transport quantile map $\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $μ$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+tδ_{x_0}$, we prove that the first-order limit $\mathbf{I}(x_0;\mathbf{Q}_P(z)) := \lim_{t\downarrow 0} [\mathbf{Q}_{(1-t)P+tδ_{x_0}}(z)-\mathbf{Q}_P(z)]/t$ exists whenever $x_0\ne \mathbf{Q}_P(z)$ and characterize it uniquely. Specifically, $\mathbf{I}(x_0;\mathbf{Q}_P(z))=\nabla G_{x_0}(z)$, where $G_{x_0}$ is characterized by a uniformly elliptic equation with a Dirac source and a Neumann boundary condition. In every dimension $d\ge 2$, this influence function has a pole-type singularity. For fixed $z\in\operatorname{int}(Ω_μ)$, it remains bounded when $\mathbf{F}_P(x_0)$ stays away from $z$, where $\mathbf{F}_P=\mathbf{Q}_P^{-1}$ is the transport-based distribution function, but diverges as $x_0\to\mathbf{Q}_P(z)$, equivalently as $\mathbf{F}_P(x_0)\to z$. In fact, $\|\mathbf{I}(x_0;\mathbf{Q}_P(z))\|\asymp\|z-\mathbf{F}_P(x_0)\|^{-(d-1)}$. This contrasts with the bounded influence function of univariate quantiles and implies that $\mathbf{I}(X;\mathbf{Q}_P(z))$, for $X\sim P$, has infinite second moment. Numerical experiments further suggest that empirical transport quantiles may exhibit stable-type non-Gaussian fluctuations.

math.ST↗

arXiv2026

Contaminated Multi-task Learning with Heterogeneity: Fundamental Limits and Optimal Algorithms

Integrating information across related tasks can improve estimation and prediction in transfer, multi-task, and federated learning, but contamination and heterogeneity make robust borrowing challenging. We study a contaminated multi-task empirical risk minimization (ERM) framework in which an $ε$ fraction of $K$ tasks, each with sample size $n$, may be arbitrarily contaminated while the remaining tasks are heterogeneous. Our goal is to estimate both the global minimizer of the average risk and the clean task-specific minimizers, thereby combining robustness and personalization. In the Gaussian mean model, we show that several common paradigms, including adaptive and robust regularization around a shared center, global matrix regularization, decomposition-based regularization, and score-based outlier-task detection, all suffer from a worst-case contamination error of order $ε\sqrt{d/n}$, which is suboptimal compared to the lower bound $ε/\sqrt{n}$. This identifies a dimension-dependent barrier for these approaches. We then establish minimax lower bounds for a general heterogeneous ERM setting and propose a computationally efficient filtering-based robust multi-task gradient descent method. Under local strong convexity, smoothness, and sub-Gaussian gradient assumptions, the proposed method attains high-probability upper bounds matching the minimax rates up to logarithmic factors over a broad regime. In particular, it removes the extra $\sqrt{d}$ contamination dependence of many regularization-based methods and score-based outlier detection, while achieving personalization to local tasks under strong heterogeneity. Simulations and a real-data analysis demonstrate strong robustness and personalization relative to a broad range of benchmark methods.

stat.ML↗

arXiv2026

The Threshold Breakdown Point

We introduce a novel approach to finite sample robustness that avoids the pessimism of traditional breakdown analyses. We define the threshold breakdown point, the smallest contamination fraction needed to induce a prescribed deviation, and the finite sample m-sensitivity, the worst-case deviation that an estimator can incur after m observations are contaminated. We derive these measures for commonly used M-estimators, their standard errors and related test statistics. This allows us to extend the decision breakdown point of Zhang (1996) to obtain general breakdown characterizations for hypothesis testing, and show how these notions correspond to finite sample counterparts of the power and level breakdown functions of He, Simpson and Portnoy (1990). We complement our work with an inferential framework for the threshold breakdown and m-sensitivity that yields consistency and asymptotic normality results, as well as a valid multiplier bootstrap for uncertainty quantification. We illustrate the practical utility of our methods in various numerical examples and an application to a two sample testing problem for a blood pressure dataset.

math.ST↗

arXiv2026

Differentially Private Hyperparameter Tuning using Local Bayesian Optimization

Hyperparameter tuning is a key component of machine learning procedures, but when validation data contain sensitive user information, search mechanisms can leak private information through the selected configuration. Existing differentially private hyperparameter tuning methods often rely on near-random search, while prior differentially private Bayesian optimization approaches are typically global and, therefore, scale poorly with the hyperparameter dimensionality. We study differentially private hyperparameter tuning using local Bayesian optimization, focusing on settings where the validation objective is available only through noisy black box evaluations and gradients are unavailable or impractical to compute. We introduce DP-GIBO, a differentially private local Bayesian optimization framework that privately approximates gradients using a Gaussian Process surrogate. Under suitable conditions, we prove that DP-GIBO converges to a locally optimal hyperparameter configuration up to a privacy-dependent error, with dimensional dependence that is polynomial rather than exponential.Empirically, we show that DP-GIBO provides scalable private hyperparameter tuning across multiple tasks, substantially outperforming non-private random search and global Bayesian optimization baselines in moderate-to-high-dimensional hyperparameter spaces.

stat.ML↗

arXiv2026

Breakdown properties of optimal transport maps: general transportation costs

Two recent works, Avella-Medina and González-Sanz (2026) and Passeggeri and Paindaveine (2026), studied the robustness of the optimal transport map through its breakdown point, i.e., the smallest fraction of contamination that can make the map take arbitrarily aberrant values. Their main finding is the following: let $P$ and $Q$ denote the target and reference measures, respectively, and let $T$ be the optimal transport map for the squared Euclidean cost. Then, the breakdown point of $T(u)$, when $P$ is perturbed and $Q$ is fixed, coincides with the Tukey depth of $u$ relative to $Q$. In this note, we extend this result to general convex cost functions, demonstrating that the cost function does not have any impact on the breakdown point of the optimal transport map. Our contribution provides a definitive characterization of the breakdown point of the optimal transport map. In particular, it shows that for a broad class of regular cost functions, all transport-based quantiles enjoy the same high breakdown point properties.

math.ST↗

arXiv2026

Differentially Private Inference for Longitudinal Linear Regression

Differential Privacy (DP) provides a rigorous framework for releasing statistics while protecting individual information present in a dataset. Although substantial progress has been made on differentially private linear regression, existing methods almost exclusively address the item-level DP setting, where each user contributes a single observation. Many scientific and economic applications instead involve longitudinal or panel data, in which each user contributes multiple dependent observations. In these settings, item-level DP offers inadequate protection, and user-level DP - shielding an individual's entire trajectory - is the appropriate privacy notion. We develop a comprehensive framework for estimation and inference in longitudinal linear regression under user-level DP. We propose a user-level private regression estimator based on aggregating local regressions, and we establish finite-sample guarantees and asymptotic normality under short-range dependence. For inference, we develop a privatized, bias-corrected covariance estimator that is automatically heteroskedasticity- and autocorrelation-consistent. These results provide the first unified framework for practical user-level DP estimation and inference in longitudinal linear regression under dependence, with strong theoretical guarantees and promising empirical performance.

math.ST↗

arXiv2026

Statistical Guarantees for Data-driven Posterior Tempering

Posterior tempering reduces the influence of the likelihood in the calculation of the posterior by raising the likelihood to a fractional power $α$. The resulting power posterior - also known as an $α$-posterior or fractional posterior - has been shown to exhibit appealing properties, including robustness to model misspecification and asymptotic normality (Bernstein-von Mises theorem). However, practical recommendations for selecting the tempering parameter and statistical guarantees for the resulting power posterior remain open questions. Cross-validation-based approaches to tuning this parameter suggest interesting asymptotic regimes for the selected $α$, which can either vanish or behave like a mixture distribution with a point mass at infinity and the remaining mass converging to zero. We formalize the asymptotic properties of the power posterior in these regimes. In particular, we provide sufficient conditions for (i) consistency of the power posterior moments and (ii) asymptotic normality of the power posterior mean. Our analysis required us to establish a new Laplace approximation that is interesting in its own right and is the key technical tool for showing a critical threshold $α\asymp 1/\sqrt{n}$ where the asymptotic normality of the posterior mean breaks. Our results allow for the power to depend on the data in an arbitrary way.

math.ST↗

arXiv2025

A theoretical framework for M-posteriors: frequentist guarantees and robustness properties

We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class M-posteriors and show that they are asymptotically normally distributed under mild conditions on the M-estimation loss and the prior. In particular, an M-posterior contracts in probability around a normal distribution centered at an M-estimator, showing frequentist consistency and suggesting some degree of robustness depending on the reference M-estimator. We formalize the robustness properties of the M-posteriors by a new characterization of the posterior influence function and a novel definition of breakdown point adapted for posterior distributions. We illustrate the wide applicability of our theory in various popular models and illustrate their empirical relevance in some numerical examples.

math.ST↗

arXiv2024

Thompson Sampling Itself is Differentially Private

In this work we first show that the classical Thompson sampling algorithm for multi-arm bandits is differentially private as-is, without any modification. We provide per-round privacy guarantees as a function of problem parameters and show composition over $T$ rounds; since the algorithm is unchanged, existing $O(\sqrt{NT\log N})$ regret bounds still hold and there is no loss in performance due to privacy. We then show that simple modifications -- such as pre-pulling all arms a fixed number of times, increasing the sampling variance -- can provide tighter privacy guarantees. We again provide privacy guarantees that now depend on the new parameters introduced in the modification, which allows the analyst to tune the privacy guarantee as desired. We also provide a novel regret analysis for this new algorithm, and show how the new parameters also impact expected regret. Finally, we empirically validate and illustrate our theoretical findings in two parameter regimes and demonstrate that tuning the new parameters substantially improve the privacy-regret tradeoff.

cs.LG↗

arXiv2024

Asymptotics for power posterior mean estimation

Power posteriors "robustify" standard Bayesian inference by raising the likelihood to a constant fractional power, effectively downweighting its influence in the calculation of the posterior. Power posteriors have been shown to be more robust to model misspecification than standard posteriors in many settings. Previous work has shown that power posteriors derived from low-dimensional, parametric locally asymptotically normal models are asymptotically normal (Bernstein-von Mises) even under model misspecification. We extend these results to show that the power posterior moments converge to those of the limiting normal distribution suggested by the Bernstein-von Mises theorem. We then use this result to show that the mean of the power posterior, a point estimator, is asymptotically equivalent to the maximum likelihood estimator.

math.ST↗

arXiv2023

Spectral learning of multivariate extremes

We propose a spectral clustering algorithm for analyzing the dependence structure of multivariate extremes. More specifically, we focus on the asymptotic dependence of multivariate extremes characterized by the angular or spectral measure in extreme value theory. Our work studies the theoretical performance of spectral clustering based on a random $k$-nearest neighbor graph constructed from an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. In particular, we derive the asymptotic distribution of extremes arising from a linear factor model and prove that, under certain conditions, spectral clustering can consistently identify the clusters of extremes arising in this model. Leveraging this result we propose a simple consistent estimation strategy for learning the angular measure. Our theoretical findings are complemented with numerical experiments illustrating the finite sample performance of our methods.

stat.ML↗

arXiv2021

On the Robustness to Misspecification of $α$-Posteriors and Their Variational Approximations

$α$-posteriors and their variational approximations distort standard posterior inference by downweighting the likelihood and introducing variational approximation errors. We show that such distortions, if tuned appropriately, reduce the Kullback-Leibler (KL) divergence from the true, but perhaps infeasible, posterior distribution when there is potential parametric model misspecification. To make this point, we derive a Bernstein-von Mises theorem showing convergence in total variation distance of $α$-posteriors and their variational approximations to limiting Gaussian distributions. We use these distributions to evaluate the KL divergence between true and reported posteriors. We show this divergence is minimized by choosing $α$ strictly smaller than one, assuming there is a vanishingly small probability of model misspecification. The optimized value becomes smaller as the the misspecification becomes more severe. The optimized KL divergence increases logarithmically in the degree of misspecification and not linearly as with the usual posterior.

stat.ML↗