arXiv:2409.14980v3 Announce Type: replace 
Abstract: We introduce a (de)-regularization of the Maximum Mean Discrepancy (DrMMD) and its Wasserstein gradient flow. Existing gradient flows that transport samples from source distribution to target distribution with only target samples, either lack tractable numerical implementation ($f$-divergence flows) or require strong assumptions, and modifications such as noise injection, to ensure convergence (Maximum Mean Discrepancy flows). In contrast, DrMMD flow can simultaneously (i) guarantee near-global convergence for a broad class of targets in both continuous and discrete time, and (ii) be implemented in closed form using only samples. The former is achieved by leveraging the connection between the DrMMD and the $\chi^2$-divergence, while the latter comes by treating DrMMD as MMD with a de-regularized kernel. Our numerical scheme uses an adaptive de-regularization schedule throughout the flow to optimally trade off between discretization errors and deviations from the $\chi^2$ regime. The potential application of the DrMMD flow is demonstrated across several numerical experiments, including a large-scale setting of training student/teacher networks.

تم تقديم نهج جديد يعرف باسم (إلغاء) تنظيم الحد الأقصى للاختلاف المتوسط (DrMMD) ، مما يعزز كفاءة تدفقات التدرج في فاسرشتاين. تتيح هذه الطريقة نقل العينات من توزيع المصدر إلى توزيع الهدف باستخدام عينات الهدف فقط ، متجاوزة القيود التي تواجه التدفقات الحالية التي تفتقر إلى القابلية للتنفيذ العددي أو تتطلب افتراضات قوية للتقارب.

Se ha introducido un nuevo enfoque conocido como (de)-regularización de la Maximum Mean Discrepancy (DrMMD), que mejora la eficiencia de los flujos de gradiente de Wasserstein. Este método permite el transporte de muestras de una distribución fuente a una distribución objetivo utilizando solo muestras objetivo, superando las limitaciones de los flujos existentes que carecen de trazabilidad numérica o requieren suposiciones fuertes para la convergencia.

Une nouvelle approche appelée (dé)-régularisation de la Maximum Mean Discrepancy (DrMMD) a été introduite, améliorant l'efficacité des flux de gradient de Wasserstein. Cette méthode permet le transport d'échantillons d'une distribution source vers une distribution cible en utilisant uniquement des échantillons cibles, surmontant les limitations des flux existants qui manquent soit de traçabilité numérique, soit nécessitent des hypothèses fortes pour la convergence.

A new approach known as (de)-regularized Maximum Mean Discrepancy (DrMMD) has been introduced, enhancing the efficiency of Wasserstein gradient flows. This method allows for the transport of samples from a source distribution to a target distribution using only target samples, overcoming limitations faced by existing flows that either lack numerical tractability or require strong assumptions for convergence.

(De)-regularized Maximum Mean Discrepancy Gradient Flow

arXiv:2601.08184v1 Announce Type: cross 
Abstract: Finite-time central limit theorem (CLT) rates play a central role in modern machine learning (ML). In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($\mathcal W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in ML: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the \textit{first optimal} $\mathcal O(n^{-1/2})$ rate in $\mathcal W_1$, as well as the first $\mathcal W_p$ ($p\ge 2$) CLT rates under mild moment assumptions, substantially improving the best previously known bounds in these dependent-data regimes. As an application of our optimal $\mathcal W_1$ rate for locally dependent sequences, we further obtain the first optimal $\mathcal W_1$--CLT rate for multivariate $U$-statistics.
  On the technical side, we derive a tractable auxiliary bound for $\mathcal W_1$ Gaussian approximation errors that is well suited to studying dependent data. For Markov chains, we further prove that the regeneration time of the split chain associated with a geometrically ergodic chain has a geometric tail without assuming strong aperiodicity or other restrictive conditions. These tools may be of independent interests and enable our optimal $\mathcal W_1$ rates and underpin our $\mathcal W_p$ ($p\ge 2$) results.

أظهرت دراسة حديثة معدلات مثالية لنظرية الحد المركزي (CLT) في الزمن المحدد للبيانات متعددة المتغيرات المعتمدة في مسافة فاسرشتاين-$p$، مع التركيز على تسلسلات معتمدة محليًا وسلاسل ماركوف ذات تآزر هندسي. تكشف النتائج عن أول معدل مثالي $	ext{O}(n^{-1/2})$ في $	ext{W}_1$ وتحسينات كبيرة لمعدلات $	ext{W}_p$ تحت افتراضات لحظية خفيفة.

Un estudio reciente ha establecido tasas óptimas del teorema central del límite (TCL) en tiempo finito para datos multivariantes dependientes en la distancia de Wasserstein-$p$, centrándose en secuencias localmente dependientes y cadenas de Markov ergódicas geométricamente. Los hallazgos revelan la primera tasa óptima de $	ext{O}(n^{-1/2})$ en $	ext{W}_1$ y mejoras significativas para las tasas de $	ext{W}_p$ bajo supuestos de momento leves.

Une étude récente a établi des taux optimaux du théorème central limite (TCL) en temps fini pour des données multivariées dépendantes dans la distance de Wasserstein-$p$, en se concentrant sur des séquences localement dépendantes et des chaînes de Markov géométriquement ergodiques. Les résultats révèlent le premier taux optimal de $	ext{O}(n^{-1/2})$ dans $	ext{W}_1$ et des améliorations significatives pour les taux de $	ext{W}_p$ sous des hypothèses de moment légères.

A recent study has established optimal finite-time central limit theorem (CLT) rates for multivariate dependent data in Wasserstein-$p$ distance, focusing on locally dependent sequences and geometrically ergodic Markov chains. The findings reveal the first optimal $	ext{O}(n^{-1/2})$ rate in $	ext{W}_1$ and significant improvements for $	ext{W}_p$ rates under mild moment assumptions.

(De)-regularized Maximum Mean Discrepancy Gradient Flow

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