arXiv subjects

Yaiza Bermudez

Publications and source records attributed to Yaiza Bermudez.

6 recordsLinked to original sources

arXiv2026

A Unified Framework for Training-Dataset Aggregation and Exact Machine Unlearning via Gibbs Probability Measures

In this paper, three operations on Gibbs probability measures are studied: renormalization, normalized log-linear combination, and nesting. The resulting measures are shown to be Gibbs probability measures with explicit expressions for their objective functions, reference measures, and regularization factors. These measures are further characterized as unique solutions to optimization problems involving linear combinations of the original objective functions, subject to either relative entropy regularization or relative entropy constraints. When the reference measures are probability measures, geometric interpretations of these three operations are established involving Pythagorean identities and exponential geodesics.A central result is the equivalence between nesting and normalized log-linear combinations of Gibbs probability measures. These results are then applied to supervised machine learning with Gibbs algorithms, which are represented by Gibbs probability measures. In this context, either through nesting or log-linear combinations, Gibbs algorithms trained on local datasets are used to form new Gibbs algorithms trained upon the aggregation of all local training datasets. Interestingly, the parameters of such operations, determine the weights of each training data point on the aggregated dataset. %Operations on local Gibbs probability measures are shown to control the contributions of training data. This observation is leveraged for achieving: (a) exact machine unlearning in centralized, decentralized, and federated learning systems; and (b) centralized-equivalent performance, in the sense of producing algorithms trained on the aggregation of all local datasets in decentralized and federated learning .

stat.ML↗

arXiv2026

Machine Unlearning for Gibbs Supervised Learning Algorithms

In this paper, a method for achieving exact unlearning for Gibbs supervised learning algorithms is proposed using a variational formulation inspired by empirical risk minimization subject to relative entropy regularization (ERM-RER). Such a method consists of maximizing the expected empirical risk over the dataset to be unlearned subject to a regularization by relative entropy with respect to the original algorithm. The optimization variable is a probability measure on the models; and the solution is another Gibbs probability measure that represents a new Gibbs supervised learning algorithm. The method guarantees exact unlearning in the sense that the new Gibbs algorithm coincides in distribution with the algorithm that would have been obtained by retraining from scratch on the dataset to be retained. As a byproduct, a framework for reweighting data points in ERM-RER by strategically choosing both the reference measure and the regularization factor is obtained. In this framework, exact unlearning is the special case in which zero-weight is assigned to the contribution of the data points to be unlearned. More generally, depending on the choice of certain parameters, data points can be up-weighted or down-weighted in ERM-RER problems for particular purposes, e.g., controlling the generalization error of Gibbs algorithms. This paves the way for new constructive or adversarial views on classical reweighting data points in ERM-RER.

stat.ML↗

arXiv2026

Equivalence Between Nested Gibbs Measures and Log-Linear Combinations of Gibbs Measures

In this paper, three operations on Gibbs probability measures are studied. The first operation, often referred to as renormalization, takes one Gibbs probability measure and generates a new Gibbs measure by normalizing a power of its density. This normalization has a twofold effect: it changes the regularization factor and concentrates the support within a subset of the original support. Interestingly, these effects can be independently controlled by different parameters. The second operation consists of a normalized log-linear combination of the densities of Gibbs probability measures. The third operation takes two Gibbs probability measures and changes the reference measure of the latter with the former. Hence, the former is said to be "nested" within the latter, yielding a new Gibbs probability measure. The resulting measures from the second and third operations are also Gibbs probability measures and are shown, respectively, to solve optimization problems involving the expectations of linear combinations of the objective functions of the given measures, subject to a relative entropy regularization. These optimization problems differ exclusively in the coefficients of the linear combinations. This leads to the conclusion that there exists a set of parameters for which nesting one Gibbs probability measure into another has the same effect as log-linearly combining them. These operations have relevant applications in statistical learning. As an example, a one-shot federated learning system in which clients send their locally trained Gibbs algorithms to the server for log-linear combination is shown to achieve the same performance as a Gibbs algorithm trained upon the aggregation of all local training datasets.

cs.IT↗

arXiv2026

Decentralized Machine Learning with Centralized Performance Guarantees via Gibbs Algorithms

In this paper, it is shown, for the first time, that centralized performance is achievable in decentralized learning without sharing the local datasets. Specifically, when clients adopt an empirical risk minimization with relative-entropy regularization (ERM-RER) learning framework and a forward-backward communication between clients is established, it suffices to share the locally obtained Gibbs measures to achieve the same performance as that of a centralized ERM-RER with access to all the datasets. The core idea is that the Gibbs measure produced by client~$k$ is used, as reference measure, by client~$k+1$. This effectively establishes a principled way to encode prior information through a reference measure. In particular, achieving centralized performance in the decentralized setting requires a specific scaling of the regularization factors with the local sample sizes. Overall, this result opens the door to novel decentralized learning paradigms that shift the collaboration strategy from sharing data to sharing the local inductive bias via the reference measures over the set of models.

stat.ML↗

arXiv2026

Equivalence of optimal transport problems to regularization on the family of f-divergences

This work establishes that an optimal transport~(OT) problem regularized by a given $f$-divergence admits the same solution as another OT problem regularized by a different $g$-divergence, under an appropriate transformation of the cost function. This structural equivalence between OT problems regularized by distinct divergences, in the sense of sharing the same unique minimizer, is demonstrated within the framework of Polish spaces with bounded cost functions.

math.ST↗

arXiv2025

Proofs for Folklore Theorems on the Radon-Nikodym Derivative

In this technical report, rigorous statements and formal proofs are presented for both foundational and advanced folklore theorems on the Radon-Nikodym derivative. The cases of conditional and marginal probability measures are carefully considered, which leads to an identity involving the sum of mutual and lautum information suggesting a new interpretation for such a sum.

cs.IT↗