arXiv subjects

Pedro Pinto

Publications and source records attributed to Pedro Pinto.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative limit theorems for generalized P\'olya urns with applications to random tree models

We establish novel quantitative limit theorems for the asymptotic distribution of colours in a generalized P\'olya urn. Concretely, we construct explicit rates of convergence for the proportion of balls of each colour in the urn, both in square-mean and almost surely, under a general condition on the replacement matrix. As an application, we revisit three models of random recursive trees studied by Janson (Random Structures & Algorithms 26 (2005), 69--83): random recursive trees, random plane recursive trees, and random recursive $d$-ary trees. For each model, we show that the corresponding outdegree statistics can be cast as generalized P\'olya urns, and thereby obtain explicit rates of convergence, both in $L^2$ and almost surely, for the proportion of nodes of each outdegree. In all three cases, the rates we obtain are of order $O(1/n)$ in $L^2$ and almost surely, and are uniform in the outdegree under consideration.

math.PR↗

arXiv2026

Training Machine Learning Models on Encrypted Data: A Privacy-Preserving Framework using Homomorphic Encryption

The use of Machine Learning (ML) for data-driven decision-making often relies on access to sensitive datasets, which introduces privacy challenges. Traditional encryption methods protect data at rest or in transit but fail to secure it during processing, exposing it to unauthorized access. Homomorphic encryption emerges as a transformative solution, enabling computations on encrypted data without decryption, thus preserving confidentiality throughout the ML pipeline. This paper addresses the challenge of training ML models on encrypted data while maintaining accuracy and efficiency by proposing a proof-of-concept for a privacy-preserving framework that leverages Cheon-Kim-Kim-Song (CKKS) for approximate real-number arithmetic. Also, it demonstrates the feasibility of training K-Nearest Neighbors (KNN) and linear regression models on encrypted data, and evaluates encrypted inference for a basic Multilayer Perceptron (MLP) architecture. Experimental results show that models trained under Homomorphic encryption achieve performance metrics comparable to plaintext-trained models, validating the approach. However, challenges such as computational overhead, noise management, and limited support for non-polynomial operations persist. This work lays the groundwork for broader adoption of privacy-preserving ML in real-world applications, balancing security with computational feasibility.

cs.CR↗

arXiv2025

From Model to Classroom: Evaluating Generated MCQs for Portuguese with Narrative and Difficulty Concerns

While MCQs are valuable for learning and evaluation, manually creating them with varying difficulty levels and targeted reading skills remains a time-consuming and costly task. Recent advances in generative AI provide an opportunity to automate MCQ generation efficiently. However, assessing the actual quality and reliability of generated MCQs has received limited attention -- particularly regarding cases where generation fails. This aspect becomes particularly important when the generated MCQs are meant to be applied in real-world settings. Additionally, most MCQ generation studies focus on English, leaving other languages underexplored. This paper investigates the capabilities of current generative models in producing MCQs for reading comprehension in Portuguese, a morphologically rich language. Our study focuses on generating MCQs that align with curriculum-relevant narrative elements and span different difficulty levels. We evaluate these MCQs through expert review and by analyzing the psychometric properties extracted from student responses to assess their suitability for elementary school students. Our results show that current models can generate MCQs of comparable quality to human-authored ones. However, we identify issues related to semantic clarity and answerability. Also, challenges remain in generating distractors that engage students and meet established criteria for high-quality MCQ option design.

cs.CL↗

arXiv2024

Products of hyperbolic spaces

The class of uniformly smooth hyperbolic spaces was recently introduced by the first author as a common generalization of both CAT(0) spaces and uniformly smooth Banach spaces, in a way that Reich's theorem on resolvent convergence could still be proven. We define products of such spaces, showing that they are reasonably well-behaved. In this manner, we provide the first example of a space for which Reich's theorem holds and which is neither a CAT(0) space, nor a convex subset of a normed space.

math.MG↗

arXiv2023

A Review on Cryptocurrency Transaction Methods for Money Laundering

Cryptocurrencies are considered relevant assets and they are currently used as an investment or to carry out transactions. However, specific characteristics commonly associated with the cryptocurrencies such as irreversibility, immutability, decentralized architecture, absence of control authority, mobility, and pseudo-anonymity make them appealing for money laundering activities. Thus, the collection and characterization of current cryptocurrency-based methods used for money laundering are paramount to understanding the circulation flows of physical and digital money and preventing this illegal activity. In this paper, a collection of cryptocurrency transaction methods is presented and distributed through the money laundering life cycle. Each method is analyzed and classified according to the phase of money laundering it corresponds to. The result of this article may in the future help design efficient strategies to prevent illegal money laundering activities.

cs.CR↗

arXiv2023

Fej\'er monotone sequences revisited

In this paper we introduce a localized and relativized generalization of the usual concept of Fej\'er monotonicity together with uniform and quantitative versions thereof and show that the main quantitative results obtained by the 1st author together with Nicolae and Leu\c{s}tean in 2018 and with L\'opez-Acedo and Nicolae in 2019 respectively, extend to this generalization. Our framework, in particular, covers the sequence generated by the Dykstra algorithm while the latter is not Fej\'er-monotone in the ordinary sense. This gives a theoretical explanation why under a metric regularity assumption one obtains an explicit rate of convergence for Dykstra's algorithm which was proved recently by the 2nd author.

math.OC↗

arXiv2023

On the finitary content of Dykstra's cyclic projections algorithm

We study the asymptotic behaviour of the well-known Dykstra's algorithm through the lens of proof-theoretical techniques. We provide an elementary proof for the convergence of Dykstra's algorithm in which the standard argument is stripped to its central features and where the original compactness principles are circumvented, additionally providing highly uniform primitive recursive rates of metastability in a full general setting. Moreover, under an additional assumption, we are even able to obtain effective general rates of convergence. We argue that such additional condition is actually necessary for the existence of general uniform rates of convergence.

math.OC↗

arXiv2023

On computational properties of Cauchy problems generated by accretive operators

In this paper, we provide quantitative versions of results on the asymptotic behavior of nonlinear semigroups generated by an accretive operator due to O. Nevanlinna and S. Reich as well as H.-K. Xu. These results themselves rely on a particular assumption on the underlying operator introduced by A. Pazy under the name of `convergence condition'. Based on logical techniques from `proof mining', a subdiscipline of mathematical logic, we derive various notions of a `convergence condition with modulus' which provide quantitative information on this condition in different ways. These techniques then also facilitate the extraction of quantitative information on the convergence results of Nevanlinna and Reich as well as Xu, in particular also in the form of rates of convergence which depend on these moduli for the convergence condition.

math.AP↗

arXiv2022

Rates of asymptotic regularity for the alternating Halpern-Mann iteration

In this paper we extend to $UCW$-hyperbolic spaces the quantitative asymptotic regularity results for the alternating Halpern-Mann iteration obtained by Dinis and the second author for CAT(0) spaces. These results are new even for uniformly convex normed spaces. Furthermore, for a particular choice of the parameter sequences, we compute linear rates of asymptotic regularity in $W$-hyperbolic spaces and quadratic rates of $T$- and $U$-asymptotic regularity in CAT(0) spaces.

math.OC↗

arXiv2021

Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces

In this paper we consider, in the general context of CAT(0) spaces, an iterative schema which alternates between Halpern and Krasnoselskii-Mann style iterations. We prove, under suitable conditions, the strong convergence of this algorithm, benefiting from ideas from the proof mining program. We give quantitative information in the form of effective rates of asymptotic regularity and of metastability (in the sense of Tao). Motivated by these results we are also able to obtain strongly convergent versions of the forward-backward and the Douglas-Rachford algorithms. Our results generalize recent work by Bo\c{t}, Csetnek and Meier, and Cheval and Leu\c{s}tean.

math.FA↗

arXiv2021

Quantitative translations for viscosity approximation methods in hyperbolic spaces

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscosity versions. We show that these transformations can be applied to earlier quantitative studies of these iterations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry and S. Chebbi, we obtain similar results. Finally, in uniformly convex Banach spaces we study a strong notion of accretive operator due to Brezis and Sibony and extract an uniform modulus of uniqueness for the property of being a zero point. In this context, we show that it is possible to obtain Cauchy rates for the Browder type and the Halpern iterations (and hence also for their viscosity versions).

math.FA↗

arXiv2021

Effective metastability for a method of alternating resolvents

A generalized method of alternating resolvents was introduced by Boikanyo and Moro{\c s}anu as a way to approximate common zeros of two maximal monotone operators. In this paper we analyse the strong convergence of this algorithm under two different sets of conditions. As a consequence we obtain effective rates of metastability (in the sense of Terence Tao) and quasi-rates of asymptotic regularity. Furthermore, we bypass the need for sequential weak compactness in the original proofs. Our quantitative results are obtained using proof-theoretical techniques in the context of the proof mining program.

math.FA↗

arXiv2020

On the convergence of algorithms with Tikhonov regularization terms

We consider the strongly convergent modified versions of the Krasnosel'ski\u{\i}-Mann, the forward-backward and the Douglas-Rachford algorithms with Tikhonov regularization terms, introduced by Radu Bo\c{t}, Ern\"{o} Csetnek and Dennis Meier. We obtain quantitative information for these modified iterations, namely rates of asymptotic regularity and metastability. Furthermore, our arguments avoid the use of sequential weak compactness and use only a weak form of the projection argument.

math.FA↗

arXiv2020

Quantitative results on a Halpern-type proximal point algorithm

We apply proof mining methods to analyse a result of Boikanyo and Moro\c{s}anu on the strong convergence of a Halpern-type proximal point algorithm. As a consequence, we obtain quantitative versions of this result, providing uniform effective rates of asymptotic regularity and metastability.

math.OC↗

arXiv2019

A rate of metastability for the Halpern type Proximal Point Algorithm

Using proof-theoretical techniques, we analyze a proof by H.-K. Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of T. Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our final quantitative result bypasses the need of the sequential weak compactness argument present in the original proof. This elimination is reflected in the extraction of primitive recursive quantitative information. This work follows from recent results in Proof Mining regarding the removal of sequential weak compactness arguments.

math.FA↗

arXiv2019

Quantitative results on the multi-parameters Proximal Point Algorithm

We give a quantitative analysis of a theorem due to Fenghui Wang and Huanhuan Cui concerning the convergence of a multi-parametric version of the proximal point algorithm. Wang and Cui's result ensures the convergence of the algorithm to a zero of the operator. Our quantitative analysis provides explicit bounds on the metastability (in the sense of Terence Tao) for the convergence and the asymptotic regularity of the iteration. Moreover, our analysis bypasses the need of sequential weak compactness and only requires a weak form of the metric projection argument.

math.FA↗

arXiv2019

Metastability of the proximal point algorithm with multi-parameters

In this article we use techniques of proof mining to analyse a result, due to Yonghong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor's result ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided by Fernando Ferreira and Paulo Oliva's bounded functional interpretation, provides a primitive recursive bound on the metastability for the convergence of the algorithm, in the sense of Terence Tao. Furthermore, we obtain quantitative information on the asymptotic regularity of the iteration. The results of this paper are made possible by an arithmetization of the $\limsup$.

math.LO↗

arXiv2019

The ANTAREX Domain Specific Language for High Performance Computing

The ANTAREX project relies on a Domain Specific Language (DSL) based on Aspect Oriented Programming (AOP) concepts to allow applications to enforce extra functional properties such as energy-efficiency and performance and to optimize Quality of Service (QoS) in an adaptive way. The DSL approach allows the definition of energy-efficiency, performance, and adaptivity strategies as well as their enforcement at runtime through application autotuning and resource and power management. In this paper, we present an overview of the key outcome of the project, the ANTAREX DSL, and some of its capabilities through a number of examples, including how the DSL is applied in the context of the project use cases.

cs.DC↗