Beyond the Principle of Proportionality: Controlling the Restriction of Rights under Factual Uncertainty

The principle of proportionality is considered the main legal tool to control restrictive measures of rights, both in ordinary courts and at a constitutional level. In addition to its general limitations, new shortcomings of the principle have played a central role during the pandemic, questioning the principleʼs efficacy in situations of factual uncertainty, especially in technically or scientifically complex contexts. This article analyses this efficacy problem and exemplifies it with specific measures adopted to prevent COVID-19. It also analyses potential ways to counter those shortcomings, such as refining the principle itself, emphasising judicial deference to legislative and executive powers, or adopting prior decisions as to the information that must be taken into account in case of uncertainty. Finally, the article proposes some additional checks that could complement the culture of justification promoted by the principle and strengthen the control of public powers when restricting rights under conditions of uncertainty.Derech

Multinomial logistic regression and stochastic natural gradient descent

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Jesús Cerquides Bueno[en] Function optimization is a widely faced problem nowadays. Its interest, in particular, lies in every learning algorithm in AI, whose achievements are measured by a Loss-Function. On one hand, Multinomial Logistic Regression is a commonly applied model to engage and simplify the problem of predicting a categorical distributed variable which depends on a set of distinct categorical distributed variables. On the other hand, Gradient Descent allows us to reach local extrema of a smooth function. Moreover, large datasets force the use of online optimization. Improving the convergence speed and reducing the computational cost of gradient based online learning algorithms will automatically translate into a significant enhancement on many machine learning processes. In this text, we present a Stochastic Gradient Descent algorithm variant, specifically designed for Multinomial Logistic Regression learning problems by taking advantage of the geometry and the intrinsic metric of the space. We compare it to current most advanced stochastic algorithms, and we provide the favorable experimental results obtained

Efficient and convergent natural gradient based optimization algorithms for machine learning

[eng] Many times machine learning is casted as an optimization problem. This is the case when an objective function assesses the success of an agent in a certain task and hence, learning is accomplished by optimizing that function. Furthermore, gradient descent is an optimization algorithm that has proven to be a powerful tool, becoming the cornerstone to solving most machine learning challenges. Among its strengths, there are the low computational complexity and the convergence guarantee property to the optimum of the function, after certain regularities on the function. Nevertheless, large dimension scenarios show sudden drops in convergence rates which inhibit further improvements in an acceptable amount of time. For this reason, the field has contemplated the natural gradient to tackle this issue. The natural gradient is defined on a Riemannian manifold (M, g). A Riemannian manifold is a manifold M equipped with a metric g. The natural gradient vector of a function f at a point p in (M, g) is a vector in the tangent space at p that points to the direction in which f locally increases its value faster taking into account the metric attached to the manifold. It turns out that the manifold of probability distributions of the same family, usually considered in machine learning, has a natural metric associated, namely the fisher information metric. While natural gradient based algorithms show a better convergence speed in some limited examples, they often fail in providing good estimates or they even diverge. Moreover, they demand more calculations than the ones performed by gradient descent algorithms, increasing the computational complexity order. This thesis explores the natural gradient descent algorithm for the function optimization task. Our research aims at designing a natural gradient based algorithm to solve a function optimization problem, whose computational complexity is comparable to those gradient based and such that it benefits from higher rates of convergence compared to standard gradient based methods. To reach our objectives, the hypothesis formulated in this thesis is that the convergence property guarantee stabilizes natural gradient algorithms and it gives access to fast rates of convergence. Furthermore, the natural gradient can be computed fast for particular manifolds named dually flat manifolds, and hence, fast natural gradient optimization methods become available. The beginning of our research is mainly focused on the convergence property for natural gradient methods. We develop some strategies to define natural gradient methods whose convergence can be proven. The main assumptions require (M, g) to be a Riemannian manifold and f to be a differentiable function on M. Moreover, it turns out that the multinomial logistic regression problem, a widely considered machine learning problem, can be adapted and solved by taking a dually flat manifolds as the model. Hence, this problem is our most promising target in which the objective of the thesis can be completely accomplished.[cat] L’aprenentatge automàtic sovint es relaciona amb un problema d’optimització. Quan l’éxit o l’error d’un agent en una determinada tasca ve donat per una funció, aprendre a realitzar correctament la tasca equival a optimitzar la funció en questió. El descens del gradient és un mètode d’optimització emprat per resoldre la majoria d’aquest tipus de problemes. Aquest algorisme és eficient i, donades certes condicions, convergeix a la solució. No obstant, la convergència pot esdevenir molt lenta en problemes de dimensió alta, on l’algorisme requerix un temps desmesurat. El gradient natural és emprat, sense gaire èxit, per tal d’evitar aquest fet. En una varietat de Riemann (M, g) amb mètrica g, el gradient natural d’una funció "f" en un punt "p" és un vector del espai tangent en "p" que assenyala la direcció on "f" creix localment més intensament, tenint en compte la mètrica del espai. En teoria, el gradient natural té propietats que podrien afavorir la velocitat de convergència, però en problemes pràctics no s’observa cap millora. Alguns algorismes basats en el gradient natural fins i tot divergeixen essent superats pel descens del gradient standard. A més a més, el gradient natural en general té una complexitat computacional més elevada. Aquesta tesis explora els algorismes basats en el gradient natural. En moltes ocasions, l’aprenentatge automàtic es du a terme en families de distribucions de probabilitat, on la mètrica associada a aquest tipus d’espais és la mètrica de Fisher. La nostra hipòtesi és que per obtenir una velocitat de convergència alta és suficient l’assoliment de la propietat de convergència. L’objectiu és definir exemples d’aquest tipus d’algorismes que siguin convergents i amb un cost computacional reduït per tal que pugui ser emprat en problemes actuals de dimensió alta. Per assolir el nostre objectiu, hem trobat indispensable limitar-nos al conjunt de varietats de Riemann anomenades varietats dualment planes. En particular, afrontem el problema de regressió logística multinomial. Aquest espai ens permet definir un algorisme efficient i convergent basat en el gradient natural gràcies a propietats intrínseques de la varietat

El romance en los documentos de la Catedral de Toledo (1171-1252): la escritura

This article is part of a broader work that aims to accomplish a complete historical revision of Toledan Castilian. For this purpose, we started from the direct analysis of the rich and mostly unedited document body of the Toledo Cathedral’s Archive and the Toledo City Hall’s Archive. As the language analysis cannot be done without considering first the multiple writing traditions present in the Toledan scene, we began examining the paleographic and graphic aspects of the documents. This study put in evidence the coexistence of very different traditions, which is in turn reflecting, without any doubt, on other analysis’ levels, from the phonetics to the lexic. We also traced the northern provenience of the Toledan writing habits, and their connections with the Castilian of Alfonso X, with surprising result

Desarrollo y explotación del "Corpus de Documentos Españoles Anteriores a 1700" (CODEA)

Este artículo tiene por objeto presentar el estado actual del "Corpus de Documentos Españoles Anteriores a 1700" (CODEA, http://www.textoshispanicos.es), elaborado por el "Grupo de Investigación de Textos para la Historia del Español" (GITHE) de la Universidad de Alcalá. El CODEA ofrece a día de hoy 1500 documentos de diferentes archivos, de todas las provincias peninsulares no bilingües de España, y de los ss. XII al XVII. La triple presentación (facsimilar, paleográfica y crítica) facilita su empleo en diversos ámbitos, de la paleografía a la sintaxis histórica. Se adelantan, además, los desarrollos previstos para un futuro inmediato, como la búsqueda por lemas, la presentación estadística directa de los resultados de las consultas y la visualización en forma de mapa de las respuestas a las búsquedas. Y se exponen las vías y posibilidades para su explotación lingüística en los diferentes niveles (grafía y fonética, sintaxis y léxico). El CODEA funda su utilidad en su característica de corpus «primario», en el que los elaboradores son responsables del proceso íntegro de edición de los textos que ofrecen.This paper aims to present the current status of the "Corpus de Documentos Españoles Anteriores a 1700" (CODEA, http://www.textoshispanicos.es) prepared by the "Grupo de Investigación de Textos para la Historia del Español" (GITHE) of the University of Alcalá. CODEA offers 1500 documents from different archives produced in all of the non-bilingual peninsular provinces of Spain, spanning from the 12th to the 17th centuries. The triple presentation (facsimile, paleographic transcription and edition with normalized spelling) facilitates its use in various disciplines, from palaeography to historical syntax. In the paper we also present further developments that will be available in the near future, such as the possibility of conducting searches by lemmata, the presentation of results in statistical graphs, and the display of maps with the geographical distribution of the forms in the search results. Likewise the paper illustrates with various examples the potential of these developments for different types of linguistic analyses (spelling and phonetics, syntax and lexical studies). The most distinctive feature of CODEA is to be a «primary» corpus in which the authors provide the texts and are fully responsible for their editorial process