Régression sur composantes principales

En statistiques, la Régression sur composantes principales est une analyse en régression sur les composantes d'une analyse en composantes principales. On utilise souvent cette technique lorsque les variables explicatives sont proches d'être colinéaires, lorsque par exemple le nombre de variables est très supérieur au nombre d'individus. La Régression sur composantes principales est souvent comparée à la Régression PLS Principe La méthode se déroule en trois phases :

Analyse en composantes principales sur les variables explicatives
Régression linéaire de la variable dépendante (ou variable à expliquer) sur la partie des composantes la plus corrélée à celle-ci
Calcul des paramètres de la régression en fonction des variables d'origines.

Notes et références Notes Références Ouvrages spécialisés Articles publiés sur internet Voir aussi Bibliographie *

Theory of Deep Learning: Neural Tangent Kernel and Beyond

Arthur Ulysse Jacot-Guillarmod

In the recent years, Deep Neural Networks (DNNs) have managed to succeed at tasks that previously appeared impossible, such as human-level object recognition, text synthesis, translation, playing games and many more. In spite of these major achievements, our understanding of these models, in particular of what happens during their training, remains very limited. This PhD started with the introduction of the Neural Tangent Kernel (NTK) to describe the evolution of the function represented by the network during training. In the infinite-width limit, i.e. when the number of neurons in the layers of the network grows to infinity, the NTK converges to a deterministic and time-independent limit, leading to a simple yet complete description of the dynamics of infinitely-wide DNNs. This allowed one to give the first general proof of convergence of DNNs to a global minimum, and yielded the first description of the limiting spectrum of the Hessian of the loss surface of DNNs throughout training.More importantly, the NTK plays a crucial role in describing the generalization abilities of DNNs, i.e. the performance of the trained network on unseen data. The NTK analysis uncovered a direct link between the function learned by infinitely wide DNNs and Kernel Ridge Regression predictors, whose generalization properties are studied in this thesis using tools of random matrix theory. Our analysis of KRR reveals the importance of the eigendecomposition of the NTK, which is affected by a number of architectural choices. In very deep networks, an ordered regime and a chaotic regime appear, determined by the choice of non-linearity and the balance between the weights and bias parameters; these two phases are characterized by different speeds of decay of the eigenvalues of the NTK, leading to a tradeoff between convergence speed and generalization. In practical contexts such as Generative Adversarial Networks or Topology Optimization, the network architecture can be chosen to guarantee certain properties of the NTK and its spectrum.These results give an almost complete description DNNs in this infinite-width limit. It is then natural to wonder how it extends to finite-width networks used in practice. In the so-called NTK regime, the discrepancy between finite- and infinite-widths DNNs is mainly a result of the variance w.r.t. to the sampling of the parameters, as shown empirically and mathematically relying on the similarity between DNNs and random feature models.In contrast to the NTK regime, where the NTK remains constant during training, there exist so-called active regimes, where the evolution of the NTK is significant, which appear in a number of settings. One such regime appears in Deep Linear Networks with a very small initialization, where the training dynamics approach a sequence of saddle-points, representing linear maps of increasing rank, leading to a low-rank bias which is absent in the NTK regime.

EPFL2022

Locality defeats the curse of dimensionality in convolutional teacher-student scenarios

Francesco Cagnetta, Alessandro Favero, Matthieu Wyart

Convolutional neural networks perform a local and translationally-invariant treatment of the data: quantifying which of these two aspects is central to their success remains a challenge. We study this problem within a teacher-student framework for kernel regression, using `convolutional' kernels inspired by the neural tangent kernel of simple convolutional architectures of given filter size. Using heuristic methods from physics, we find in the ridgeless case that locality is key in determining the learning curve exponent

\beta

(that relates the test error

\epsilon_t\sim P^{-\beta}

to the size of the training set

P

), whereas translational invariance is not. In particular, if the filter size of the teacher

t

is smaller than that of the student

s

\beta

is a function of

s

only and does not depend on the input dimension. We confirm our predictions on

\beta

empirically. We conclude by proving, under a natural universality assumption, that performing kernel regression with a ridge that decreases with the size of the training set leads to similar learning curve exponents to those we obtain in the ridgeless case.

2021