Andrés R. Masegosa

Andrés R. Masegosa, Luis A. Ortega

This paper introduces a distribution-dependent PAC-Chernoff bound that exhibits perfect tightness for interpolators, even within over-parameterized model classes. This bound, which relies on basic principles of Large Deviation Theory, defines a natural measure of the smoothness of a model, characterized by simple real-valued functions. Building upon this bound and the new concept of smoothness, we present an unified theoretical framework revealing why certain interpolators show an exceptional generalization, while others falter. We theoretically show how a wide spectrum of modern learning methodologies, encompassing techniques such as $\ell_2$-norm, distance-from-initialization and input-gradient regularization, in combination with data augmentation, invariant architectures, and over-parameterization, collectively guide the optimizer toward smoother interpolators, which, according to our theoretical framework, are the ones exhibiting superior generalization performance. This study shows that distribution-dependent bounds serve as a powerful tool to understand the complex dynamics behind the generalization capabilities of over-parameterized interpolators.

Yi-Shan Wu, Andrés R. Masegosa, Stephan S. Lorenzen et al.

We present a new second-order oracle bound for the expected risk of a weighted majority vote. The bound is based on a novel parametric form of the Chebyshev- Cantelli inequality (a.k.a. one-sided Chebyshev's), which is amenable to efficient minimization. The new form resolves the optimization challenge faced by prior oracle bounds based on the Chebyshev-Cantelli inequality, the C-bounds [Germain et al., 2015], and, at the same time, it improves on the oracle bound based on second order Markov's inequality introduced by Masegosa et al. [2020]. We also derive a new concentration of measure inequality, which we name PAC-Bayes-Bennett, since it combines PAC-Bayesian bounding with Bennett's inequality. We use it for empirical estimation of the oracle bound. The PAC-Bayes-Bennett inequality improves on the PAC-Bayes-Bernstein inequality of Seldin et al. [2012]. We provide an empirical evaluation demonstrating that the new bounds can improve on the work of Masegosa et al. [2020]. Both the parametric form of the Chebyshev-Cantelli inequality and the PAC-Bayes-Bennett inequality may be of independent interest for the study of concentration of measure in other domains.

Andrés R. Masegosa, Stephan S. Lorenzen, Christian Igel et al.

We present a novel analysis of the expected risk of weighted majority vote in multiclass classification. The analysis takes correlation of predictions by ensemble members into account and provides a bound that is amenable to efficient minimization, which yields improved weighting for the majority vote. We also provide a specialized version of our bound for binary classification, which allows to exploit additional unlabeled data for tighter risk estimation. In experiments, we apply the bound to improve weighting of trees in random forests and show that, in contrast to the commonly used first order bound, minimization of the new bound typically does not lead to degradation of the test error of the ensemble.

Andres R. Masegosa

Virtually any model we use in machine learning to make predictions does not perfectly represent reality. So, most of the learning happens under model misspecification. In this work, we present a novel analysis of the generalization performance of Bayesian model averaging under model misspecification and i.i.d. data using a new family of second-order PAC-Bayes bounds. This analysis shows, in simple and intuitive terms, that Bayesian model averaging provides suboptimal generalization performance when the model is misspecified. In consequence, we provide strong theoretical arguments showing that Bayesian methods are not optimal for learning predictive models, unless the model class is perfectly specified. Using novel second-order PAC-Bayes bounds, we derive a new family of Bayesian-like algorithms, which can be implemented as variational and ensemble methods. The output of these algorithms is a new posterior distribution, different from the Bayesian posterior, which induces a posterior predictive distribution with better generalization performance. Experiments with Bayesian neural networks illustrate these findings.

Javier Cózar, Rafael Cabañas, Antonio Salmerón et al.

InferPy is a Python package for probabilistic modeling with deep neural networks. It defines a user-friendly API that trades-off model complexity with ease of use, unlike other libraries whose focus is on dealing with very general probabilistic models at the cost of having a more complex API. In particular, this package allows to define, learn and evaluate general hierarchical probabilistic models containing deep neural networks in a compact and simple way. InferPy is built on top of Tensorflow Probability and Keras.

Andrés R. Masegosa, Rafael Cabañas, Helge Langseth et al.

Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to (i) very restricted model classes where exact or approximate probabilistic inference were feasible, and (ii) small or medium-sized data sets which fit within the main memory of the computer. However, developments in variational inference, a general form of approximate probabilistic inference originated in statistical physics, are allowing probabilistic modeling to overcome these restrictions: (i) Approximate probabilistic inference is now possible over a broad class of probabilistic models containing a large number of parameters, and (ii) scalable inference methods based on stochastic gradient descent and distributed computation engines allow to apply probabilistic modeling over massive data sets. One important practical consequence of these advances is the possibility to include deep neural networks within a probabilistic model to capture complex non-linear stochastic relationships between random variables. These advances in conjunction with the release of novel probabilistic modeling toolboxes have greatly expanded the scope of application of probabilistic models, and allow these models to take advantage of the recent strides made by the deep learning community. In this paper we review the main concepts, methods and tools needed to use deep neural networks within a probabilistic modeling framework.

Andres Masegosa, Thomas D. Nielsen, Helge Langseth et al.

Making inferences from data streams is a pervasive problem in many modern data analysis applications. But it requires to address the problem of continuous model updating and adapt to changes or drifts in the underlying data generating distribution. In this paper, we approach these problems from a Bayesian perspective covering general conjugate exponential models. Our proposal makes use of non-conjugate hierarchical priors to explicitly model temporal changes of the model parameters. We also derive a novel variational inference scheme which overcomes the use of non-conjugate priors while maintaining the computational efficiency of variational methods over conjugate models. The approach is validated on three real data sets over three latent variable models.