Pierre-Antoine Manzagol

Information matrices and generalization

Valentin Thomas

Fabian Pedregosa

Bart Van Merriënboer

Yoshua Bengio

This work revisits the use of information criteria to characterize the generalization of deep learning models. In particular, we empirically… (see more) demonstrate the effectiveness of the Takeuchi information criterion (TIC), an extension of the Akaike information criterion (AIC) for misspecified models, in estimating the generalization gap, shedding light on why quantities such as the number of parameters cannot quantify generalization. The TIC depends on both the Hessian of the loss H and the covariance of the gradients C. By exploring the similarities and differences between these two matrices as well as the Fisher information matrix F, we study the interplay between noise and curvature in deep models. We also address the question of whether C is a reasonable approximation to F, as is commonly assumed.

2019-06-18

arXiv.org (preprint)

dblp.uni-trier.de

Information matrices and generalization

Valentin Thomas

Fabian Pedregosa

Bart Van Merriënboer

Yoshua Bengio

This work revisits the use of information criteria to characterize the generalization of deep learning models. In particular, we empirically… (see more) demonstrate the effectiveness of the Takeuchi information criterion (TIC), an extension of the Akaike information criterion (AIC) for misspecified models, in estimating the generalization gap, shedding light on why quantities such as the number of parameters cannot quantify generalization. The TIC depends on both the Hessian of the loss H and the covariance of the gradients C. By exploring the similarities and differences between these two matrices as well as the Fisher information matrix F, we study the interplay between noise and curvature in deep models. We also address the question of whether C is a reasonable approximation to F, as is commonly assumed.

2019-06-18

arXiv.org (preprint)

dblp.uni-trier.de

Reducing the variance in online optimization by transporting past gradients

Sébastien M. R. Arnold

Reza Babanezhad Harikandeh

Ioannis Mitliagkas

Most stochastic optimization methods use gradients once before discarding them. While variance reduction methods have shown that reusing pas… (see more)t gradients can be beneficial when there is a finite number of datapoints, they do not easily extend to the online setting. One issue is the staleness due to using past gradients. We propose to correct this staleness using the idea of implicit gradient transport (IGT) which transforms gradients computed at previous iterates into gradients evaluated at the current iterate without using the Hessian explicitly. In addition to reducing the variance and bias of our updates over time, IGT can be used as a drop-in replacement for the gradient estimate in a number of well-understood methods such as heavy ball or Adam. We show experimentally that it achieves state-of-the-art results on a wide range of architectures and benchmarks. Additionally, the IGT gradient estimator yields the optimal asymptotic convergence rate for online stochastic optimization in the restricted setting where the Hessians of all component functions are equal.

Online variance-reducing optimization

Reza Babanezhad Harikandeh

Reza Babanezhad

2018-02-12

International Conference on Learning Representations (published)

openreview.net

Negative eigenvalues of the Hessian in deep neural networks

Guillaume Alain

Nicolas Boulanger-Lewandowski

The loss function of deep networks is known to be non-convex but the precise nature of this nonconvexity is still an active area of research… (see more). In this work, we study the loss landscape of deep networks through the eigendecompositions of their Hessian matrix. In particular, we examine how important the negative eigenvalues are and the benefits one can observe in handling them appropriately.

2018-01-01

ICLR (Workshop) (published)

openreview.net

Theano: A Python framework for fast computation of mathematical expressions

Rami Al-rfou'

Guillaume Alain

Amjad Almahairi

Christof Angermüller

Dzmitry Bahdanau

Nicolas Ballas

Frédéric Bastien

Justin S. Bayer

A. Belikov

A. Belopolsky

J. Bergstra

Josh Bleecher Snyder

Xavier Bouthillier

Alexandre De Brébisson

Olivier Breuleux … (see 92 more)

Paul F. Christiano

Marc-Alexandre Côté

Myriam Côté

Julien Demouth

Sander Dieleman

M'elanie Ducoffe

Samira Ebrahimi Kahou

Dumitru Erhan

Ziye Fan

Orhan Firat

Mathieu Germain

Xavier Glorot

Ian J. Goodfellow

Matthew Graham

Balázs Hidasi

Arjun Jain

S'ebastien Jean

Kai Jia

Mikhail V. Korobov

Vivek Kulkarni

Alex Lamb

Pascal Lamblin

Eric P. Larsen

César Laurent

S. Lee

Simon-mark Lefrancois

Simon Lemieux

Nicholas Léonard

Zhouhan Lin

J. Livezey

Cory R. Lorenz

Jeremiah L. Lowin

Qianli M. Ma

R. McGibbon

Mehdi Mirza

Alberto Orlandi

Chris Pal

Razvan Pascanu

Mohammad Pezeshki

Colin Raffel

Daniel Renshaw

Matthew David Rocklin

Adriana Romero Soriano

Markus Dr. Roth

Peter Sadowski

John Salvatier

François Savard

Jan Schlüter

John D. Schulman

Gabriel Schwartz

Iulian V. Serban

Dmitriy Serdyuk

Samira Shabanian

Etienne Simon

Sigurd Spieckermann

S. Subramanyam

Jakub Sygnowski

Jeremie Tanguay

Gijs van Tulder

Joseph P. Turian

Sebastian Urban

Dustin J. Webb

M. Willson

Lijun Xue

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficie… (see more)ntly. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.

2016-05-09

ArXiv (preprint)

Nicolas Boulanger-Lewandowski

Theano: A Python framework for fast computation of mathematical expressions

Rami Al-rfou'

Guillaume Alain

Amjad Almahairi

Christof Angermüller

Dzmitry Bahdanau

Nicolas Ballas

Frédéric Bastien

Justin S. Bayer

A. Belikov

A. Belopolsky

Josh Bleecher Snyder

Xavier Bouthillier

Alexandre De Brébisson

Olivier Breuleux … (see 92 more)

Paul F. Christiano

Marc-Alexandre Côté

Myriam Côté

Julien Demouth

Sander Dieleman

M'elanie Ducoffe

Samira Ebrahimi Kahou

Dumitru Erhan

Ziye Fan

Orhan Firat

Mathieu Germain

Xavier Glorot

Ian G Goodfellow

Matthew Graham

Balázs Hidasi

Arjun Jain

Kai Jia

Mikhail V. Korobov

Vivek Kulkarni

Alex Lamb

Pascal Lamblin

Eric Larsen

César Laurent

S. Lee

Simon-mark Lefrancois

Simon Lemieux

Nicholas Léonard

Zhouhan Lin

J. Livezey

Cory R. Lorenz

Jeremiah L. Lowin

Qianli M. Ma

R. McGibbon

Mehdi Mirza

Alberto Orlandi

Chris Pal

Razvan Pascanu

Mohammad Pezeshki

Colin Raffel

Daniel Renshaw

Matthew David Rocklin

Adriana Romero Soriano

Markus Dr. Roth

Peter Sadowski

John Salvatier

François Savard

Jan Schlüter

John D. Schulman

Gabriel Schwartz

Iulian V. Serban

Dmitriy Serdyuk

Samira Shabanian

Etienne Simon

Sigurd Spieckermann

S. Subramanyam

Gijs van Tulder

Sebastian Urban

Dustin J. Webb

M. Willson

Lijun Xue

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficie… (see more)ntly. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.

2016-05-09

ArXiv (preprint)