Nicolas Le Roux

Ross Goroshin

Image pre-processing in the frequency domain has traditionally played a vital role in computer vision and was even part of the standard pipe… (see more)line in the early days of deep learning. However, with the advent of large datasets, many practitioners concluded that this was unnecessary due to the belief that these priors can be learned from the data itself. Frequency aliasing is a phenomenon that may occur when sub-sampling any signal, such as an image or feature map, causing distortion in the sub-sampled output. We show that we can mitigate this effect by placing non-trainable blur filters and using smooth activation functions at key locations, particularly where networks lack the capacity to learn them. These simple architectural changes lead to substantial improvements in out-of-distribution generalization on both image classification under natural corruptions on ImageNet-C [10] and few-shot learning on Meta-Dataset [17], without introducing additional trainable parameters and using the default hyper-parameters of open source codebases.

2020-11-20

ArXiv (preprint)

To Each Optimizer a Norm, To Each Norm its Generalization

Sharan Vaswani

Reza Babanezhad Harikandeh

Jose Gallego

Aaron Mishkin

Simon Lacoste-Julien

We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and… (see more) over-parametrized regimes. Since it is difficult to determine whether an optimizer converges to solutions that minimize a known norm, we flip the problem and investigate what is the corresponding norm minimized by an interpolating solution. Using this reasoning, we prove that for over-parameterized linear regression, projections onto linear spans can be used to move between different interpolating solutions. For under-parameterized linear classification, we prove that for any linear classifier separating the data, there exists a family of quadratic norms ||.||_P such that the classifier's direction is the same as that of the maximum P-margin solution. For linear classification, we argue that analyzing convergence to the standard maximum l2-margin is arbitrary and show that minimizing the norm induced by the data results in better generalization. Furthermore, for over-parameterized linear classification, projections onto the data-span enable us to use techniques from the under-parameterized setting. On the empirical side, we propose techniques to bias optimizers towards better generalizing solutions, improving their test performance. We validate our theoretical results via synthetic experiments, and use the neural tangent kernel to handle non-linear models.

2020-06-11

ArXiv (preprint)

The Geometry of Sign Gradient Descent

Lukas Balles

Fabian Pedregosa

2020-02-19

ArXiv (preprint)

openreview.net

How to make your optimizer generalize better

Sharan Vaswani

Reza Babenzhad

Sait AI Lab

Montreal.

Jose Gallego

Aaron Mishkin

Simon Lacoste-Julien

We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and… (see more) over-parametrized regimes. For over-parameterized linear regression, where there are inﬁnitely many interpolating solutions, different optimization methods can converge to solutions with varying generalization performance. In this setting, we show that projections onto linear spans can be used to move between solutions. Furthermore, via a simple reparameterization, we can ensure that an arbitrary optimizer converges to the minimum (cid:96) 2 -norm solution with favourable generalization properties. For under-parameterized linear clas-siﬁcation, optimizers can converge to different decision boundaries separating the data. We prove that for any such classiﬁer, there exists a family of quadratic norms (cid:107)·(cid:107) P such that the classiﬁer’s direction is the same as that of the maximum P -margin solution. We argue that analyzing convergence to the standard maximum (cid:96) 2 -margin is arbitrary and show that minimizing the norm induced by the data can result in better generalization. We validate our theoretical results via experiments on synthetic and real datasets.

2020-01-01

(published)

www.semanticscholar.org

An operator view of policy gradient methods

Dibya Ghosh

Marlos C. Machado

We cast policy gradient methods as the repeated application of two operators: a policy improvement operator …

On the interplay between noise and curvature and its effect on optimization and generalization

Valentin Thomas

Fabian Pedregosa

Bart van Merriënboer

Pierre-Antoine Manzagol

The speed at which one can minimize an expected loss using stochastic methods depends on two properties: the curvature of the loss and the v… (see more)ariance of the gradients. While most previous works focus on one or the other of these properties, we explore how their interaction affects optimization speed. Further, as the ultimate goal is good generalization performance, we clarify how both curvature and noise are relevant to properly estimate the generalization gap. Realizing that the limitations of some existing works stems from a confusion between these matrices, we also clarify the distinction between the Fisher matrix, the Hessian, and the covariance matrix of the gradients.

2020-01-01

AISTATS (published)

proceedings.mlr.press

On the interplay between noise and curvature and its effect on optimization and generalization

Valentin Thomas

Fabian Pedregosa

Bart van Merriënboer

Pierre-Antoine Mangazol

The speed at which one can minimize an expected loss using stochastic methods depends on two properties: the curvature of the loss and the v… (see more)ariance of the gradients. While most previous works focus on one or the other of these properties, we explore how their interaction affects optimization speed. Further, as the ultimate goal is good generalization performance, we clarify how both curvature and noise are relevant to properly estimate the generalization gap. Realizing that the limitations of some existing works stems from a confusion between these matrices, we also clarify the distinction between the Fisher matrix, the Hessian, and the covariance matrix of the gradients.

2019-06-19

ArXiv (preprint)

Information matrices and generalization

Valentin Thomas

Fabian Pedregosa

Bart van Merriënboer

Pierre-Antoine Manzagol

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

Pierre-Antoine Manzagol

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

The Value Function Polytope in Reinforcement Learning

Robert Dadashi

Adrien Ali Taiga

Dale Schuurmans

Marc Gendron-Bellemare

We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main… (see more) contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structural relationship between policies and value functions including the line theorem, which shows that the value functions of policies constrained on all but one state describe a line segment. Finally, we use this novel perspective to introduce visualizations to enhance the understanding of the dynamics of reinforcement learning algorithms.

2019-05-24

Proceedings of the 36th International Conference on Machine Learning (published)

proceedings.mlr.press

Understanding the impact of entropy on policy optimization

Zafarali Ahmed

Mohammad Norouzi

Dale Schuurmans

Entropy regularization is commonly used to improve policy optimization in reinforcement learning. It is believed to help with \emph{explorat… (see more)ion} by encouraging the selection of more stochastic policies. In this work, we analyze this claim using new visualizations of the optimization landscape based on randomly perturbing the loss function. We first show that even with access to the exact gradient, policy optimization is difficult due to the geometry of the objective function. Then, we qualitatively show that in some environments, a policy with higher entropy can make the optimization landscape smoother, thereby connecting local optima and enabling the use of larger learning rates. This paper presents new tools for understanding the optimization landscape, shows that policy entropy serves as a regularizer, and highlights the challenge of designing general-purpose policy optimization algorithms.

2019-05-24

Proceedings of the 36th International Conference on Machine Learning (published)

proceedings.mlr.press