On the Inductive Bias of Neural Tangent Kernels

State-of-the-art neural networks are heavily over-parameterized, making the optimization algorithm a crucial ingredient for learning predictive models with good generalization properties. A recent line of work has shown that in a certain over-parameterized regime, the learning dynamics of gradient descent are governed by a certain kernel obtained at initialization, called the neural tangent kernel. We study the inductive bias of learning in such a regime by analyzing this kernel and the corresponding function space (RKHS). In particular, we study smoothness, approximation, and stability properties of functions with finite norm, including stability to image deformations in the case of convolutional networks, and compare to other known kernels for similar architectures.Comment: NeurIPS 201

Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations

The success of deep convolutional architectures is often attributed in part to their ability to learn multiscale and invariant representations of natural signals. However, a precise study of these properties and how they affect learning guarantees is still missing. In this paper, we consider deep convolutional representations of signals; we study their invariance to translations and to more general groups of transformations, their stability to the action of diffeomorphisms, and their ability to preserve signal information. This analysis is carried by introducing a multilayer kernel based on convolutional kernel networks and by studying the geometry induced by the kernel mapping. We then characterize the corresponding reproducing kernel Hilbert space (RKHS), showing that it contains a large class of convolutional neural networks with homogeneous activation functions. This analysis allows us to separate data representation from learning, and to provide a canonical measure of model complexity, the RKHS norm, which controls both stability and generalization of any learned model. In addition to models in the constructed RKHS, our stability analysis also applies to convolutional networks with generic activations such as rectified linear units, and we discuss its relationship with recent generalization bounds based on spectral norms

Stochastic Optimization with Variance Reduction for Infinite Datasets with Finite-Sum Structure

Stochastic optimization algorithms with variance reduction have proven successful for minimizing large finite sums of functions. Unfortunately, these techniques are unable to deal with stochastic perturbations of input data, induced for example by data augmentation. In such cases, the objective is no longer a finite sum, and the main candidate for optimization is the stochastic gradient descent method (SGD). In this paper, we introduce a variance reduction approach for these settings when the objective is composite and strongly convex. The convergence rate outperforms SGD with a typically much smaller constant factor, which depends on the variance of gradient estimates only due to perturbations on a single example.Comment: Advances in Neural Information Processing Systems (NIPS), Dec 2017, Long Beach, CA, United State

A Contextual Bandit Bake-off

Contextual bandit algorithms are essential for solving many real-world interactive machine learning problems. Despite multiple recent successes on statistically and computationally efficient methods, the practical behavior of these algorithms is still poorly understood. We leverage the availability of large numbers of supervised learning datasets to empirically evaluate contextual bandit algorithms, focusing on practical methods that learn by relying on optimization oracles from supervised learning. We find that a recent method (Foster et al., 2018) using optimism under uncertainty works the best overall. A surprisingly close second is a simple greedy baseline that only explores implicitly through the diversity of contexts, followed by a variant of Online Cover (Agarwal et al., 2014) which tends to be more conservative but robust to problem specification by design. Along the way, we also evaluate various components of contextual bandit algorithm design such as loss estimators. Overall, this is a thorough study and review of contextual bandit methodology

A Kernel Perspective for Regularizing Deep Neural Networks

We propose a new point of view for regularizing deep neural networks by using the norm of a reproducing kernel Hilbert space (RKHS). Even though this norm cannot be computed, it admits upper and lower approximations leading to various practical strategies. Specifically, this perspective (i) provides a common umbrella for many existing regularization principles, including spectral norm and gradient penalties, or adversarial training, (ii) leads to new effective regularization penalties, and (iii) suggests hybrid strategies combining lower and upper bounds to get better approximations of the RKHS norm. We experimentally show this approach to be effective when learning on small datasets, or to obtain adversarially robust models.Comment: ICM

Online learning for audio clustering and segmentation

International audienceAudio segmentation is an essential problem in many audio signal processing tasks which tries to segment an audio signal into homogeneous chunks, or segments. Most current approaches rely on a change-point detection phase for finding segment boundaries, followed by a similarity matching phase which identifies similar segments. In this thesis, we focus instead on joint segmentation and clustering algorithms which solve both tasks simultaneously, through the use of unsupervised learning techniques in sequential models. Hidden Markov and semi-Markov models are a natural choice for this modeling task, and we present their use in the context of audio segmentation. We then explore the use of online learning techniques in sequential models and their application to real-time audio segmentation tasks. We present an existing online EM algorithm for hidden Markov models and extend it to hidden semi-Markov models by introducing a different parameterization of semi-Markov chains. Finally, we develop new online learning algorithms for sequential models based on incremental optimization of surrogate functions.Le problème de la segmentation audio, essentiel dans de nombreuses tâches de traitement du signal audio, cherche à décomposer un signal audio en courts segments de contenu homogène. La plupart des approches courantes en segmentation sont basées sur une phase de détection de rupture qui trouve les limites entre segments, suivie d'une phase de calcul de similarité qui identifie les segments similaires. Dans ce rapport, nous nous intéressons à une approche différente, qui cherche à effectuer les deux tâches -- segmentation et clustering -- simultanément, avec des méthodes d'apprentissage non supervisé dans des modèles séquentiels. Les modèles de Markov et de semi-Markov cachés sont des choix naturels dans ce contexte de modélisation, et nous présentons leur utilisation en segmentation audio. Nous nous intéressons ensuite à l'utilisation de méthodes d'apprentissage en ligne dans des modèles séquentiels, et leur application à la segmentation audio en temps réel. Nous présentons un modèle existant de online EM pour les modèles de Markov cachés, et l'étendons aux modèles de semi-Markov cachés grâce à une nouvelle paramétrisation des chaines de semi-Markov. Enfin, nous introduisons de nouveaux algorithmes en ligne pour les modèles séquentiels qui s'appuient sur une optimisation incrémentale de fonctions surrogées

On minimal variations for unsupervised representation learning

Unsupervised representation learning aims at describing raw data efficiently to solve various downstream tasks. It has been approached with many techniques, such as manifold learning, diffusion maps, or more recently self-supervised learning. Those techniques are arguably all based on the underlying assumption that target functions, associated with future downstream tasks, have low variations in densely populated regions of the input space. Unveiling minimal variations as a guiding principle behind unsupervised representation learning paves the way to better practical guidelines for self-supervised learning algorithms.Comment: 5 pages, 1 figure; 1 tabl

On the Inductive Bias of Neural Tangent Kernels

International audienceState-of-the-art neural networks are heavily over-parameterized, making the optimization algorithm a crucial ingredient for learning predictive models with good generalization properties. A recent line of work has shown that in a certain over-parameterized regime, the learning dynamics of gradient descent are governed by a certain kernel obtained at initialization, called the neural tangent kernel. We study the inductive bias of learning in such a regime by analyzing this kernel and the corresponding function space (RKHS). In particular, we study smoothness, approximation, and stability properties of functions with finite norm, including stability to image deformations in the case of convolutional networks, and compare to other known kernels for similar architectures

Invariance and Stability of Deep Convolutional Representations

International audienceIn this paper, we study deep signal representations that are near-invariant to groups of transformations and stable to the action of diffeomorphisms without losing signal information. This is achieved by generalizing the multilayer kernel introduced in the context of convolutional kernel networks and by studying the geometry of the corresponding reproducing kernel Hilbert space. We show that the signalrepresentation is stable, and that models from this functional space, such as a large class of convolutional neural networks, may enjoy the same stability