Theoretical Deep Learning

Deep learning has long been criticised as a black-box model for lacking sound theoretical explanation. During the PhD course, I explore and establish theoretical foundations for deep learning. In this thesis, I present my contributions positioned upon existing literature: (1) analysing the generalizability of the neural networks with residual connections via complexity and capacity-based hypothesis complexity measures; (2) modeling stochastic gradient descent (SGD) by stochastic differential equations (SDEs) and their dynamics, and further characterizing the generalizability of deep learning; (3) understanding the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems, which sheds light in reconciling the over-representation and excellent generalizability of deep learning; and (4) discovering the interplay between generalization, privacy preservation, and adversarial robustness, which have seen rising concerns in deep learning deployment

When Are Solutions Connected in Deep Networks?

The question of how and why the phenomenon of mode connectivity occurs in training deep neural networks has gained remarkable attention in the research community. From a theoretical perspective, two possible explanations have been proposed: (i) the loss function has connected sublevel sets, and (ii) the solutions found by stochastic gradient descent are dropout stable. While these explanations provide insights into the phenomenon, their assumptions are not always satisfied in practice. In particular, the first approach requires the network to have one layer with order of $N$ neurons ($N$ being the number of training samples), while the second one requires the loss to be almost invariant after removing half of the neurons at each layer (up to some rescaling of the remaining ones). In this work, we improve both conditions by exploiting the quality of the features at every intermediate layer together with a milder over-parameterization condition. More specifically, we show that: (i) under generic assumptions on the features of intermediate layers, it suffices that the last two hidden layers have order of $\sqrt{N}$ neurons, and (ii) if subsets of features at each layer are linearly separable, then no over-parameterization is needed to show the connectivity. Our experiments confirm that the proposed condition ensures the connectivity of solutions found by stochastic gradient descent, even in settings where the previous requirements do not hold.Comment: Accepted at NeurIPS 202

Parameterizing and Aggregating Activation Functions in Deep Neural Networks

The nonlinear activation functions applied by each neuron in a neural network are essential for making neural networks powerful representational models. If these are omitted, even deep neural networks reduce to simple linear regression due to the fact that a linear combination of linear combinations is still a linear combination. In much of the existing literature on neural networks, just one or two activation functions are selected for the entire network, even though the use of heterogenous activation functions has been shown to produce superior results in some cases. Even less often employed are activation functions that can adapt their nonlinearities as network parameters along with standard weights and biases. This dissertation presents a collection of papers that advance the state of heterogenous and parameterized activation functions. Contributions of this dissertation include three novel parametric activation functions and applications of each, a study evaluating the utility of the parameters in parametric activation functions, an aggregated activation approach to modeling time-series data as an alternative to recurrent neural networks, and an improvement upon existing work that aggregates neuron inputs using product instead of sum

Sparse, hierarchical and shared-factors priors for representation learning

La représentation en caractéristiques est une préoccupation centrale des systèmes d’apprentissage automatique d’aujourd’hui. Une représentation adéquate peut faciliter une tâche d’apprentissage complexe. C’est le cas lorsque par exemple cette représentation est de faible dimensionnalité et est constituée de caractéristiques de haut niveau. Mais comment déterminer si une représentation est adéquate pour une tâche d’apprentissage ? Les récents travaux suggèrent qu’il est préférable de voir le choix de la représentation comme un problème d’apprentissage en soi. C’est ce que l’on nomme l’apprentissage de représentation. Cette thèse présente une série de contributions visant à améliorer la qualité des représentations apprises. La première contribution élabore une étude comparative des approches par dictionnaire parcimonieux sur le problème de la localisation de points de prises (pour la saisie robotisée) et fournit une analyse empirique de leurs avantages et leurs inconvénients. La deuxième contribution propose une architecture réseau de neurones à convolution (CNN) pour la détection de points de prise et la compare aux approches d’apprentissage par dictionnaire. Ensuite, la troisième contribution élabore une nouvelle fonction d’activation paramétrique et la valide expérimentalement. Finalement, la quatrième contribution détaille un nouveau mécanisme de partage souple de paramètres dans un cadre d’apprentissage multitâche.Feature representation is a central concern of today’s machine learning systems. A proper representation can facilitate a complex learning task. This is the case when for instance the representation has low dimensionality and consists of high-level characteristics. But how can we determine if a representation is adequate for a learning task? Recent work suggests that it is better to see the choice of representation as a learning problem in itself. This is called Representation Learning. This thesis presents a series of contributions aimed at improving the quality of the learned representations. The first contribution elaborates a comparative study of Sparse Dictionary Learning (SDL) approaches on the problem of grasp detection (for robotic grasping) and provides an empirical analysis of their advantages and disadvantages. The second contribution proposes a Convolutional Neural Network (CNN) architecture for grasp detection and compares it to SDL. Then, the third contribution elaborates a new parametric activation function and validates it experimentally. Finally, the fourth contribution details a new soft parameter sharing mechanism for multitasking learning

Mildly Overparameterized ReLU Networks Have a Favorable Loss Landscape

We study the loss landscape of both shallow and deep, mildly overparameterized ReLU neural networks on a generic finite input dataset for the squared error loss. We show both by count and volume that most activation patterns correspond to parameter regions with no bad local minima. Furthermore, for one-dimensional input data, we show most activation regions realizable by the network contain a high dimensional set of global minima and no bad local minima. We experimentally confirm these results by finding a phase transition from most regions having full rank Jacobian to many regions having deficient rank depending on the amount of overparameterization.Comment: 40 page