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Publication# Theory of Deep Learning: Neural Tangent Kernel and Beyond

Résumé

In the recent years, Deep Neural Networks (DNNs) have managed to succeed at tasks that previously appeared impossible, such as human-level object recognition, text synthesis, translation, playing games and many more. In spite of these major achievements, our understanding of these models, in particular of what happens during their training, remains very limited. This PhD started with the introduction of the Neural Tangent Kernel (NTK) to describe the evolution of the function represented by the network during training. In the infinite-width limit, i.e. when the number of neurons in the layers of the network grows to infinity, the NTK converges to a deterministic and time-independent limit, leading to a simple yet complete description of the dynamics of infinitely-wide DNNs. This allowed one to give the first general proof of convergence of DNNs to a global minimum, and yielded the first description of the limiting spectrum of the Hessian of the loss surface of DNNs throughout training.More importantly, the NTK plays a crucial role in describing the generalization abilities of DNNs, i.e. the performance of the trained network on unseen data. The NTK analysis uncovered a direct link between the function learned by infinitely wide DNNs and Kernel Ridge Regression predictors, whose generalization properties are studied in this thesis using tools of random matrix theory. Our analysis of KRR reveals the importance of the eigendecomposition of the NTK, which is affected by a number of architectural choices. In very deep networks, an ordered regime and a chaotic regime appear, determined by the choice of non-linearity and the balance between the weights and bias parameters; these two phases are characterized by different speeds of decay of the eigenvalues of the NTK, leading to a tradeoff between convergence speed and generalization. In practical contexts such as Generative Adversarial Networks or Topology Optimization, the network architecture can be chosen to guarantee certain properties of the NTK and its spectrum.These results give an almost complete description DNNs in this infinite-width limit. It is then natural to wonder how it extends to finite-width networks used in practice. In the so-called NTK regime, the discrepancy between finite- and infinite-widths DNNs is mainly a result of the variance w.r.t. to the sampling of the parameters, as shown empirically and mathematically relying on the similarity between DNNs and random feature models.In contrast to the NTK regime, where the NTK remains constant during training, there exist so-called active regimes, where the evolution of the NTK is significant, which appear in a number of settings. One such regime appears in Deep Linear Networks with a very small initialization, where the training dynamics approach a sequence of saddle-points, representing linear maps of increasing rank, leading to a low-rank bias which is absent in the NTK regime.

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Deep neural networks have been empirically successful in a variety of tasks, however their theoretical understanding is still poor. In particular, modern deep neural networks have many more parameters than training data. Thus, in principle they should overfit the training samples and exhibit poor generalization to the complete data distribution. Counter intuitively however, they manage to achieve both high training accuracy and high testing accuracy. One can prove generalization using a validation set, however this can be difficult when training samples are limited and at the same time we do not obtain any information about why deep neural networks generalize well. Another approach is to estimate the complexity of the deep neural network. The hypothesis is that if a network with high training accuracy has high complexity it will have memorized the data, while if it has low complexity it will have learned generalizable patterns. In the first part of this thesis we explore Spectral Complexity, a measure of complexity that depends on combinations of norms of the weight matrices of the deep neural network. For a dataset that is difficult to classify, with no underlying model and/or no recurring pattern, for example one where the labels have been chosen randomly, spectral complexity has a large value, reflecting that the network needs to memorize the labels, and will not generalize well. Putting back the real labels, the spectral complexity becomes lower reflecting that some structure is present and the network has learned patterns that might generalize to unseen data. Spectral complexity results in vacuous estimates of the generalization error (the difference between the training and testing accuracy), and we show that it can lead to counterintuitive results when comparing the generalization error of different architectures. In the second part of the thesis we explore non-vacuous estimates of the generalization error. In Chapter 2 we analyze the case of PAC-Bayes where a posterior distribution over the weights of a deep neural network is learned using stochastic variational inference, and the generalization error is the KL divergence between this posterior and a prior distribution. We find that a common approximation where the posterior is constrained to be Gaussian with diagonal covariance, known as the mean-field approximation, limits significantly any gains in bound tightness. We find that, if we choose the prior mean to be the random network initialization, the generalization error estimate tightens significantly. In Chapter 3 we explore an existing approach to learning the prior mean, in PAC-Bayes, from the training set. Specifically, we explore differential privacy, which ensures that the training samples contribute only a limited amount of information to the prior, making it distribution and not training set dependent. In this way the prior should generalize well to unseen data (as it hasn't memorized individual samples) and at the same time any posterior distribution that is close to it in terms of the KL divergence will also exhibit good generalization.

With ever greater computational resources and more accessible software, deep neural networks have become ubiquitous across industry and academia.
Their remarkable ability to generalize to new samples defies the conventional view, which holds that complex, over-parameterized networks would be prone to overfitting.
This apparent discrepancy is exacerbated by our inability to inspect and interpret the high-dimensional, non-linear, latent representations they learn, which has led many to refer to neural networks as

`black-boxes''. The Law of Parsimony states that `

simpler solutions are more likely to be correct than complex ones''. Since they perform quite well in practice, a natural question to ask, then, is in what way are neural networks simple?
We propose that compression is the answer. Since good generalization requires invariance to irrelevant variations in the input, it is necessary for a network to discard this irrelevant information. As a result, semantically similar samples are mapped to similar representations in neural network deep feature space, where they form simple, low-dimensional structures.
Conversely, a network that overfits relies on memorizing individual samples. Such a network cannot discard information as easily.
In this thesis we characterize the difference between such networks using the non-negative rank of activation matrices. Relying on the non-negativity of rectified-linear units, the non-negative rank is the smallest number that admits an exact non-negative matrix factorization.
We derive an upper bound on the amount of memorization in terms of the non-negative rank, and show it is a natural complexity measure for rectified-linear units.
With a focus on deep convolutional neural networks trained to perform object recognition, we show that the two non-negative factors derived from deep network layers decompose the information held therein in an interpretable way. The first of these factors provides heatmaps which highlight similarly encoded regions within an input image or image set. We find that these networks learn to detect semantic parts and form a hierarchy, such that parts are further broken down into sub-parts.
We quantitatively evaluate the semantic quality of these heatmaps by using them to perform semantic co-segmentation and co-localization. In spite of the convolutional network we use being trained solely with image-level labels, we achieve results comparable or better than domain-specific state-of-the-art methods for these tasks.
The second non-negative factor provides a bag-of-concepts representation for an image or image set. We use this representation to derive global image descriptors for images in a large collection. With these descriptors in hand, we perform two variations content-based image retrieval, i.e. reverse image search. Using information from one of the non-negative matrix factors we obtain descriptors which are suitable for finding semantically related images, i.e., belonging to the same semantic category as the query image. Combining information from both non-negative factors, however, yields descriptors that are suitable for finding other images of the specific instance depicted in the query image, where we again achieve state-of-the-art performance.The way our brain learns to disentangle complex signals into unambiguous concepts is fascinating but remains largely unknown. There is evidence, however, that hierarchical neural representations play a key role in the cortex. This thesis investigates biologically plausible models of unsupervised learning of hierarchical representations as found in the brain and modern computer vision models. We use computational modeling to address three main questions at the intersection of artificial intelligence (AI) and computational neuroscience.The first question is: What are useful neural representations and when are deep hierarchical representations needed? We approach this point with a systematic study of biologically plausible unsupervised feature learning in a shallow 2-layer networks on digit (MNIST) and object (CIFAR10) classification. Surprisingly, random features support high performance, especially for large hidden layers. When combined with localized receptive fields, random feature networks approach the performance of supervised backpropagation on MNIST, but not on CIFAR10. We suggest that future models of biologically plausible learning should outperform such random feature benchmarks on MNIST, or that such models should be evaluated in different ways.The second question is: How can hierarchical representations be learned with mechanisms supported by neuroscientific evidence? We cover this question by proposing a unifying Hebbian model, inspired by common models of V1 simple and complex cells based on unsupervised sparse coding and temporal invariance learning. In shallow 2-layer networks, our model reproduces learning of simple and complex cell receptive fields, as found in V1. In deeper networks, we stack multiple layers of Hebbian learning but find that it does not yield hierarchical representations of increasing usefulness. From this, we hypothesise that standard Hebbian rules are too constrained to build increasingly useful representations, as observed in higher areas of the visual cortex or deep artificial neural networks.The third question is: Can AI inspire learning models that build deep representations and are still biologically plausible? We address this question by proposing a learning rule that takes inspiration from neuroscience and recent advances in self-supervised deep learning. The proposed rule is Hebbian, i.e. only depends on pre- and post-synaptic neuronal activity, but includes additional local factors, namely predictive dendritic input and widely broadcasted modulation factors. Algorithmically, this rule applies self-supervised contrastive predictive learning to a causal, biological setting using saccades. We find that networks trained with this generalised Hebbian rule build deep hierarchical representations of images, speech and video.We see our modeling as a potential starting point for both, new hypotheses, that can be tested experimentally, and novel AI models that could benefit from added biological realism.