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Publication# Subquadratic Overparameterization for Shallow Neural Networks

Résumé

Overparameterization refers to the important phenomenon where the width of a neural network is chosen such that learning algorithms can provably attain zero loss in nonconvex training. The existing theory establishes such global convergence using various initialization strategies, training modifications, and width scalings. In particular, the state-of-the-art results require the width to scale quadratically with the number of training data under standard initialization strategies used in practice for best generalization performance. In contrast, the most recent results obtain linear scaling either with requiring initializations that lead to the “lazy-training”, or training only a single layer. In this work, we provide an analytical framework that allows us to adopt standard initialization strategies, possibly avoid lazy training, and train all layers simultaneously in basic shallow neural networks while attaining a desirable subquadratic scaling on the network width. We achieve the desiderata via Polyak-Łojasiewicz condition, smoothness, and standard assumptions on data, and use tools from random matrix theory.

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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.

In this thesis, we explore the use of machine learning techniques for information retrieval. More specifically, we focus on ad-hoc retrieval, which is concerned with searching large corpora to identify the documents relevant to user queries. This identification is performed through a ranking task. Given a user query, an ad-hoc retrieval system ranks the corpus documents, so that the documents relevant to the query ideally appear above the others. In a machine learning framework, we are interested in proposing learning algorithms that can benefit from limited training data in order to identify a ranker likely to achieve high retrieval performance over unseen documents and queries. This problem presents novel challenges compared to traditional learning tasks, such as regression or classification. First, our task is a ranking problem, which means that the loss for a given query cannot be measured as a sum of an individual loss suffered for each corpus document. Second, most retrieval queries present a highly unbalanced setup, with a set of relevant documents accounting only for a very small fraction of the corpus. Third, ad-hoc retrieval corresponds to a kind of "double" generalization problem, since the learned model should not only generalize to new documents but also to new queries. Finally, our task also presents challenging efficiency constraints, since ad-hoc retrieval is typically applied to large corpora. The main objective of this thesis is to investigate the discriminative learning of ad-hoc retrieval models. For that purpose, we propose different models based on kernel machines or neural networks adapted to different retrieval contexts. The proposed approaches rely on different online learning algorithms that allow efficient learning over large corpora. The first part of the thesis focuses on text retrieval. In this case, we adopt a classical approach to the retrieval ranking problem, and order the text documents according to their estimated similarity to the text query. The assessment of semantic similarity between text items plays a key role in that setup and we propose a learning approach to identify an effective measure of text similarity. This identification is not performed relying on a set of queries with their corresponding relevant document sets, since such data are especially expensive to label and hence rare. Instead, we propose to rely on hyperlink data, since hyperlinks convey semantic proximity information that is relevant to similarity learning. This setup is hence a transfer learning setup, where we benefit from the proximity information encoded by hyperlinks to improve the performance over the ad-hoc retrieval task. We then investigate another retrieval problem, i.e. the retrieval of images from text queries. Our approach introduces a learning procedure optimizing a criterion related to the ranking performance. This criterion adapts our previous learning objective for learning textual similarity to the image retrieval problem. This yields an image ranking model that addresses the retrieval problem directly. This approach contrasts with previous research that relies on an intermediate image annotation task. Moreover, our learning procedure builds upon recent work on the online learning of kernel-based classifiers. This yields an efficient, scalable algorithm, which can benefit from recent kernels developed for image comparison. In the last part of the thesis, we show that the objective function used in the previous retrieval problems can be applied to the task of keyword spotting, i.e. the detection of given keywords in speech utterances. For that purpose, we formalize this problem as a ranking task: given a keyword, the keyword spotter should order the utterances so that the utterances containing the keyword appear above the others. Interestingly, this formulation yields an objective directly maximizing the area under the receiver operating curve, the most common keyword spotter evaluation measure. This objective is then used to train a model adapted to this intrinsically sequential problem. This model is then learned with a procedure derived from the algorithm previously introduced for the image retrieval task. To conclude, this thesis introduces machine learning approaches for ad-hoc retrieval. We propose learning models for various multi-modal retrieval setups, i.e. the retrieval of text documents from text queries, the retrieval of images from text queries and the retrieval of speech recordings from written keywords. Our approaches rely on discriminative learning and enjoy efficient training procedures, which yields effective and scalable models. In all cases, links with prior approaches were investigated and experimental comparisons were conducted.

One of the main goal of Artificial Intelligence is to develop models capable of providing valuable predictions in real-world environments. In particular, Machine Learning (ML) seeks to design such models by learning from examples coming from this same environment. However, the real world is most of the time not static, and the environment in which the model will be used can differ from the one in which it is trained. It is hence desirable to design models that are robust to changes of environments. This encapsulates a large family of topics in ML, such as adversarial robustness, meta-learning, domain adaptation and others, depending on the way the environment is perturbed.In this dissertation, we focus on methods for training models whose performance does not drastically degrade when applied to environments differing from the one the model has been trained in. Various types of environmental changes will be treated, differing in their structure or magnitude. Each setup defines a certain kind of robustness to certain environmental changes, and leads to a certain optimization problem to be solved. We consider 3 different setups, and propose algorithms for solving each associated problem using 3 different types of methods, namely, min-max optimization (Chapter 2), regularization (Chapter 3) and variable selection (Chapter 4).Leveraging the framework of distributionally robust optimization, which phrases the problem of robust training as a min-max optimization problem, we first aim to train robust models by directly solving the associated min-max problem. This is done by exploiting recent work on game theory as well as first-order sampling algorithms based on Langevin dynamics. Using this approach, we propose a method for training robust agents in the scope of Reinforcement Learning.We then treat the case of adversarial robustness, i.e., robustness to small arbitrary perturbation of the model's input. It is known that neural networks trained using classical optimization methods are particularly sensitive to this type of perturbations. The adversarial robustness of a model is tightly connected to its smoothness, which is quantified by its so-called Lipschitz constant. This constant measures how much the model's output changes upon any bounded input perturbation. We hence develop a method to estimate an upper bound on the Lipschitz constant of neural networks via polynomial optimization, which can serve as a robustness certificate against adversarial attacks. We then propose to penalize the Lipschitz constant during training by minimizing the 1-path-norm of the neural network, and we develop an algorithm for solving the resulting regularized problem by efficiently computing the proximal operator of the 1-path-norm term, which is non-smooth and non-convex.Finally, we consider a scenario where the environmental changes can be arbitrary large (as opposed to adversarial robustness), but need to preserve a certain causal structure. Recent works have demonstrated interesting connections between robustness and the use of causal variables. Assuming that certain mechanisms remain invariant under some change of the environment, it has been shown that knowing the underlying causal structure of the data at hand allows to train models that are invariant to such changes. Unfortunately, in many cases, the causal structure is unknown. We thus propose a causal discovery algorithm from observational data in the case of non-linear additive model.