This dissertation studies a fundamental open challenge in deep learning theory: why do deep networks generalize well even while being overparameterized, unregularized and fitting the training data to zero error?
In the first part of the thesis, we will empirically study how training deep networks via stochastic gradient descent implicitly controls the networks' capacity. Subsequently, to show how this leads to better generalization, we will derive {\em data-dependent} {\em uniform-convergence-based} generalization bounds with improved dependencies on the parameter count.
Uniform convergence has in fact been the most widely used tool in deep learning literature, thanks to its simplicity and generality. Given its popularity, in this thesis, we will also take a step back to identify the fundamental limits of uniform convergence as a tool to explain generalization. In particular, we will show that in some example overparameterized settings, {\em any} uniform convergence bound will provide only a vacuous generalization bound.
With this realization in mind, in the last part of the thesis, we will change course and introduce an {\em empirical} technique to estimate generalization using unlabeled data. Our technique does not rely on any notion of uniform-convergece-based complexity and is remarkably precise. We will theoretically show why our technique enjoys such precision.
We will conclude by discussing how future work could explore novel ways to incorporate distributional assumptions in generalization bounds (such as in the form of unlabeled data) and explore other tools to derive bounds, perhaps by modifying uniform convergence or by developing completely new tools altogether.
Boosting is one of the most successful ideas in machine learning, achieving great practical performance with little fine-tuning. The success of boosted classifiers is most often attributed to improvements in margins. The focus on margin explanations was pioneered in the seminal work by Schapire et al. (1998) and has culminated in the $k$'th margin generalization bound by Gao and Zhou (2013), which was recently proved to be near-tight for some data distributions (Gronlund et al. 2019). In this work, we first demonstrate that the $k$'th margin bound is inadequate in explaining the performance of…
Miami University（迈阿密大学）；University of Alabama（阿拉巴马大学）
Empirical evidence shows that ensembles, such as bagging, boosting, random and rotation forests, generally perform better in terms of their generalization error than individual classifiers. To explain this performance, Schapire et al. (1998) developed an upper bound on the generalization error of an ensemble based on the margins of the training data, from which it was concluded that larger margins should lead to lower generalization error, everything else being equal. Many other researchers have backed this assumption and presented tighter bounds on the generalization error based on either the…
Miami University（迈阿密大学）；University of Alabama（阿拉巴马大学）
Boosting and other ensemble methods combine a large number of weak classifiers through weighted voting to produce stronger predictive models. To explain the successful performance of boosting algorithms, Schapire et al. (1998) showed that AdaBoost is especially effective at increasing the margins of the training data. Schapire et al. (1998) also developed an upper bound on the generalization error of any ensemble based on the margins of the training data, from which it was concluded that larger margins should lead to lower generalization error, everything else being equal (sometimes referred t…
Element AI（元素AI）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
One of the principal scientific challenges in deep learning is explaining generalization, i.e., why the particular way the community now trains networks to achieve small training error also leads to small error on held-out data from the same population. It is widely appreciated that some worst-case theories -- such as those based on the VC dimension of the class of predictors induced by modern neural network architectures -- are unable to explain empirical performance. A large volume of work aims to close this gap, primarily by developing bounds on generalization error, optimization error, and…
Coherent Gradients (CGH) is a recently proposed hypothesis to explain why over-parameterized neural networks trained with gradient descent generalize well even though they have sufficient capacity to memorize the training set. The key insight of CGH is that, since the overall gradient for a single step of SGD is the sum of the per-example gradients, it is strongest in directions that reduce the loss on multiple examples if such directions exist. In this paper, we validate CGH on ResNet, Inception, and VGG models on ImageNet. Since the techniques presented in the original paper do not scale bey…
Tsinghua University（清华大学）；University of Michigan（密歇根大学）；Shanghai Qi Zhi Institute（上海期智研究院）
Generalization is one of the fundamental issues in machine learning. However, traditional techniques like uniform convergence may be unable to explain generalization under overparameterization. As alternative approaches, techniques based on stability analyze the training dynamics and derive algorithm-dependent generalization bounds. Unfortunately, the stability-based bounds are still far from explaining the surprising generalization in deep learning since neural networks usually suffer from unsatisfactory stability. This paper proposes a novel decomposition framework to improve the stability-b…
Institute of Electrical Engineering, EPFL（洛桑联邦理工学院电气工程研究所）
Explaining how overparametrized neural networks simultaneously achieve low risk and zero empirical risk on benchmark datasets is an open problem. PAC-Bayes bounds optimized using variational inference (VI) have been recently proposed as a promising direction in obtaining non-vacuous bounds. We show empirically that this approach gives negligible gains when modeling the posterior as a Gaussian with diagonal covariance--known as the mean-field approximation. We investigate common explanations, such as the failure of VI due to problems in optimization or choosing a suboptimal prior. Our results s…
Stochastic gradient descent with a large initial learning rate is widely used for training modern neural net architectures. Although a small initial learning rate allows for faster training and better test performance initially, the large learning rate achieves better generalization soon after the learning rate is annealed. Towards explaining this phenomenon, we devise a setting in which we can prove that a two layer network trained with large initial learning rate and annealing provably generalizes better than the same network trained with a small learning rate from the start. The key insight…
While there has been progress in developing non-vacuous generalization bounds for deep neural networks, these bounds tend to be uninformative about why deep learning works. In this paper, we develop a compression approach based on quantizing neural network parameters in a linear subspace, profoundly improving on previous results to provide state-of-the-art generalization bounds on a variety of tasks, including transfer learning. We use these tight bounds to better understand the role of model size, equivariance, and the implicit biases of optimization, for generalization in deep learning. Nota…
University of Science and Technology of China（中国科学技术大学）；Tencent America（腾讯美国）
Despite the recent progress in Graph Neural Networks (GNNs), it remains challenging to explain the predictions made by GNNs. Existing explanation methods mainly focus on post-hoc explanations where another explanatory model is employed to provide explanations for a trained GNN. The fact that post-hoc methods fail to reveal the original reasoning process of GNNs raises the need of building GNNs with built-in interpretability. In this work, we propose Prototype Graph Neural Network (ProtGNN), which combines prototype learning with GNNs and provides a new perspective on the explanations of GNNs. …
The Pennsylvania State University（宾夕法尼亚州立大学）；NEC Labs America（NEC美国实验室）
Despite recent progress in Graph Neural Networks (GNNs), explaining predictions made by GNNs remains a challenging open problem. The leading method independently addresses the local explanations (i.e., important subgraph structure and node features) to interpret why a GNN model makes the prediction for a single instance, e.g. a node or a graph. As a result, the explanation generated is painstakingly customized for each instance. The unique explanation interpreting each instance independently is not sufficient to provide a global understanding of the learned GNN model, leading to a lack of gene…
IBM Quantum, IBM Research – Zurich（IBM量子计算，IBM苏黎世研究院）；University of KwaZulu-Natal, Durban（夸祖鲁-纳塔尔大学，德班）；Department of Mathematics, ETH Zurich（苏黎世联邦理工学院数学系）
Making statements about the performance of trained models on tasks involving new data is one of the primary goals of machine learning, i.e., to understand the generalization power of a model. Various capacity measures try to capture this ability, but usually fall short in explaining important characteristics of models that we observe in practice. In this study, we propose the local effective dimension as a capacity measure which seems to correlate well with generalization error on standard data sets. Importantly, we prove that the local effective dimension bounds the generalization error and d…
Given that Neural Networks generalize unreasonably well in the IID setting (with benign overfitting and betterment in performance with more parameters), OOD presents a consistent failure case to better the understanding of how they learn. This paper focuses on Domain Generalization (DG), which is perceived as the front face of OOD generalization. We find that the presence of multiple domains incentivizes domain agnostic learning and is the primary reason for generalization in Tradition DG. We show that the state-of-the-art results can be obtained by borrowing ideas from IID generalization and …
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of …
Carnegie Mellon University（卡内基梅隆大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
Aimed at explaining the surprisingly good generalization behavior of overparameterized deep networks, recent works have developed a variety of generalization bounds for deep learning, all based on the fundamental learning-theoretic technique of uniform convergence. While it is well-known that many of these existing bounds are numerically large, through numerous experiments, we bring to light a more concerning aspect of these bounds: in practice, these bounds can {\em increase} with the training dataset size. Guided by our observations, we then present examples of overparameterized linear class…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing …
Towards Mitigation of Hallucination for LLM-empowered Agents: Progressive Generalization Bound Exploration and Watchdog Monitor
College of Computer Science, Nankai University（南开大学计算机科学学院）；Haihe Lab of ITAI College of intelligent Science and Technology, Inner Mongolia University of Technology（内蒙古工业大学智能科学与技术学院海河实验室）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；Department of Electrical and Computer Engineering, Stony Brook University（石溪大学电气与计算机工程系）；Department of Computer Science, Michigan Technological University（密歇根技术大学计算机科学系）
Optimization dependent generalization bound for ReLU networks based on sensitivity in the tangent bundle
Institute for Computer Science and Control (SZTAKI)（计算机科学与控制研究所（SZTAKI））；Eötvös Loránd University（厄特沃什·罗兰大学）；CNRS CRIStAL（法国国家科学研究中心信号与自动化信息处理实验室）；Université de Lille（里尔大学）；Széchenyi István University（塞切尼·伊什特万大学）
The ``Neural Tangent Kernel'' (NTK) (Jacot et al 2018), and its empirical variants have been proposed as a proxy to capture certain behaviors of real neural networks. In this work, we study NTKs through the lens of scaling laws, and demonstrate that they fall short of explaining important aspects of neural network generalization. In particular, we demonstrate realistic settings where finite-width neural networks have significantly better data scaling exponents as compared to their corresponding empirical and infinite NTKs at initialization. This reveals a more fundamental difference between th…
Explaining the surprising generalization performance of deep neural networks is an active and important line of research in theoretical machine learning. Influential work by Arora et al. (ICML'18) showed that, noise stability properties of deep nets occurring in practice can be used to provably compress model representations. They then argued that the small representations of compressed networks imply good generalization performance albeit only of the compressed nets. Extending their compression framework to yield generalization bounds for the original uncompressed networks remains elusive. Ou…
Bounding the generalization error of learning algorithms has a long history, which yet falls short in explaining various generalization successes including those of deep learning. Two important difficulties are (i) exploiting the dependencies between the hypotheses, (ii) exploiting the dependence between the algorithm's input and output. Progress on the first point was made with the chaining method, originating from the work of Kolmogorov, and used in the VC-dimension bound. More recently, progress on the second point was made with the mutual information method by Russo and Zou '15. Yet, these…
Modern machine learning systems based on neural networks have shown great success in learning complex data patterns while being able to make good predictions on unseen data points. However, the limited interpretability of these systems hinders further progress and application to several domains in the real world. This predicament is exemplified by time consuming model selection and the difficulties faced in predictive explainability, especially in the presence of adversarial examples. In this paper, we take a step towards better understanding of neural networks by introducing a local polytope …
Understanding Generalization from Embedding Dimension and Distributional Convergence
Department of Biomedical Engineering, Southern University of Science and Technology（生物医学工程系，南方科技大学）；Department of Electronic and Electrical Engineering, Southern University of Science and Technology（电子与电气工程系，南方科技大学）
University of Cambridge（剑桥大学）；University of Toronto（多伦多大学）
One of the defining properties of deep learning is that models are chosen to have many more parameters than available training data. In light of this capacity for overfitting, it is remarkable that simple algorithms like SGD reliably return solutions with low test error. One roadblock to explaining these phenomena in terms of implicit regularization, structural properties of the solution, and/or easiness of the data is that many learning bounds are quantitatively vacuous when applied to networks learned by SGD in this "deep learning" regime. Logically, in order to explain generalization, we ne…
CSAIL, MIT（麻省理工学院计算机科学与人工智能实验室）；GoogleX（谷歌X实验室）；Center for Brains, Minds and Machines, MIT（麻省理工学院大脑、心智与机器中心）；IIT（意大利理工学院）；Claremont Graduate University（克莱蒙特研究生大学）；Alphabet (Google) X（Alphabet（谷歌）X实验室）
A main puzzle of deep networks revolves around the absence of overfitting despite large overparametrization and despite the large capacity demonstrated by zero training error on randomly labeled data. In this note, we show that the dynamics associated to gradient descent minimization of nonlinear networks is topologically equivalent, near the asymptotically stable minima of the empirical error, to linear gradient system in a quadratic potential with a degenerate (for square loss) or almost degenerate (for logistic or crossentropy loss) Hessian. The proposition depends on the qualitative theory…
University of New Brunswick（新不伦瑞克大学）；University of Toronto（多伦多大学）
Explaining the generalization characteristics of deep learning is an emerging topic in advanced machine learning. There are several unanswered questions about how learning under stochastic optimization really works and why certain strategies are better than others. In this paper, we address the following question: \textit{can we probe intermediate layers of a deep neural network to identify and quantify the learning quality of each layer?} With this question in mind, we propose new explainability metrics that measure the redundant information in a network's layers using a low-rank factorizatio…
Duke University（杜克大学）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
Mode connectivity is a surprising phenomenon in the loss landscape of deep nets. Optima -- at least those discovered by gradient-based optimization -- turn out to be connected by simple paths on which the loss function is almost constant. Often, these paths can be chosen to be piece-wise linear, with as few as two segments. We give mathematical explanations for this phenomenon, assuming generic properties (such as dropout stability and noise stability) of well-trained deep nets, which have previously been identified as part of understanding the generalization properties of deep nets. Our expla…
Deep neural networks perform exceptionally well on various learning tasks with state-of-the-art results. While these models are highly expressive and achieve impressively accurate solutions with excellent generalization abilities, they are susceptible to minor perturbations. Samples that suffer such perturbations are known as "adversarial examples". Even though deep learning is an extensively researched field, many questions about the nature of deep learning models remain unanswered. In this paper, we introduce a new conceptual framework attached with a formal description that aims to shed lig…
Jožef Stefan Institute（约瑟夫·斯特凡研究所）；Jožef Stefan International Postgraduate School（约瑟夫·斯特凡国际研究生院）
The competition "Predicting Generalization in Deep Learning (PGDL)" aims to provide a platform for rigorous study of generalization of deep learning models and offer insight into the progress of understanding and explaining these models. This report presents the solution that was submitted by the user \emph{smeznar} which achieved the eight place in the competition. In the proposed approach, we create simple metrics and find their best combination with automatic testing on the provided dataset, exploring how combinations of various properties of the input neural network architectures can be us…
Microsoft Research（微软研究院）；Cornell University（康奈尔大学）
Large language models (LLMs) have displayed an impressive ability to harness natural language to perform complex tasks. In this work, we explore whether we can leverage this learned ability to find and explain patterns in data. Specifically, given a pre-trained LLM and data examples, we introduce interpretable autoprompting (iPrompt), an algorithm that generates a natural-language string explaining the data. iPrompt iteratively alternates between generating explanations with an LLM and reranking them based on their performance when used as a prompt. Experiments on a wide range of datasets, fro…
We consider unsupervised domain adaptation (UDA), where labeled data from a source domain (e.g., photographs) and unlabeled data from a target domain (e.g., sketches) are used to learn a classifier for the target domain. Conventional UDA methods (e.g., domain adversarial training) learn domain-invariant features to improve generalization to the target domain. In this paper, we show that contrastive pre-training, which learns features on unlabeled source and target data and then fine-tunes on labeled source data, is competitive with strong UDA methods. However, we find that contrastive pre-trai…
While recent progress has spawned very powerful machine learning systems, those agents remain extremely specialized and fail to transfer the knowledge they gain to similar yet unseen tasks. In this paper, we study a simple reinforcement learning problem and focus on learning policies that encode the proper invariances for generalization to different settings. We evaluate three potential methods for policy generalization: data augmentation, meta-learning and adversarial training. We find our data augmentation method to be effective, and study the potential of meta-learning and adversarial learn…
The key distinguishing property of a Bayesian approach is marginalization, rather than using a single setting of weights. Bayesian marginalization can particularly improve the accuracy and calibration of modern deep neural networks, which are typically underspecified by the data, and can represent many compelling but different solutions. We show that deep ensembles provide an effective mechanism for approximate Bayesian marginalization, and propose a related approach that further improves the predictive distribution by marginalizing within basins of attraction, without significant overhead. We…
Understanding how large neural networks avoid memorizing training data is key to explaining their high generalization performance. To examine the structure of when and where memorization occurs in a deep network, we use a recently developed replica-based mean field theoretic geometric analysis method. We find that all layers preferentially learn from examples which share features, and link this behavior to generalization performance. Memorization predominately occurs in the deeper layers, due to decreasing object manifolds' radius and dimension, whereas early layers are minimally affected. Thi…
In attempts to produce ML models less reliant on spurious patterns in NLP datasets, researchers have recently proposed curating counterfactually augmented data (CAD) via a human-in-the-loop process in which given some documents and their (initial) labels, humans must revise the text to make a counterfactual label applicable. Importantly, edits that are not necessary to flip the applicable label are prohibited. Models trained on the augmented data appear, empirically, to rely less on semantically irrelevant words and to generalize better out of domain. While this work draws loosely on causal th…
Noisy labels are very common in deep supervised learning. Although many studies tend to improve the robustness of deep training for noisy labels, rare works focus on theoretically explaining the training behaviors of learning with noisily labeled data, which is a fundamental principle in understanding its generalization. In this draft, we study its two phenomena, clean data first and phase transition, by explaining them from a theoretical viewpoint. Specifically, we first show that in the first epoch training, the examples with clean labels will be learned first. We then show that after the le…
In recent years, neural networks (NNs) have made giant leaps in a wide variety of domains. NNs are often referred to as black box algorithms due to how little we can explain their empirical success. Our foundational research seeks to explain why neural networks generalize. A recent advancement derived a mutual information measure for explaining the performance of deep NNs through a sequence of increasingly complex functions. We show deep NNs learn a series of boosted classifiers whose generalization is popularly attributed to self-averaging over an increasing number of interpolating sub-classi…
Princeton University（普林斯顿大学）；Microsoft Research India（微软印度研究院）
Simple recurrent neural networks (RNNs) and their more advanced cousins LSTMs etc. have been very successful in sequence modeling. Their theoretical understanding, however, is lacking and has not kept pace with the progress for feedforward networks, where a reasonably complete understanding in the special case of highly overparametrized one-hidden-layer networks has emerged. In this paper, we make progress towards remedying this situation by proving that RNNs can learn functions of sequences. In contrast to the previous work that could only deal with functions of sequences that are sums of fun…
