Deep Learning Generalization, Extrapolation, and Over-parameterization
Yale Center for Medical Informatics（耶鲁医学信息学中心）；VA Connecticut Healthcare System（康涅狄格州退伍军人医疗系统）
Roozbeh Yousefzadeh
备注：
Abstract accepted and presented at the Workshop on Theory of Over-parameterized Machine Learning, April 2021. arXiv admin note: text overlap with arXiv:2101.09849
We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These models have orders of magnitude more parameters than their training samples, and they can achieve perfect accuracy on their training sets, even when training images are randomly labeled, or the contents of images are replaced with random noise. The training loss function of these models has infinite number of near zero minimizers, where only a small subset of those minimizers generalize well. Overall, it is not clear why models need to be over-parameterized, why we should use a very specific training regime to train them, and why their classifications are so susceptible to imperceivable adversarial perturbations (phenomenon known as adversarial vulnerability) \cite{papernot2016limitations,shafahi2018adversarial,tsipras2018robustness}. Some recent studies have made advances in answering these questions, however, they only consider interpolation. We show that interpolation is not adequate to understand generalization of deep networks and we should broaden our perspective.
There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. However, the extrapolation of both of these parameterizations to infinite width is problematic. The standard parameterization leads to a divergent neural tangent kernel while the NTK parameterization fails to capture crucial aspects of finite width networks such as: the dependence of training dynamics on relative layer widths, the relative training dynamics of weights and biases, and ove…
Yale Center for Medical Informatics（耶鲁医学信息学中心）；VA Connecticut Healthcare System（康涅狄格州退伍军人医疗系统）
In this work, we study over-parameterization as a necessary condition for having the ability for the models to extrapolate outside the convex hull of training set. We specifically, consider classification models, e.g., image classification and other applications of deep learning. Such models are classification functions that partition their domain and assign a class to each partition \cite{strang2019linear}. Partitions are defined by decision boundaries and so is the classification model/function. Convex hull of training set may occupy only a subset of the domain, but trained model may partiti…
Constructing accurate and generalizable approximators for complex physico-chemical processes exhibiting highly non-smooth dynamics is challenging. In this work, we propose new developments and perform comparisons for two promising approaches: manifold-based polynomial chaos expansion (m-PCE) and the deep neural operator (DeepONet), and we examine the effect of over-parameterization on generalization. We demonstrate the performance of these methods in terms of generalization accuracy by solving the 2D time-dependent Brusselator reaction-diffusion system with uncertainty sources, modeling an aut…
Universitat Politècnica de Catalunya-Barcelona Tech（加泰罗尼亚理工大学-巴塞罗那理工）；University of Minnesota（明尼苏达大学）
Prior information can be incorporated in matrix completion to improve estimation accuracy and extrapolate the missing entries. Reproducing kernel Hilbert spaces provide tools to leverage the said prior information, and derive more reliable algorithms. This paper analyzes the generalization error of such approaches, and presents numerical tests confirming the theoretical results.
Energy Research Institute, Nanyang Technological University（南洋理工大学能源研究院）
An artificial neural network architecture, parameterization networks, is proposed for simulating extrapolated dynamics beyond observed data in dynamical systems. Parameterization networks are used to ensure the long term integrity of extrapolated dynamics, while careful tuning of model hyperparameters against validation errors controls overfitting. A parameterization network is demonstrated on the logistic map, where chaos and other nonlinear phenomena consistent with the underlying model can be extrapolated from non-chaotic training time series with good fidelity. The stated results are a lot…
A popular assumption for out-of-distribution generalization is that the training data comprises sub-datasets, each drawn from a distinct distribution; the goal is then to "interpolate" these distributions and "extrapolate" beyond them -- this objective is broadly known as domain generalization. A common belief is that ERM can interpolate but not extrapolate and that the latter is considerably more difficult, but these claims are vague and lack formal justification. In this work, we recast generalization over sub-groups as an online game between a player minimizing risk and an adversary present…
Institute of Computational Perception, Johannes Kepler University Linz（林茨约翰内斯·开普勒大学计算感知研究所）；LIT AI Lab（LIT人工智能实验室）
Convolutional Neural Networks (CNNs) have been dominating classification tasks in various domains, such as machine vision, machine listening, and natural language processing. In machine listening, while generally exhibiting very good generalization capabilities, CNNs are sensitive to the specific audio recording device used, which has been recognized as a substantial problem in the acoustic scene classification (DCASE) community. In this study, we investigate the relationship between over-parameterization of acoustic scene classification models, and their resulting generalization abilities. Sp…
Rule Extrapolation in Language Models: A Study of Compositional Generalization on OOD Prompts
University of Cambridge（剑桥大学）；AI Center, UCL（伦敦大学学院人工智能中心）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；ELLIS Institute Tübingen（蒂宾根ELLIS研究所）；Tübingen AI Center（蒂宾根人工智能中心）
Understanding how agents learn to generalize -- and, in particular, to extrapolate -- in high-dimensional, naturalistic environments remains a challenge for both machine learning and the study of biological agents. One approach to this has been the use of function learning paradigms, which allow peoples' empirical patterns of generalization for smooth scalar functions to be described precisely. However, to date, such work has not succeeded in identifying mechanisms that acquire the kinds of general purpose representations over which function learning can operate to exhibit the patterns of gene…
We introduce an approach to training a given compact network. To this end, we leverage over-parameterization, which typically improves both neural network optimization and generalization. Specifically, we propose to expand each linear layer of the compact network into multiple consecutive linear layers, without adding any nonlinearity. As such, the resulting expanded network, or ExpandNet, can be contracted back to the compact one algebraically at inference. In particular, we introduce two convolutional expansion strategies and demonstrate their benefits on several tasks, including image class…
Yale Center for Medical Informatics, Yale University（耶鲁大学耶鲁医学信息学中心）；VA Connecticut Healthcare System（康涅狄格州退伍军人医疗保健系统）；Yale School of Medicine（耶鲁大学医学院）
We study the functional task of deep learning image classification models and show that image classification requires extrapolation capabilities. This suggests that new theories have to be developed for the understanding of deep learning as the current theory assumes models are solely interpolating, leaving many questions about them unanswered. We investigate the pixel space and also the feature spaces extracted from images by trained models (in their hidden layers, including the 64-dimensional feature space in the last hidden layer of pre-trained residual neural networks), and also the featur…
In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimiza…
Centre d’Analyse et de Mathématique Sociales（社会分析与数学中心）；École des Hautes Études en Sciences Sociales（社会科学高等学院）
There has been a long history of works showing that neural networks have hard time extrapolating beyond the training set. A recent study by Balestriero et al. (2021) challenges this view: defining interpolation as the state of belonging to the convex hull of the training set, they show that the test set, either in input or neural space, cannot lie for the most part in this convex hull, due to the high dimensionality of the data, invoking the well known curse of dimensionality. Neural networks are then assumed to necessarily work in extrapolative mode. We here study the neural activities of the…
Mila（米拉研究所）；University of Montreal（蒙特利尔大学）；Vector（Vector研究所）；University of Toronto（多伦多大学）；McGill University（麦吉尔大学）；Facebook AI Research（脸书人工智能研究院）
Distributional shift is one of the major obstacles when transferring machine learning prediction systems from the lab to the real world. To tackle this problem, we assume that variation across training domains is representative of the variation we might encounter at test time, but also that shifts at test time may be more extreme in magnitude. In particular, we show that reducing differences in risk across training domains can reduce a model's sensitivity to a wide range of extreme distributional shifts, including the challenging setting where the input contains both causal and anti-causal ele…
Multi-task learning aims at solving multiple machine learning tasks at the same time. A good solution to a multi-task learning problem should be generalizable in addition to being Pareto optimal. In this paper, we provide some insights on understanding the trade-off between Pareto efficiency and generalization as a result of parameterization in multi-task deep learning models. As a multi-objective optimization problem, enough parameterization is needed for handling task conflicts in a constrained solution space; however, from a multi-task generalization perspective, over-parameterization under…
Deep learning networks are typically trained by Stochastic Gradient Descent (SGD) methods that iteratively improve the model parameters by estimating a gradient on a very small fraction of the training data. A major roadblock faced when increasing the batch size to a substantial fraction of the training data for improving training time is the persistent degradation in performance (generalization gap). To address this issue, recent work propose to add small perturbations to the model parameters when computing the stochastic gradients and report improved generalization performance due to smoothi…
A recent line of research on deep learning focuses on the extremely over-parameterized setting, and shows that when the network width is larger than a high degree polynomial of the training sample size $n$ and the inverse of the target error $ε^{-1}$, deep neural networks learned by (stochastic) gradient descent enjoy nice optimization and generalization guarantees. Very recently, it is shown that under certain margin assumptions on the training data, a polylogarithmic width condition suffices for two-layer ReLU networks to converge and generalize (Ji and Telgarsky, 2019). However, whether dee…
Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks~(RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that arise in its training. Specifically, we parameterize the transition matrix by its singular value decomposition(SVD), which allows us to explicitly track and control its singular values. We attain efficiency by using tools that are common in numerical linear algebra, namely Househ…
We study the generalization of deep learning models in relation to the convex hull of their training sets. A trained image classifier basically partitions its domain via decision boundaries and assigns a class to each of those partitions. The location of decision boundaries inside the convex hull of training set can be investigated in relation to the training samples. However, our analysis shows that in standard image classification datasets, all testing images are considerably outside that convex hull, in the pixel space, in the wavelet space, and in the internal representations learned by de…
Quantifying Hyperparameter Transfer and the Importance of Embedding Layer Learning Rate
Department of Physics, University of Maryland, College Park（马里兰大学物理系）；Department of Computer Science, University of Maryland, College Park（马里兰大学计算机科学系）；Joint Quantum Institute, University of Maryland, College Park（马里兰大学联合量子研究所）；Meta Superintelligence Labs, Fundamental AI Research（Meta超智能实验室，基础人工智能研究）
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 …
University of Cambridge（剑桥大学）；University of Amsterdam（阿姆斯特丹大学）；Universitat Pompeu Fabra（庞培法布拉大学）
Neural networks are surprisingly good at interpolating and perform remarkably well when the training set examples resemble those in the test set. However, they are often unable to extrapolate patterns beyond the seen data, even when the abstractions required for such patterns are simple. In this paper, we first review the notion of extrapolation, why it is important and how one could hope to tackle it. We then focus on a specific type of extrapolation which is especially useful for natural language processing: generalization to sequences that are longer than the training ones. We hypothesize t…
Joint Tensor-Train Parameterization for Efficient and Expressive Low-Rank Adaptation
Georgia Institute of Technology（佐治亚理工学院）；National Taiwan University（国立台湾大学）；University of Palermo（帕尔米奥大学）；NVIDIA Research（NVIDIA研究）；Hon-Hai Quantum Computing Research Center（鸿海量子计算研究中心）
Neural tangent kernel (NTK) is a powerful tool to analyze training dynamics of neural networks and their generalization bounds. The study on NTK has been devoted to typical neural network architectures, but it is incomplete for neural networks with Hadamard products (NNs-Hp), e.g., StyleGAN and polynomial neural networks (PNNs). In this work, we derive the finite-width NTK formulation for a special class of NNs-Hp, i.e., polynomial neural networks. We prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK. Based on our res…
Over-parameterization and adaptive methods have played a crucial role in the success of deep learning in the last decade. The widespread use of over-parameterization has forced us to rethink generalization by bringing forth new phenomena, such as implicit regularization of optimization algorithms and double descent with training progression. A series of recent works have started to shed light on these areas in the quest to understand -- why do neural networks generalize well? The setting of over-parameterized linear regression has provided key insights into understanding this mysterious behavi…
We investigate how the final parameters found by stochastic gradient descent are influenced by over-parameterization. We generate families of models by increasing the number of channels in a base network, and then perform a large hyper-parameter search to study how the test error depends on learning rate, batch size, and network width. We find that the optimal SGD hyper-parameters are determined by a "normalized noise scale," which is a function of the batch size, learning rate, and initialization conditions. In the absence of batch normalization, the optimal normalized noise scale is directly…
Strategic Over-Parameterization for Generalizable Low-Rank Adaptation
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China（1 基础物理与数学科学学院，杭州先进研究院，UCAS，杭州 310024，中国）；School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China（2 物理科学学院，中国科学院大学，玉泉路19A号，北京 100049，中国）；CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China（3 中国科学院理论物理重点实验室，理论物理研究所，中国科学院，北京 100190，中国）；School of Physics and Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, China（4 物理学院和量子态构造与操控（教育部）重点实验室，中国人民大学，北京 100872，中国）
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Boston University（波士顿大学）
Flexible neural sequence models outperform grammar- and automaton-based counterparts on a variety of tasks. However, neural models perform poorly in settings requiring compositional generalization beyond the training data -- particularly to rare or unseen subsequences. Past work has found symbolic scaffolding (e.g. grammars or automata) essential in these settings. We describe R&R, a learned data augmentation scheme that enables a large category of compositional generalizations without appeal to latent symbolic structure. R&R has two components: recombination of original training examples via …
