Deep Neural Networks can generalize despite being significantly overparametrized. Recent research has tried to examine this phenomenon from various view points and to provide bounds on the generalization error or measures predictive of the generalization gap based on these viewpoints, such as norm-based, PAC-Bayes based, and margin-based analysis. In this work, we provide an interpretation of generalization from the perspective of quality of internal representations of deep neural networks, based on neuroscientific theories of how the human visual system creates invariant and untangled object representations. Instead of providing theoretical bounds, we demonstrate practical complexity measures which can be computed ad-hoc to uncover generalization behaviour in deep models. We also provide a detailed description of our solution that won the NeurIPS competition on Predicting Generalization in Deep Learning held at NeurIPS 2020. An implementation of our solution is available at https://github.com/parthnatekar/pgdl.
Google Research（谷歌研究院）；University of Toronto（多伦多大学）；Element AI；Microsoft Research（微软研究院）；University Paris-Saclay（巴黎-萨克雷大学）；ChaLearn
Understanding generalization in deep learning is arguably one of the most important questions in deep learning. Deep learning has been successfully adopted to a large number of problems ranging from pattern recognition to complex decision making, but many recent researchers have raised many concerns about deep learning, among which the most important is generalization. Despite numerous attempts, conventional statistical learning approaches have yet been able to provide a satisfactory explanation on why deep learning works. A recent line of works aims to address the problem by trying to predict…
Generalization is the ability of a model to predict on unseen domains and is a fundamental task in machine learning. Several generalization bounds, both theoretical and empirical have been proposed but they do not provide tight bounds .In this work, we propose a simple yet effective method to predict the generalization performance of a model by using the concept that models that are robust to augmentations are more generalizable than those which are not. We experiment with several augmentations and composition of augmentations to check the generalization capacity of a model. We also provide a …
This paper aims at studying the sample complexity of graph convolutional networks (GCNs), by providing tight upper bounds of Rademacher complexity for GCN models with a single hidden layer. Under regularity conditions, theses derived complexity bounds explicitly depend on the largest eigenvalue of graph convolution filter and the degree distribution of the graph. Again, we provide a lower bound of Rademacher complexity for GCNs to show optimality of our derived upper bounds. Taking two commonly used examples as representatives, we discuss the implications of our results in designing graph conv…
We derive generalization error bounds for stationary univariate autoregressive (AR) models. We show that imposing stationarity is enough to control the Gaussian complexity without further regularization. This lets us use structural risk minimization for model selection. We demonstrate our methods by predicting interest rate movements.
Leveraging PAC-Bayes Theory and Gibbs Distributions for Generalization Bounds with Complexity Measures
Univ Rennes（雷恩大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；IRISA（雷恩信息与自动化研究所）；Université Jean Monnet Saint-Etienne（圣艾蒂安让·莫奈大学）；Institut d Optique Graduate School（光学研究所研究生院）；Laboratoire Hubert Curien（于贝尔·居里实验室）；Institut Universitaire de France (IUF)（法国大学研究院）；ServiceNow Research（ServiceNow研究院）
Generalization Guarantees via Algorithm-dependent Rademacher Complexity
University of Amsterdam（阿姆斯特丹大学）；University of Melbourne（墨尔本大学）；Purdue University（普渡大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
JD Explore Academy, JD.com Inc.（京东探索研究院）；Faculty of Science, University of Hong Kong（香港大学理学院）
Complex-valued neural networks (CVNNs) have been widely applied to various fields, especially signal processing and image recognition. However, few works focus on the generalization of CVNNs, albeit it is vital to ensure the performance of CVNNs on unseen data. This paper is the first work that proves a generalization bound for the complex-valued neural network. The bound scales with the spectral complexity, the dominant factor of which is the spectral norm product of weight matrices. Further, our work provides a generalization bound for CVNNs when training data is sequential, which is also af…
Deep convolutional neural networks (CNNs) have been shown to be able to fit a random labeling over data while still being able to generalize well for normal labels. Describing CNN capacity through a posteriori measures of complexity has been recently proposed to tackle this apparent paradox. These complexity measures are usually validated by showing that they correlate empirically with GE; being empirically larger for networks trained on random vs normal labels. Focusing on the case of spectral complexity we investigate theoretically and empirically the insensitivity of the complexity measure …
Institute for Advanced Study（高等研究院）；Princeton University（普林斯顿大学）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；New York University（纽约大学）；Facebook AI Research & Courant Institute（脸书人工智能研究和库朗研究所）
Despite existing work on ensuring generalization of neural networks in terms of scale sensitive complexity measures, such as norms, margin and sharpness, these complexity measures do not offer an explanation of why neural networks generalize better with over-parametrization. In this work we suggest a novel complexity measure based on unit-wise capacities resulting in a tighter generalization bound for two layer ReLU networks. Our capacity bound correlates with the behavior of test error with increasing network sizes, and could potentially explain the improvement in generalization with over-par…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
Recent papers have shown that sufficiently overparameterized neural networks can perfectly fit even random labels. Thus, it is crucial to understand the underlying reason behind the generalization performance of a network on real-world data. In this work, we propose several measures to predict the generalization error of a network given the training data and its parameters. Using one of these measures, based on noise resilience of the network, we secured 5th position in the predicting generalization in deep learning (PGDL) competition at NeurIPS 2020.
University College London（伦敦大学学院）；Tel-Aviv University（特拉维夫大学）；Duke University（杜克大学）
This paper studies the generalization error of invariant classifiers. In particular, we consider the common scenario where the classification task is invariant to certain transformations of the input, and that the classifier is constructed (or learned) to be invariant to these transformations. Our approach relies on factoring the input space into a product of a base space and a set of transformations. We show that whereas the generalization error of a non-invariant classifier is proportional to the complexity of the input space, the generalization error of an invariant classifier is proportion…
Stability, Complexity and Data-Dependent Worst-Case Generalization Bounds
INRIA, CNRS, Département d’Informatique de l’Ecole Normale Supérieure / PSL, France（法国国家信息与自动化技术研究所、国家科学研究中心、巴黎高等师范学院计算机系/巴黎社会科学实验室）；Department of Computing, Imperial College London, United Kingdom（伦敦帝国理工学院计算机系、英国）；School of Computation and Technology, Technical University of Munich, Germany（慕尼黑技术大学计算与技术学院、德国）；Munich Center for Machine Learning, Germany（慕尼黑机器学习中心、德国）
Massachusetts Institute of Technology（麻省理工学院）；Spark Wave（火花波公司）；Google Research（谷歌研究院）；University of Southern California（南加利福尼亚大学）；Courant Institute（库朗研究所）；Cornell University（康奈尔大学）
We present a study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main result is a generalization bound for data-dependent hypothesis sets expressed in terms of a notion of hypothesis set stability and a notion of Rademacher complexity for data-dependent hypothesis sets that we introduce. This bound admits as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds. We also illustrate the use of these learni…
We present a unifying picture of PAC-Bayesian and mutual information-based upper bounds on the generalization error of randomized learning algorithms. As we show, Tong Zhang's information exponential inequality (IEI) gives a general recipe for constructing bounds of both flavors. We show that several important results in the literature can be obtained as simple corollaries of the IEI under different assumptions on the loss function. Moreover, we obtain new bounds for data-dependent priors and unbounded loss functions. Optimizing the bounds gives rise to variants of the Gibbs algorithm, for whi…
Providing users with alternatives to choose from is an essential component in many online platforms, making the accurate prediction of choice vital to their success. A renewed interest in learning choice models has led to significant progress in modeling power, but most current methods are either limited in the types of choice behavior they capture, cannot be applied to large-scale data, or both. Here we propose a learning framework for predicting choice that is accurate, versatile, theoretically grounded, and scales well. Our key modeling point is that to account for how humans choose, predic…
While many classical notions of learnability (e.g., PAC learnability) are distribution-free, utilizing the specific structures of an input distribution may improve learning performance. For example, a product distribution on a multi-dimensional input space has a much simpler structure than a correlated distribution. A recent paper [GHTZ21] shows that the sample complexity of a general learning problem on product distributions is polynomial in the input dimension, which is exponentially smaller than that on correlated distributions. However, the learning algorithm they use is not the standard E…
Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral
RIKEN AIP（日本理化学研究所AIP）；CyberAgent Inc.（CyberAgent公司）；OMRON SINIC X Corporation（OMRON SINIC X株式会社）；University College Cork（科克大学）；The University of Tokyo（东京大学）
KTH（瑞典皇家理工学院）；Science for Life Laboratory（生命科学实验室）
In this preregistration submission, we propose an empirical study of how networks handle changes in complexity of the data. We investigate the effect of network capacity on generalization performance in the face of increasing data complexity. For this, we measure the generalization error for an image classification task where the number of classes steadily increases. We compare a number of modern architectures at different scales in this setting. The methodology, setup, and hypotheses described in this proposal were evaluated by peer review before experiments were conducted.
We introduce a new notion of complexity of functions and we show that it has the following properties: (i) it governs a PAC Bayes-like generalization bound, (ii) for neural networks it relates to natural notions of complexity of functions (such as the variation), and (iii) it explains the generalization gap between neural networks and linear schemes. While there is a large set of papers which describes bounds that have each such property in isolation, and even some that have two, as far as we know, this is a first notion that satisfies all three of them. Moreover, in contrast to previous works…
Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data, there has been a growing literature on learning and generalization in settings where data is sampled from an ergodic process. This work has also developed complexity measures, which appropriately extend the notion of Rademacher complexity to bound the generalization error and learning rates of hypothesis classes in this setting. Rather than time-series data, our work is motivated by settings where da…
Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms
School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；INRIA（法国国家信息与自动化研究所）；Imperial College London（伦敦帝国学院）；CNRS, Ecole Normale Supérieure, PSL Research University（法国国家科学研究中心、巴黎高等师范学院、巴黎文理研究大学）
Lancaster University（兰卡斯特大学）；New York University（纽约大学）；Google Research（谷歌研究院）
This paper studies the problem of learning weighted automata from a finite labeled training sample. We consider several general families of weighted automata defined in terms of three different measures: the norm of an automaton's weights, the norm of the function computed by an automaton, or the norm of the corresponding Hankel matrix. We present new data-dependent generalization guarantees for learning weighted automata expressed in terms of the Rademacher complexity of these families. We further present upper bounds on these Rademacher complexities, which reveal key new data-dependent terms…
While standard statistical inference techniques and machine learning generalization bounds assume that tests are run on data selected independently of the hypotheses, practical data analysis and machine learning are usually iterative and adaptive processes where the same holdout data is often used for testing a sequence of hypotheses (or models), which may each depend on the outcome of the previous tests on the same data. In this work, we present RadaBound a rigorous, efficient and practical procedure for controlling the generalization error when using a holdout sample for multiple adaptive te…
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…
University of California, San Francisco（加州大学旧金山分校）；University of California, Santa Barbara（加州大学圣巴巴拉分校）
We study generalization in deep learning by appealing to complexity measures originally developed in approximation and information theory. While these concepts are challenged by the high-dimensional and data-defined nature of deep learning, we show that simple vector quantization approaches such as PCA, GMMs, and SVMs capture their spirit when applied layer-wise to deep extracted features giving rise to relatively inexpensive complexity measures that correlate well with generalization performance. We discuss our results in 2020 NeurIPS PGDL challenge.
Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
Research on generalization bounds for deep networks seeks to give ways to predict test error using just the training dataset and the network parameters. While generalization bounds can give many insights about architecture design, training algorithms, etc., what they do not currently do is yield good predictions for actual test error. A recently introduced Predicting Generalization in Deep Learning competition~\citep{jiang2020neurips} aims to encourage discovery of methods to better predict test error. The current paper investigates a simple idea: can test error be predicted using {\em synthet…
Insight Centre for Data Analytics（洞察数据分析中心）；University College Dublin（都柏林大学学院）；University College Cork（科克大学学院）
Smoothed analysis is a framework for analyzing the complexity of an algorithm, acting as a bridge between average and worst-case behaviour. For example, Quicksort and the Simplex algorithm are widely used in practical applications, despite their heavy worst-case complexity. Smoothed complexity aims to better characterize such algorithms. Existing theoretical bounds for the smoothed complexity of sorting algorithms are still quite weak. Furthermore, empirically computing the smoothed complexity via its original definition is computationally infeasible, even for modest input sizes. In this paper…
The field of Deep Learning is rich with empirical evidence of human-like performance on a variety of prediction tasks. However, despite these successes, the recent Predicting Generalization in Deep Learning (PGDL) NeurIPS 2020 competition suggests that there is a need for more robust and efficient measures of network generalization. In this work, we propose a new framework for evaluating the generalization capabilities of trained networks. We use perturbation response (PR) curves that capture the accuracy change of a given network as a function of varying levels of training sample perturbation…
The estimation of the generalization error of classifiers often relies on a validation set. Such a set is hardly available in few-shot learning scenarios, a highly disregarded shortcoming in the field. In these scenarios, it is common to rely on features extracted from pre-trained neural networks combined with distance-based classifiers such as nearest class mean. In this work, we introduce a Gaussian model of the feature distribution. By estimating the parameters of this model, we are able to predict the generalization error on new classification tasks with few samples. We observe that accura…
Developing a robust generalization measure for the performance of machine learning models is an important and challenging task. A lot of recent research in the area focuses on the model decision boundary when predicting generalization. In this paper, as part of the "Predicting Generalization in Deep Learning" competition, we analyse the complexity of decision boundaries using the concept of fractal dimension and develop a generalization measure based on that technique.
Predicting Critical Heat Flux with Uncertainty Quantification and Domain Generalization Using Conditional Variational Autoencoders and Deep Neural Networks
IMT Atlantique（大西洋高等电信学校）；Lab-STICC（布列塔尼电子与电信通信科学实验室）；MILA, Université de Montréal（蒙特利尔大学MILA人工智能研究所）；Université Cote d’Azur（蔚蓝海岸大学）；École Normale Supérieure de Lyon（里昂高等师范学院）
In the context of few-shot learning, one cannot measure the generalization ability of a trained classifier using validation sets, due to the small number of labeled samples. In this paper, we are interested in finding alternatives to answer the question: is my classifier generalizing well to previously unseen data? We first analyze the reasons for the variability of generalization performances. We then investigate the case of using transfer-based solutions, and consider three settings: i) supervised where we only have access to a few labeled samples, ii) semi-supervised where we have access to…
For a long time, designing neural architectures that exhibit high performance was considered a dark art that required expert hand-tuning. One of the few well-known guidelines for architecture design is the avoidance of exploding gradients, though even this guideline has remained relatively vague and circumstantial. We introduce the nonlinearity coefficient (NLC), a measurement of the complexity of the function computed by a neural network that is based on the magnitude of the gradient. Via an extensive empirical study, we show that the NLC is a powerful predictor of test error and that attaini…
As shown in recent research, deep neural networks can perfectly fit randomly labeled data, but with very poor accuracy on held out data. This phenomenon indicates that loss functions such as cross-entropy are not a reliable indicator of generalization. This leads to the crucial question of how generalization gap should be predicted from the training data and network parameters. In this paper, we propose such a measure, and conduct extensive empirical studies on how well it can predict the generalization gap. Our measure is based on the concept of margin distribution, which are the distances of…
Predicting Alignment Generalization with Value Representations
Carnegie Mellon University（卡内基梅隆大学）；Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；McGill University（麦吉尔大学）；ETH Zurich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）；University of British Columbia（不列颠哥伦比亚大学）；Vector Institute（矢量研究所）
Predicting Out-of-Domain Generalization with Neighborhood Invariance
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Massachusetts Institute of Technology（麻省理工学院）；New York University（纽约大学）；Prescient Design, Genentech（基因泰克Prescient Design）
Generalization with Reverse-Calibration of Well and Seismic Data Using Machine Learning Methods for Complex Reservoirs Predicting During Early-Stage Geological Exploration Oil Field
AITRICS；KAIST（韩国科学技术院）；BARO AI；Seoul South Korea（首尔（韩国））；Daejeon South Korea（大田（韩国））
While various complexity measures for deep neural networks exist, specifying an appropriate measure capable of predicting and explaining generalization in deep networks has proven challenging. We propose Neural Complexity (NC), a meta-learning framework for predicting generalization. Our model learns a scalar complexity measure through interactions with many heterogeneous tasks in a data-driven way. The trained NC model can be added to the standard training loss to regularize any task learner in a standard supervised learning scenario. We contrast NC's approach against existing manually-design…
Rethinking Generalization in Graph Neural Networks: A Structural Complexity Perspective
Institute of Intelligent Information Processing（智能信息处理研究院）；Shanxi University（山西大学）；Alliance Manchester Business School University of Manchester（曼彻斯特大学曼彻斯特商业学院）；University of Manchester（曼彻斯特大学）；Department of Mathematics（数学系）；Hong Kong Baptist University（香港 Baptist 大学）
