Lund University（隆德大学）；CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）；Monash University（莫纳什大学）；Ruhr University Bochum（波鸿鲁尔大学）；Fraunhofer IAIS（弗劳恩霍夫智能分析和信息系统研究所）；Google Research（谷歌研究院）
Henning Petzka, Michael Kamp, Linara Adilova, Cristian Sminchisescu, Mario Boley
备注：
The first two authors made equal contribution; Accepted for publication at NeurIPS 2021; arXiv admin note: substantial text overlap with arXiv:1912.00058
Flatness of the loss curve is conjectured to be connected to the generalization ability of machine learning models, in particular neural networks. While it has been empirically observed that flatness measures consistently correlate strongly with generalization, it is still an open theoretical problem why and under which circumstances flatness is connected to generalization, in particular in light of reparameterizations that change certain flatness measures but leave generalization unchanged. We investigate the connection between flatness and generalization by relating it to the interpolation from representative data, deriving notions of representativeness, and feature robustness. The notions allow us to rigorously connect flatness and generalization and to identify conditions under which the connection holds. Moreover, they give rise to a novel, but natural relative flatness measure that correlates strongly with generalization, simplifies to ridge regression for ordinary least squares, and solves the reparameterization issue.
University of Oxford（牛津大学）；Hertford college（赫特福德学院）
The intuition that local flatness of the loss landscape is correlated with better generalization for deep neural networks (DNNs) has been explored for decades, spawning many different flatness measures. Recently, this link with generalization has been called into question by a demonstration that many measures of flatness are vulnerable to parameter re-scaling which arbitrarily changes their value without changing neural network outputs. Here we show that, in addition, some popular variants of SGD such as Adam and Entropy-SGD, can also break the flatness-generalization correlation. As an altern…
Lund University（隆德大学）；Fraunhofer Center for Machine Learning（弗劳恩霍夫机器学习中心）；Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）；Monash University（莫纳什大学）；Google Research（谷歌研究院）
The performance of deep neural networks is often attributed to their automated, task-related feature construction. It remains an open question, though, why this leads to solutions with good generalization, even in cases where the number of parameters is larger than the number of samples. Back in the 90s, Hochreiter and Schmidhuber observed that flatness of the loss surface around a local minimum correlates with low generalization error. For several flatness measures, this correlation has been empirically validated. However, it has recently been shown that existing measures of flatness cannot t…
Flatness is Necessary, Neural Collapse is Not: Rethinking Generalization via Grokking
Ting Han Lamarr Institute, TU Dortmund, Germany（廷汉拉马尔研究所，杜伊斯堡-艾森大学，德国）；Institute for AI in Medicine, UK Essen（人工智能医学研究所，英国埃森）；Research Center Trustworthy Data Science and Security of the University Alliance Ruhr, TU Dortmund, Germany（trustworthy数据科学与安全研究所以及鲁尔大学联盟，杜伊斯堡-艾森大学，德国）；Ruhr University Bochum, Germany（波恩鲁尔大学，德国）；Department of Physics, TU Dortmund, Germany（物理系，杜伊斯堡-艾森大学，德国）；Lamarr Institute, TU Dortmund, Germany（廷汉拉马尔研究所，杜伊斯堡-艾森大学，德国）
The notion of flat minima has played a key role in the generalization studies of deep learning models. However, existing definitions of the flatness are known to be sensitive to the rescaling of parameters. The issue suggests that the previous definitions of the flatness might not be a good measure of generalization, because generalization is invariant to such rescalings. In this paper, from the PAC-Bayesian perspective, we scrutinize the discussion concerning the flat minima and introduce the notion of normalized flat minima, which is free from the known scale dependence issues. Additionally,…
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland Informatics Campus（萨尔兰信息学园区）；University of Tübingen（图宾根大学）
Adversarial training (AT) has become the de-facto standard to obtain models robust against adversarial examples. However, AT exhibits severe robust overfitting: cross-entropy loss on adversarial examples, so-called robust loss, decreases continuously on training examples, while eventually increasing on test examples. In practice, this leads to poor robust generalization, i.e., adversarial robustness does not generalize well to new examples. In this paper, we study the relationship between robust generalization and flatness of the robust loss landscape in weight space, i.e., whether robust loss…
It has been empirically observed that the flatness of minima obtained from training deep networks seems to correlate with better generalization. However, for deep networks with positively homogeneous activations, most measures of sharpness/flatness are not invariant to rescaling of the network parameters, corresponding to the same function. This means that the measure of flatness/sharpness can be made as small or as large as possible through rescaling, rendering the quantitative measures meaningless. In this paper we show that for deep networks with positively homogenous activations, these res…
The Uncanny Valley: Exploring Adversarial Robustness from a Flatness Perspective
CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）；Ruhr University Bochum（波鸿鲁尔大学）；Institute for AI in medicine (IKIM)（医学人工智能研究所）；University Hospital Essen（埃森大学医院）；Monash University（莫纳什大学）
Existing generalization measures that aim to capture a model's simplicity based on parameter counts or norms fail to explain generalization in overparameterized deep neural networks. In this paper, we introduce a new, theoretically motivated measure of a network's simplicity which we call prunability: the smallest \emph{fraction} of the network's parameters that can be kept while pruning without adversely affecting its training loss. We show that this measure is highly predictive of a model's generalization performance across a large set of convolutional networks trained on CIFAR-10, does not …
AI Lab, Institute for Data Science and Analytics, Bocconi University（博科尼大学数据科学与分析研究所人工智能实验室）；Dept. Applied Science and Technology, Politecnico di Torino（都灵理工大学应用科学与技术系）
The properties of flat minima in the empirical risk landscape of neural networks have been debated for some time. Increasing evidence suggests they possess better generalization capabilities with respect to sharp ones. First, we discuss Gaussian mixture classification models and show analytically that there exist Bayes optimal pointwise estimators which correspond to minimizers belonging to wide flat regions. These estimators can be found by applying maximum flatness algorithms either directly on the classifier (which is norm independent) or on the differentiable loss function used in learning…
Korea Advanced Institute of Science and Technology（韩国科学技术院）；Samsung Medical Center（三星医疗中心）；Sungkyunkwan University School of Medicine（成均馆大学医学院）；AITRICS
Explaining generalizations and preventing over-confident predictions are central goals of studies on the loss landscape of neural networks. Flatness, defined as loss invariability on perturbations of a pre-trained solution, is widely accepted as a predictor of generalization in this context. However, the problem that flatness and generalization bounds can be changed arbitrarily according to the scale of a parameter was pointed out, and previous studies partially solved the problem with restrictions: Counter-intuitively, their generalization bounds were still variant for the function-preserving…
Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good generalization performance. Moreover, when we measure generalization as a function of parameters, we see double descent behaviour, where the test error decreases, increases, and then again decreases. We show that many of these properties become understandable when viewed through the lens of effective dimensionality, which measures the dimensionality of the parameter…
Bilateral Sharpness-Aware Minimization for Flatter Minima
Faculty of Information Technology, Beijing University of Technology（北京工业大学信息技术学院）；School of Artificial Intelligence, BeiHang University（北京航空航天大学人工智能学院）；University of Chinese Academy of Sciences（中国科学院大学）；Chinese Academy of Sciences (CAS)（中国科学院）；Institute of Computing Technology, CAS（中国科学院计算技术研究所）
Beihang University（北京航空航天大学）；University of Bonn（波恩大学）；Amazon（亚马逊）；University at Buffalo（布法罗大学）；Baidu（百度）
Research on the generalization ability of deep neural networks (DNNs) has recently attracted a great deal of attention. However, due to their complex architectures and large numbers of parameters, measuring the generalization ability of specific DNN models remains an open challenge. In this paper, we propose to use multiple factors to measure and rank the relative generalization of DNNs based on a new concept of confidence dimension (CD). Furthermore, we provide a feasible framework in our CD to theoretically calculate the upper bound of generalization based on the conventional Vapnik-Chervone…
