A key challenge for deploying deep neural networks (DNNs) in safety critical settings is the need to provide rigorous ways to quantify their uncertainty. In this paper, we propose a novel algorithm for constructing predicted classification confidences for DNNs that comes with provable correctness guarantees. Our approach uses Clopper-Pearson confidence intervals for the Binomial distribution in conjunction with the histogram binning approach to calibrated prediction. In addition, we demonstrate how our predicted confidences can be used to enable downstream guarantees in two settings: (i) fast DNN inference, where we demonstrate how to compose a fast but inaccurate DNN with an accurate but slow DNN in a rigorous way to improve performance without sacrificing accuracy, and (ii) safe planning, where we guarantee safety when using a DNN to predict whether a given action is safe based on visual observations. In our experiments, we demonstrate that our approach can be used to provide guarantees for state-of-the-art DNNs.
DeepMind（深度思维）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
Modern deep learning based classifiers show very high accuracy on test data but this does not provide sufficient guarantees for safe deployment, especially in high-stake AI applications such as medical diagnosis. Usually, predictions are obtained without a reliable uncertainty estimate or a formal guarantee. Conformal prediction (CP) addresses these issues by using the classifier's predictions, e.g., its probability estimates, to predict confidence sets containing the true class with a user-specified probability. However, using CP as a separate processing step after training prevents the under…
Decision tree classifiers are a widely used tool in data stream mining. The use of confidence intervals to estimate the gain associated with each split leads to very effective methods, like the popular Hoeffding tree algorithm. From a statistical viewpoint, the analysis of decision tree classifiers in a streaming setting requires knowing when enough new information has been collected to justify splitting a leaf. Although some of the issues in the statistical analysis of Hoeffding trees have been already clarified, a general and rigorous study of confidence intervals for splitting criteria is m…
Poznan University of Technology（波兹南理工大学）；Polish Academy of Sciences（波兰科学院）；MNM Bioscience（MNM生物科学公司）
Machine learning applications, especially in the fields of me\-di\-cine and social sciences, are slowly being subjected to increasing scrutiny. Similarly to sample size planning performed in clinical and social studies, lawmakers and funding agencies may expect statistical uncertainty estimations in machine learning applications that impact society. In this paper, we present an easy-to-use python package and web application for estimating prediction confidence intervals. The package offers eight different procedures to determine and justify the sample size and confidence of predictions from ho…
This paper presents the R package MCS which implements the Model Confidence Set (MCS) procedure recently developed by Hansen et al. (2011). The Hansen's procedure consists on a sequence of tests which permits to construct a set of 'superior' models, where the null hypothesis of Equal Predictive Ability (EPA) is not rejected at a certain confidence level. The EPA statistic tests is calculated for an arbitrary loss function, meaning that we could test models on various aspects, for example punctual forecasts. The relevance of the package is shown using an example which aims at illustrating in de…
This paper describes an R package implementing large sample tests and confidence intervals (based on the central limit theorem) for various parameters. The one and two sample mean and variance contexts are considered. The statistics for all the tests are expressed in the same form, which facilitates their presentation. In the variance parameter cases, the asymptotic robustness of the classical tests depends on the departure of the data distribution from normality measured in terms of the kurtosis of the distribution.
Point estimation of class prevalences in the presence of data set shift has been a popular research topic for more than two decades. Less attention has been paid to the construction of confidence and prediction intervals for estimates of class prevalences. One little considered question is whether or not it is necessary for practical purposes to distinguish confidence and prediction intervals. Another question so far not yet conclusively answered is whether or not the discriminatory power of the classifier or score at the basis of an estimation method matters for the accuracy of the estimates …
Johns Hopkins University（约翰斯·霍普金斯大学）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
We propose a novel confidence scoring mechanism for deep neural networks based on a two-model paradigm involving a base model and a meta-model. The confidence score is learned by the meta-model observing the base model succeeding/failing at its task. As features to the meta-model, we investigate linear classifier probes inserted between the various layers of the base model. Our experiments demonstrate that this approach outperforms various baselines in a filtering task, i.e., task of rejecting samples with low confidence. Experimental results are presented using CIFAR-10 and CIFAR-100 dataset …
We propose an algorithm combining calibrated prediction and generalization bounds from learning theory to construct confidence sets for deep neural networks with PAC guarantees---i.e., the confidence set for a given input contains the true label with high probability. We demonstrate how our approach can be used to construct PAC confidence sets on ResNet for ImageNet, a visual object tracking model, and a dynamics model for the half-cheetah reinforcement learning problem.
A novel confidence interval estimator is proposed for the risk difference in noninferiority binomial trials. The confidence interval is consistent with an exact unconditional test that preserves the type-I error, and has improved power, particularly for smaller sample sizes, compared to the confidence interval by Chan & Zhang (1999). The improved performance of the proposed confidence interval is theoretically justified and demonstrated with simulations and examples. An R package is also distributed that implements the proposed methods along with other confidence interval estimators.
We present the calibrated-projection MATLAB package implementing the method to construct confidence intervals proposed by Kaido, Molinari and Stoye (2017). This manual provides details on how to use the package for inference on projections of partially identified parameters. It also explains how to use the MATLAB functions we developed to compute confidence intervals on solutions of nonlinear optimization problems with estimated constraints.
To increase the trustworthiness of deep neural network (DNN) classifiers, an accurate prediction confidence that represents the true likelihood of correctness is crucial. Towards this end, many post-hoc calibration methods have been proposed to leverage a lightweight model to map the target DNN's output layer into a calibrated confidence. Nonetheless, on an out-of-distribution (OOD) dataset in practice, the target DNN can often mis-classify samples with a high confidence, creating significant challenges for the existing calibration methods to produce an accurate confidence. In this paper, we p…
Often domain adaptation is performed using a discriminator (domain classifier) to learn domain-invariant feature representations so that a classifier trained on labeled source data will generalize well to unlabeled target data. A line of research stemming from semi-supervised learning uses pseudo labeling to directly generate "pseudo labels" for the unlabeled target data and trains a classifier on the now-labeled target data, where the samples are selected or weighted based on some measure of confidence. In this paper, we propose multi-purposing the discriminator to not only aid in producing d…
We consider using an ensemble of binary classifiers for transductive prediction, when unlabeled test data are known in advance. We derive minimax optimal rules for confidence-rated prediction in this setting. By using PAC-Bayes analysis on these rules, we obtain data-dependent performance guarantees without distributional assumptions on the data. Our analysis techniques are readily extended to a setting in which the predictor is allowed to abstain.
Randomized smoothing has been shown to provide good certified-robustness guarantees for high-dimensional classification problems. It uses the probabilities of predicting the top two most-likely classes around an input point under a smoothing distribution to generate a certified radius for a classifier's prediction. However, most smoothing methods do not give us any information about the confidence with which the underlying classifier (e.g., deep neural network) makes a prediction. In this work, we propose a method to generate certified radii for the prediction confidence of the smoothed classi…
The reliable measurement of confidence in classifiers' predictions is very important for many applications and is, therefore, an important part of classifier design. Yet, although deep learning has received tremendous attention in recent years, not much progress has been made in quantifying the prediction confidence of neural network classifiers. Bayesian models offer a mathematically grounded framework to reason about model uncertainty, but usually come with prohibitive computational costs. In this paper we propose a simple, scalable method to achieve a reliable confidence score, based on the…
Using the mathematical framework of information geometry, we introduce a novel method which allows one to efficiently determine the exact shape of simultaneous confidence regions for non-linearly parametrised models. Furthermore, we show how pointwise confidence bands around the model predictions can be constructed from detailed knowledge of the exact confidence region with little additional computational effort. We exemplify our methods using inference problems in cosmology and epidemic modelling. An open source implementation of the developed schemes is publicly available via the Information…
The area under the curve (AUC) of summary receiver operating characteristic (SROC) curve is a primary statistical outcome for meta-analysis of diagnostic test accuracy studies (DTA). However, its confidence interval has not been reported in most of DTA meta-analyses, because no certain methods and statistical packages have been provided. In this article, we provide a bootstrap algorithm for computing the confidence interval of the AUC. Also, using the bootstrap framework, we can conduct a bootstrap test for assessing significance of the difference of AUCs for multiple diagnostic tests. In addi…
PAC-Bayesian Generalization Guarantees for Fairness on Stochastic and Deterministic Classifiers
CNRS（法国国家科学研究中心）；Institut d Optique Graduate School（高等光学学院）；Inria（法国国家信息与自动化研究所）；Institut Universitaire de France（法国大学研究院）；Universite Claude Bernard Lyon 1（里昂第一大学）；Univ Rennes（雷恩大学）
Korea Advanced Institute of Science and Technology（韩国科学技术院）；University of Michigan（密歇根大学）；Google Brain（谷歌大脑）
The problem of detecting whether a test sample is from in-distribution (i.e., training distribution by a classifier) or out-of-distribution sufficiently different from it arises in many real-world machine learning applications. However, the state-of-art deep neural networks are known to be highly overconfident in their predictions, i.e., do not distinguish in- and out-of-distributions. Recently, to handle this issue, several threshold-based detectors have been proposed given pre-trained neural classifiers. However, the performance of prior works highly depends on how to train the classifiers s…
Abstract We present PAC-Bayesian bounds for the generalisation error of the K-nearest-neighbour classifier (K-NN). This is achieved by casting the K-NN classifier into a kernel space framework in the limit of vanishing kernel bandwidth. We establish a relation between prior measures over the coefficients in the kernel expansion and the induced measure on the weight vectors in kernel space. Defining a sparse prior over the coefficients allows the application of a PAC-Bayesian folk theorem that leads to a generalisation bound that is a function of the number of redundant training examples: those…
In this paper, we provide two main contributions in PAC-Bayesian theory for domain adaptation where the objective is to learn, from a source distribution, a well-performing majority vote on a different target distribution. On the one hand, we propose an improvement of the previous approach proposed by Germain et al. (2013), that relies on a novel distribution pseudodistance based on a disagreement averaging, allowing us to derive a new tighter PAC-Bayesian domain adaptation bound for the stochastic Gibbs classifier. We specialize it to linear classifiers, and design a learning algorithm which …
KAIST（韩国科学技术院）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；Kakao Enterprise；Inha University（仁荷大学）
Randomized smoothing is currently a state-of-the-art method to construct a certifiably robust classifier from neural networks against $\ell_2$-adversarial perturbations. Under the paradigm, the robustness of a classifier is aligned with the prediction confidence, i.e., the higher confidence from a smoothed classifier implies the better robustness. This motivates us to rethink the fundamental trade-off between accuracy and robustness in terms of calibrating confidences of a smoothed classifier. In this paper, we propose a simple training scheme, coined SmoothMix, to control the robustness of sm…
Peter the Great St.Petersburg Polytechnic University (SPbPU)（彼得大帝圣彼得堡理工大学（SPbPU））；St.Petersburg Clinical Research Center for Special Types of Medical Care (Oncology-oriented)（圣彼得堡特殊医疗类型临床研究中心（肿瘤导向））
A modification of the confidence screening mechanism based on adaptive weighing of every training instance at each cascade level of the Deep Forest is proposed. The idea underlying the modification is very simple and stems from the confidence screening mechanism idea proposed by Pang et al. to simplify the Deep Forest classifier by means of updating the training set at each level in accordance with the classification accuracy of every training instance. However, if the confidence screening mechanism just removes instances from training and testing processes, then the proposed modification is m…
Miami University（迈阿密大学）；Air Force Research Lab（空军研究实验室）
Assessing the predictive accuracy of black box classifiers is challenging in the absence of labeled test datasets. In these scenarios we may need to rely on a human oracle to evaluate individual predictions; presenting the challenge to create query algorithms to guide the search for points that provide the most information about the classifier's predictive characteristics. Previous works have focused on developing utility models and query algorithms for discovering unknown unknowns --- misclassifications with a predictive confidence above some arbitrary threshold. However, if misclassification…
Toyota Motor Corporation（丰田汽车公司）；RIKEN（理化学研究所）；The University of Tokyo（东京大学）
Positive-confidence (Pconf) classification [Ishida et al., 2018] is a promising weakly-supervised learning method which trains a binary classifier only from positive data equipped with confidence. However, in practice, the confidence may be skewed by bias arising in an annotation process. The Pconf classifier cannot be properly learned with skewed confidence, and consequently, the classification performance might be deteriorated. In this paper, we introduce the parameterized model of the skewed confidence, and propose the method for selecting the hyperparameter which cancels out the negative i…
PAC-Bayesian set up involves a stochastic classifier characterized by a posterior distribution on a classifier set, offers a high probability bound on its averaged true risk and is robust to the training sample used. For a given posterior, this bound captures the trade off between averaged empirical risk and KL-divergence based model complexity term. Our goal is to identify an optimal posterior with the least PAC-Bayesian bound. We consider a finite classifier set and 5 distance functions: KL-divergence, its Pinsker's and a sixth degree polynomial approximations; linear and squared distances. …
PRECISE Center（PRECISE中心）；University of Pennsylvania（宾夕法尼亚大学）
Providing reliable model uncertainty estimates is imperative to enabling robust decision making by autonomous agents and humans alike. While recently there have been significant advances in confidence calibration for trained models, examples with poor calibration persist in most calibrated models. Consequently, multiple techniques have been proposed that leverage label-invariant transformations of the input (i.e., an input manifold) to improve worst-case confidence calibration. However, manifold-based confidence calibration techniques generally do not scale and/or require expensive retraining …
We study the tradeoff between computational effort and classification accuracy in a cascade of deep neural networks. During inference, the user sets the acceptable accuracy degradation which then automatically determines confidence thresholds for the intermediate classifiers. As soon as the confidence threshold is met, inference terminates immediately without having to compute the output of the complete network. Confidence levels are derived directly from the softmax outputs of intermediate classifiers, as we do not train special decision functions. We show that using a softmax output as a con…
Can we learn a binary classifier from only positive data, without any negative data or unlabeled data? We show that if one can equip positive data with confidence (positive-confidence), one can successfully learn a binary classifier, which we name positive-confidence (Pconf) classification. Our work is related to one-class classification which is aimed at "describing" the positive class by clustering-related methods, but one-class classification does not have the ability to tune hyper-parameters and their aim is not on "discriminating" positive and negative classes. For the Pconf classificatio…
Google Research（谷歌研究院）；Google Brain（谷歌大脑）；Stanford University（斯坦福大学）
Knowing when a classifier's prediction can be trusted is useful in many applications and critical for safely using AI. While the bulk of the effort in machine learning research has been towards improving classifier performance, understanding when a classifier's predictions should and should not be trusted has received far less attention. The standard approach is to use the classifier's discriminant or confidence score; however, we show there exists an alternative that is more effective in many situations. We propose a new score, called the trust score, which measures the agreement between the …
Confidence measures for the generalization error are crucial when small training samples are used to construct classifiers. A common approach is to estimate the generalization error by resampling and then assume the resampled estimator follows a known distribution to form a confidence set [Kohavi 1995, Martin 1996,Yang 2006]. Alternatively, one might bootstrap the resampled estimator of the generalization error to form a confidence set. Unfortunately, these methods do not reliably provide sets of the desired confidence. The poor performance appears to be due to the lack of smoothness of the ge…
In this work, we propose a PAC-Bayes bound for the generalization risk of the Gibbs classifier in the multi-class classification framework. The novelty of our work is the critical use of the confusion matrix of a classifier as an error measure; this puts our contribution in the line of work aiming at dealing with performance measure that are richer than mere scalar criterion such as the misclassification rate. Thanks to very recent and beautiful results on matrix concentration inequalities, we derive two bounds showing that the true confusion risk of the Gibbs classifier is upper-bounded by it…
California Institute of Technology（加州理工学院）；University of California, Irvine（加州大学欧文分校）
How many labeled examples are needed to estimate a classifier's performance on a new dataset? We study the case where data is plentiful, but labels are expensive. We show that by making a few reasonable assumptions on the structure of the data, it is possible to estimate performance curves, with confidence bounds, using a small number of ground truth labels. Our approach, which we call Semisupervised Performance Evaluation (SPE), is based on a generative model for the classifier's confidence scores. In addition to estimating the performance of classifiers on new datasets, SPE can be used to re…
Khoury College of Computer Science, Northeastern University（东北大学库利计算机学院）；University of Virginia（弗吉尼亚大学）；Biocomplexity Institute and Initiative, University of Virginia（弗吉尼亚大学生物复杂性研究所）
The study of strategic or adversarial manipulation of testing data to fool a classifier has attracted much recent attention. Most previous works have focused on two extreme situations where any testing data point either is completely adversarial or always equally prefers the positive label. In this paper, we generalize both of these through a unified framework for strategic classification, and introduce the notion of strategic VC-dimension (SVC) to capture the PAC-learnability in our general strategic setup. SVC provably generalizes the recent concept of adversarial VC-dimension (AVC) introduc…
We study a recent model of collaborative PAC learning where $k$ players with $k$ different tasks collaborate to learn a single classifier that works for all tasks. Previous work showed that when there is a classifier that has very small error on all tasks, there is a collaborative algorithm that finds a single classifier for all tasks and has $O((\ln (k))^2)$ times the worst-case sample complexity for learning a single task. In this work, we design new algorithms for both the realizable and the non-realizable setting, having sample complexity only $O(\ln (k))$ times the worst-case sample compl…
Air Force Research Lab（空军研究实验室）；Miami University（迈阿密大学）；General Mills（通用磨坊公司）
Given a black-box classification model and an unlabeled evaluation dataset from some application domain, efficient strategies need to be developed to evaluate the model. Random sampling allows a user to estimate metrics like accuracy, precision, and recall, but may not provide insight to high-confidence errors. High-confidence errors are rare events for which the model is highly confident in its prediction, but is wrong. Such errors can represent costly mistakes and should be explicitly searched for. In this paper we propose a generalization to the Adversarial Distance search that leverages co…
Mixed-signal machine-learning classification has recently been demonstrated as an efficient alternative for classification with power expensive digital circuits. In this paper, a high-COnfidence high-REsolution (CORE) mixed-signal classifier is proposed for classifying high-dimensional input data into multi-class output space with less power and area than state-of-the-art classifiers. A high-resolution multiplication is facilitated within a single-MOSFET by feeding the features and feature weights into, respectively, the body and gate inputs. High-resolution classifier that considers the confi…
University of Tübingen（图宾根大学）；Saarland University（萨尔大学）
Classifiers used in the wild, in particular for safety-critical systems, should not only have good generalization properties but also should know when they don't know, in particular make low confidence predictions far away from the training data. We show that ReLU type neural networks which yield a piecewise linear classifier function fail in this regard as they produce almost always high confidence predictions far away from the training data. For bounded domains like images we propose a new robust optimization technique similar to adversarial training which enforces low confidence predictions…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院（KAIST））
Any classifier can be "smoothed out" under Gaussian noise to build a new classifier that is provably robust to $\ell_2$-adversarial perturbations, viz., by averaging its predictions over the noise via randomized smoothing. Under the smoothed classifiers, the fundamental trade-off between accuracy and (adversarial) robustness has been well evidenced in the literature: i.e., increasing the robustness of a classifier for an input can be at the expense of decreased accuracy for some other inputs. In this paper, we propose a simple training method leveraging this trade-off to obtain robust smoothed…
We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound's prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid g…
Deep Neural Network Calibration by Reducing Classifier Shift with Stochastic Masking
School of Artificial Intelligence, Jilin University（吉林大学人工智能学院）；CSIRO's Data61 and Monash University (adjunct)（CSIRO的数据61与墨尔本大学（兼职））；King Abdullah University of Science and Technology（国王阿卜杜勒阿齐兹科技大学）
Classifiers and other statistics-based machine learning (ML) techniques generalize, or learn, based on various statistical properties of the training data. The assumption underlying statistical ML resulting in theoretical or empirical performance guarantees is that the distribution of the training data is representative of the production data distribution. This assumption often breaks; for instance, statistical distributions of the data may change. We term changes that affect ML performance `data drift' or `drift'. Many classification techniques compute a measure of confidence in their results…
China Agricultural University（中国农业大学）；Chongqing University（重庆大学）；Southwest University（西南大学）；Nanyang Technological University（南洋理工大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；The University of Tokyo（东京大学）
Can we learn a multi-class classifier from only data of a single class? We show that without any assumptions on the loss functions, models, and optimizers, we can successfully learn a multi-class classifier from only data of a single class with a rigorous consistency guarantee when confidences (i.e., the class-posterior probabilities for all the classes) are available. Specifically, we propose an empirical risk minimization framework that is loss-/model-/optimizer-independent. Instead of constructing a boundary between the given class and other classes, our method can conduct discriminative cl…
A standard approach in pattern classification is to estimate the distributions of the label classes, and then to apply the Bayes classifier to the estimates of the distributions in order to classify unlabeled examples. As one might expect, the better our estimates of the label class distributions, the better the resulting classifier will be. In this paper we make this observation precise by identifying risk bounds of a classifier in terms of the quality of the estimates of the label class distributions. We show how PAC learnability relates to estimates of the distributions that have a PAC guar…
Aix-Marseille Univ（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；Univ. of St-Etienne（圣艾蒂安大学）
A lot of attention has been devoted to multimedia indexing over the past few years. In the literature, we often consider two kinds of fusion schemes: The early fusion and the late fusion. In this paper we focus on late classifier fusion, where one combines the scores of each modality at the decision level. To tackle this problem, we investigate a recent and elegant well-founded quadratic program named MinCq coming from the Machine Learning PAC-Bayes theory. MinCq looks for the weighted combination, over a set of real-valued functions seen as voters, leading to the lowest misclassification rate…
Department of Mathematics and Computer Science（数学与计算机科学系）；Saarland University（萨尔大学）
Recent work has shown that state-of-the-art classifiers are quite brittle, in the sense that a small adversarial change of an originally with high confidence correctly classified input leads to a wrong classification again with high confidence. This raises concerns that such classifiers are vulnerable to attacks and calls into question their usage in safety-critical systems. We show in this paper for the first time formal guarantees on the robustness of a classifier by giving instance-specific lower bounds on the norm of the input manipulation required to change the classifier decision. Based …
Department of Computer Science, University of Copenhagen（哥本哈根大学计算机科学系）
Existing guarantees in terms of rigorous upper bounds on the generalization error for the original random forest algorithm, one of the most frequently used machine learning methods, are unsatisfying. We discuss and evaluate various PAC-Bayesian approaches to derive such bounds. The bounds do not require additional hold-out data, because the out-of-bag samples from the bagging in the training process can be exploited. A random forest predicts by taking a majority vote of an ensemble of decision trees. The first approach is to bound the error of the vote by twice the error of the corresponding G…
Laboratoire d’Informatique Fondamentale de Marseille（马赛基础信息学实验室）；CNRS（法国国家科学研究中心）
Pac-Bayes bounds are among the most accurate generalization bounds for classifiers learned from independently and identically distributed (IID) data, and it is particularly so for margin classifiers: there have been recent contributions showing how practical these bounds can be either to perform model selection (Ambroladze et al., 2007) or even to directly guide the learning of linear classifiers (Germain et al., 2009). However, there are many practical situations where the training data show some dependencies and where the traditional IID assumption does not hold. Stating generalization bound…
