It is a common practice in the machine learning community to assume that the observed data are noise-free in the input attributes. Nevertheless, scenarios with input noise are common in real problems, as measurements are never perfectly accurate. If this input noise is not taken into account, a supervised machine learning method is expected to perform sub-optimally. In this paper, we focus on multi-class classification problems and use Gaussian processes (GPs) as the underlying classifier. Motivated by a data set coming from the astrophysics domain, we hypothesize that the observed data may contain noise in the inputs. Therefore, we devise several multi-class GP classifiers that can account for input noise. Such classifiers can be efficiently trained using variational inference to approximate the posterior distribution of the latent variables of the model. Moreover, in some situations, the amount of noise can be known before-hand. If this is the case, it can be readily introduced in the proposed methods. This prior information is expected to lead to better performance results. We have evaluated the proposed methods by carrying out several experiments, involving synthetic and real data. These include several data sets from the UCI repository, the MNIST data set and a data set coming from astrophysics. The results obtained show that, although the classification error is similar across methods, the predictive distribution of the proposed methods is better, in terms of the test log-likelihood, than the predictive distribution of a classifier based on GPs that ignores input noise.
Mines Saint-Étienne（圣埃蒂安矿业学院）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；BRGM（法国地质矿产调查局）
Adding inequality constraints (e.g. boundedness, monotonicity, convexity) into Gaussian processes (GPs) can lead to more realistic stochastic emulators. Due to the truncated Gaussianity of the posterior, its distribution has to be approximated. In this work, we consider Monte Carlo (MC) and Markov Chain Monte Carlo (MCMC) methods. However, strictly interpolating the observations may entail expensive computations due to highly restrictive sample spaces. Furthermore, having (constrained) GP emulators when data are actually noisy is also of interest for real-world implementations. Hence, we intro…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
Variational Bayesian neural networks combine the flexibility of deep learning with Bayesian uncertainty estimation. However, inference procedures for flexible variational posteriors are computationally expensive. A recently proposed method, noisy natural gradient, is a surprisingly simple method to fit expressive posteriors by adding weight noise to regular natural gradient updates. Noisy K-FAC is an instance of noisy natural gradient that fits a matrix-variate Gaussian posterior with minor changes to ordinary K-FAC. Nevertheless, a matrix-variate Gaussian posterior does not capture an accurat…
École Polytechnique（巴黎综合理工学院）；University of California, Berkeley（加州大学伯克利分校）
Class labels are often imperfectly observed, due to mistakes and to genuine ambiguity among classes. We propose a new semi-supervised deep generative model that explicitly models noisy labels, called the Mislabeled VAE (M-VAE). The M-VAE can perform better than existing deep generative models which do not account for label noise. Additionally, the derivation of M-VAE gives new theoretical insights into the popular M1+M2 semi-supervised model.
University of California at Santa Barbara（加州大学圣巴巴拉分校）
We develop adaptive replicated designs for Gaussian process metamodels of stochastic experiments. Adaptive batching is a natural extension of sequential design heuristics with the benefit of replication growing as response features are learned, inputs concentrate, and the metamodeling overhead rises. Motivated by the problem of learning the level set of the mean simulator response we develop four novel schemes: Multi-Level Batching (MLB), Ratchet Batching (RB), Adaptive Batched Stepwise Uncertainty Reduction (ABSUR), Adaptive Design with Stepwise Allocation (ADSA) and Deterministic Design with…
We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus is on strongly stochastic samplers, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. To guard against noise misspecification, we assess the performance of three variants: (i) GPs with Student-$t$ observations; (ii) Student-$t$ processes (TPs); and (iii) classification GPs modeling the sign of the response. In conjunction with these metamodels, we analy…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所）；Aalto University（阿尔托大学）；University of Edinburgh（爱丁堡大学）
We consider Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained. This occurs for example when complex simulator-based statistical models are fitted to data, and synthetic likelihood (SL) method is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a sequential Bayesian experimental design problem, where the log-likelihood function is modelled with a hierarchical Gaussian process (GP) surrogate model, which is used to efficiently select additional log-likelihood evaluation…
Osaka Prefecture University（大阪府立大学）；The University of New Mexico（新墨西哥大学）
Koopman Mode Decomposition (KMD) is a technique of nonlinear time-series analysis that originates from point spectrum of the Koopman operator defined for an underlying nonlinear dynamical system. We present a numerical algorithm of KMD based on Gaussian process regression that is capable of handling noisy finite-time data. The algorithm is applied to short-term swing dynamics of a multi-machine power grid in order to estimate oscillatory modes embedded in the dynamics, and thereby the effectiveness of the algorithm is evaluated.
We study the Gaussian Process regression model in the context of training data with noise in both input and output. The presence of two sources of noise makes the task of learning accurate predictive models extremely challenging. However, in some instances additional constraints may be available that can reduce the uncertainty in the resulting predictive models. In particular, we consider the case of monotonically ordered latent input, which occurs in many application domains that deal with temporal data. We present a novel inference and learning approach based on non-parametric Gaussian varia…
The trimming scheme with a prefixed cutoff portion is known as a method of improving the robustness of statistical models such as multivariate Gaussian mixture models (MG- MMs) in small scale tests by alleviating the impacts of outliers. However, when this method is applied to real- world data, such as noisy speech processing, it is hard to know the optimal cut-off portion to remove the outliers and sometimes removes useful data samples as well. In this paper, we propose a new method based on measuring the dispersion degree (DD) of the training data to avoid this problem, so as to realise auto…
This paper presents a machine learning framework (GP-NODE) for Bayesian systems identification from partial, noisy and irregular observations of nonlinear dynamical systems. The proposed method takes advantage of recent developments in differentiable programming to propagate gradient information through ordinary differential equation solvers and perform Bayesian inference with respect to unknown model parameters using Hamiltonian Monte Carlo sampling and Gaussian Process priors over the observed system states. This allows us to exploit temporal correlations in the observed data, and efficientl…
Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs), using noisy and sparse data is a vital task in many fields. We propose a fast and accurate method, MAGI (MAnifold-constrained Gaussian process Inference), for this task. MAGI uses a Gaussian process model over time-series data, explicitly conditioned on the manifold constraint that derivatives of the Gaussian process must satisfy the ODE system. By doing so, we completely bypass the need for numerical integration and achieve substantial savings in computational time. MAGI is also su…
In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-expon…
Astronomical data often suffer from noise and incompleteness. We extend the common mixtures-of-Gaussians density estimation approach to account for situations with a known sample incompleteness by simultaneous imputation from the current model. The method, called GMMis, generalizes existing Expectation-Maximization techniques for truncated data to arbitrary truncation geometries and probabilistic rejection processes, as long as they can be specified and do not depend on the density itself. The method accounts for independent multivariate normal measurement errors for each of the observed sampl…
Multi-fidelity Gaussian process regression for noisy outputs and non-nested experimental designs: a comparison between the recursive and non-recursive formulations
Université Paris-Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）；Service d’Études Mécaniques et Thermiques（机械与热力学研究服务部）；CMAP（数学与应用数学中心）；CNRS（法国国家科学研究中心）；École polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；CEA, DAM, DIF（法国原子能和替代能源委员会，国防应用管理部，国防创新部）
This work proposes a nonparametric method to compare the underlying mean functions given two noisy datasets. The motivation for the work stems from an application of comparing wind turbine power curves. Comparing wind turbine data presents new problems, namely the need to identify the regions of difference in the input space and to quantify the extent of difference that is statistically significant. Our proposed method, referred to as funGP, estimates the underlying functions for different data samples using Gaussian process models. We build a confidence band using the probability law of the e…
It is common that a trained classification model is applied to the operating data that is deviated from the training data because of noise. This paper demonstrates that an ensemble classifier, Diversified Multiple Tree (DMT), is more robust in classifying noisy data than other widely used ensemble methods. DMT is tested on three real world biomedical data sets from different laboratories in comparison with four benchmark ensemble classifiers. Experimental results show that DMT is significantly more accurate than other benchmark ensemble classifiers on noisy test data. We also discuss a limitat…
University of Sydney（悉尼大学）；Xidian University（西安电子科技大学）；RIKEN（理化学研究所）；Hong Kong Baptist University（香港浸会大学）；The University of Melbourne（墨尔本大学）；Brain-Inspired Technology Co., Ltd.（脑启发科技有限公司）
A similarity label indicates whether two instances belong to the same class while a class label shows the class of the instance. Without class labels, a multi-class classifier could be learned from similarity-labeled pairwise data by meta classification learning. However, since the similarity label is less informative than the class label, it is more likely to be noisy. Deep neural networks can easily remember noisy data, leading to overfitting in classification. In this paper, we propose a method for learning from only noisy-similarity-labeled data. Specifically, to model the noise, we employ…
Faculty of Engineering and Information Technology, University of Technology, Sydney（悉尼科技大学工程与信息技术学院）
In this paper, we study a classification problem in which sample labels are randomly corrupted. In this scenario, there is an unobservable sample with noise-free labels. However, before being observed, the true labels are independently flipped with a probability $ρ\in[0,0.5)$, and the random label noise can be class-conditional. Here, we address two fundamental problems raised by this scenario. The first is how to best use the abundant surrogate loss functions designed for the traditional classification problem when there is label noise. We prove that any surrogate loss function can be used fo…
Microsoft Research Asia（微软亚洲研究院）；College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）；Tsinghua University（清华大学）
Existing relation classification methods that rely on distant supervision assume that a bag of sentences mentioning an entity pair are all describing a relation for the entity pair. Such methods, performing classification at the bag level, cannot identify the mapping between a relation and a sentence, and largely suffers from the noisy labeling problem. In this paper, we propose a novel model for relation classification at the sentence level from noisy data. The model has two modules: an instance selector and a relation classifier. The instance selector chooses high-quality sentences with rein…
The Exploration of Error Bounds in Classification with Noisy Labels
The Hong Kong University of Science and Technology（香港科学与技术大学）；School of Mathematics and Statistics（数学与统计学学院）；Huazhong University of Science and Technology（华中科技大学）；The University of Hong Kong（香港大学）
Universidad de Granada（格拉纳达大学）；Universidad de Sevilla（塞维利亚大学）
In this paper we propose a novel methodology to construct Optimal Classification Trees that takes into account that noisy labels may occur in the training sample. Our approach rests on two main elements: (1) the splitting rules for the classification trees are designed to maximize the separation margin between classes applying the paradigm of SVM; and (2) some of the labels of the training sample are allowed to be changed during the construction of the tree trying to detect the label noise. Both features are considered and integrated together to design the resulting Optimal Classification Tree…
A deep neural network trained on noisy labels is known to quickly lose its power to discriminate clean instances from noisy ones. After the early learning phase has ended, the network memorizes the noisy instances, which leads to a significant degradation in its generalization performance. To resolve this issue, we propose MARVEL (MARgins Via Early Learning), a new robust learning method where the memorization of the noisy instances is curbed. We propose a new test statistic that tracks the goodness of "fit" of every instance based on the epoch-history of its classification margins. If its cla…
Industry datasets used for text classification are rarely created for that purpose. In most cases, the data and target predictions are a by-product of accumulated historical data, typically fraught with noise, present in both the text-based document, as well as in the targeted labels. In this work, we address the question of how well performance metrics computed on noisy, historical data reflect the performance on the intended future machine learning model input. The results demonstrate the utility of dirty training datasets used to build prediction models for cleaner (and different) predictio…
Noisy PN learning is the problem of binary classification when training examples may be mislabeled (flipped) uniformly with noise rate rho1 for positive examples and rho0 for negative examples. We propose Rank Pruning (RP) to solve noisy PN learning and the open problem of estimating the noise rates, i.e. the fraction of wrong positive and negative labels. Unlike prior solutions, RP is time-efficient and general, requiring O(T) for any unrestricted choice of probabilistic classifier with T fitting time. We prove RP has consistent noise estimation and equivalent expected risk as learning with u…
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；CEA（法国原子能和替代能源委员会）；Institut de physique théorique（理论物理研究所）；Laboratoire de Physique de l’Ecole normale supérieure（巴黎高等师范学院物理实验室）；ENS（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）
We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number $n$ of samples and their dimension $d$ go to infinity while their ratio is fixed to $α= n/d$. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also …
We present an optimization framework for learning a fair classifier in the presence of noisy perturbations in the protected attributes. Compared to prior work, our framework can be employed with a very general class of linear and linear-fractional fairness constraints, can handle multiple, non-binary protected attributes, and outputs a classifier that comes with provable guarantees on both accuracy and fairness. Empirically, we show that our framework can be used to attain either statistical rate or false positive rate fairness guarantees with a minimal loss in accuracy, even when the noise is…
Stanford University School of Medicine（斯坦福大学医学院）；Stanford University（斯坦福大学）
Heart diseases constitute a global health burden, and the problem is exacerbated by the error-prone nature of listening to and interpreting heart sounds. This motivates the development of automated classification to screen for abnormal heart sounds. Existing machine learning-based systems achieve accurate classification of heart sound recordings but rely on expert features that have not been thoroughly evaluated on noisy recordings. Here we propose a segmental convolutional neural network architecture that achieves automatic feature learning from noisy heart sound recordings. Our experiments s…
A comprehensive methodology is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers which are typical in GPS signals. We demonstrate the effectiveness of our methods on GPS trajectory data obtained from oceanographic floating instruments known as drifters.
Randomizing the Fourier-transform (FT) phases of temporal-spatial data generates surrogates that approximate examples from the data-generating distribution. We propose such FT surrogates as a novel tool to augment and analyze training of neural networks and explore the approach in the example of sleep-stage classification. By computing FT surrogates of raw EEG, EOG, and EMG signals of under-represented sleep stages, we balanced the CAPSLPDB sleep database. We then trained and tested a convolutional neural network for sleep stage classification, and found that our surrogate-based augmentation i…
