As neural networks become more popular, the need for accompanying uncertainty estimates increases. There are currently two main approaches to test the quality of these estimates. Most methods output a density. They can be compared by evaluating their loglikelihood on a test set. Other methods output a prediction interval directly. These methods are often tested by examining the fraction of test points that fall inside the corresponding prediction intervals. Intuitively both approaches seem logical. However, we demonstrate through both theoretical arguments and simulations that both ways of evaluating the quality of uncertainty estimates have serious flaws. Firstly, both approaches cannot disentangle the separate components that jointly create the predictive uncertainty, making it difficult to evaluate the quality of the estimates of these components. Secondly, a better loglikelihood does not guarantee better prediction intervals, which is what the methods are often used for in practice. Moreover, the current approach to test prediction intervals directly has additional flaws. We show why it is fundamentally flawed to test a prediction or confidence interval on a single test set. At best, marginal coverage is measured, implicitly averaging out overconfident and underconfident predictions. A much more desirable property is pointwise coverage, requiring the correct coverage for each prediction. We demonstrate through practical examples that these effects can result in favoring a method, based on the predictive uncertainty, that has undesirable behaviour of the confidence or prediction intervals. Finally, we propose a simulation-based testing approach that addresses these problems while still allowing easy comparison between different methods.
Calibrated uncertainty estimates in machine learning are crucial to many fields such as autonomous vehicles, medicine, and weather and climate forecasting. While there is extensive literature on uncertainty calibration for classification, the classification findings do not always translate to regression. As a result, modern models for predicting uncertainty in regression settings typically produce uncalibrated and overconfident estimates. To address these gaps, we present a calibration method for regression settings that does not assume a particular uncertainty distribution over the error: Cal…
IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Generating high quality uncertainty estimates for sequential regression, particularly deep recurrent networks, remains a challenging and open problem. Existing approaches often make restrictive assumptions (such as stationarity) yet still perform poorly in practice, particularly in presence of real world non-stationary signals and drift. This paper describes a flexible method that can generate symmetric and asymmetric uncertainty estimates, makes no assumptions about stationarity, and outperforms competitive baselines on both drift and non drift scenarios. This work helps make sequential regre…
An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression
LMU Munich（慕尼黑莱布尼茨大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；DLR-German Aerospace Center University of Tromsø（德国航空航天中心图勒大学）；German Research Center for Artificial Intelligence (DFKI, DSA)（德国人工智能研究中心）
Massachusetts Institute of Technology（麻省理工学院）；Raytheon BBN Technologies（雷神BBN技术公司）
While deep neural networks are highly performant and successful in a wide range of real-world problems, estimating their predictive uncertainty remains a challenging task. To address this challenge, we propose and implement a loss function for regression uncertainty estimation based on the Bayesian Validation Metric (BVM) framework while using ensemble learning. The proposed loss reproduces maximum likelihood estimation in the limiting case. A series of experiments on in-distribution data show that the proposed method is competitive with existing state-of-the-art methods. Experiments on out-of…
The ever-growing size of the datasets renders well-studied learning techniques, such as Kernel Ridge Regression, inapplicable, posing a serious computational challenge. Divide-and-conquer is a common remedy, suggesting to split the dataset into disjoint partitions, obtain the local estimates and average them, it allows to scale-up an otherwise ineffective base approach. In the current study we suggest a fully data-driven approach to quantify uncertainty of the averaged estimator. Namely, we construct simultaneous element-wise confidence bands for the predictions yielded by the averaged estimat…
While several methods for predicting uncertainty on deep networks have been recently proposed, they do not readily translate to large and complex datasets. In this paper we utilize a simplified form of the Mixture Density Networks (MDNs) to produce a one-shot approach to quantify uncertainty in regression problems. We show that our uncertainty bounds are on-par or better than other reported existing methods. When applied to standard regression benchmark datasets, we show an improvement in predictive log-likelihood and root-mean-square-error when compared to existing state-of-the-art methods. W…
Neural Linear Models (NLM) are deep Bayesian models that produce predictive uncertainties by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused on methodically evaluating the predictive uncertainties of these models. In this work, we demonstrate that traditional training procedures for NLMs drastically underestimate uncertainty on out-of-distribution inputs, and that they therefore cannot be naively deployed in risk-sensitive applications. We identify the underlying reasons for this behavior and p…
Prognostics or Remaining Useful Life (RUL) Estimation from multi-sensor time series data is useful to enable condition-based maintenance and ensure high operational availability of equipment. We propose a novel deep learning based approach for Prognostics with Uncertainty Quantification that is useful in scenarios where: (i) access to labeled failure data is scarce due to rarity of failures (ii) future operational conditions are unobserved and (iii) inherent noise is present in the sensor readings. All three scenarios mentioned are unavoidable sources of uncertainty in the RUL estimation proce…
Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）；University of Bochum（波鸿大学）；Fraunhofer Center for Machine Learning（弗劳恩霍夫机器学习中心）；University of Bonn（波恩大学）
Quantification of uncertainty is one of the most promising approaches to establish safe machine learning. Despite its importance, it is far from being generally solved, especially for neural networks. One of the most commonly used approaches so far is Monte Carlo dropout, which is computationally cheap and easy to apply in practice. However, it can underestimate the uncertainty. We propose a new objective, referred to as second-moment loss (SML), to address this issue. While the full network is encouraged to model the mean, the dropout networks are explicitly used to optimize the model varianc…
We propose a framework for the assessment of uncertainty quantification in deep regression. The framework is based on regression problems where the regression function is a linear combination of nonlinear functions. Basically, any level of complexity can be realized through the choice of the nonlinear functions and the dimensionality of their domain. Results of an uncertainty quantification for deep regression are compared against those obtained by a statistical reference method. The reference method utilizes knowledge of the underlying nonlinear functions and is based on a Bayesian linear reg…
School of Information Technology, Deakin University（迪肯大学信息技术学院）
Deep neural networks tend to underestimate uncertainty and produce overly confident predictions. Recently proposed solutions, such as MC Dropout and SDENet, require complex training and/or auxiliary out-of-distribution data. We propose a simple solution by extending the time-tested iterative reweighted least square (IRLS) in generalised linear regression. We use two sub-networks to parametrise the prediction and uncertainty estimation, enabling easy handling of complex inputs and nonlinear response. The two sub-networks have shared representations and are trained via two complementary loss fun…
University of Pennsylvania（宾夕法尼亚大学）；Purdue University（普渡大学）；University of Michigan（密歇根大学）
Deep neural network (DNN) models have achieved state-of-the-art predictive accuracy in a wide range of supervised learning applications. However, accurately quantifying the uncertainty in DNN predictions remains a challenging task. For continuous outcome variables, an even more difficult problem is to estimate the predictive density function, which not only provides a natural quantification of the predictive uncertainty, but also fully captures the random variation in the outcome. In this work, we propose the Bayesian Deep Noise Neural Network (B-DeepNoise), which generalizes standard Bayesian…
A fundamental principle of learning theory is that there is a trade-off between the complexity of a prediction rule and its ability to generalize. Modern machine learning models do not obey this paradigm: They produce an accurate prediction even with a perfect fit to the training set. We investigate over-parameterized linear regression models focusing on the minimum norm solution: This is the solution with the minimal norm that attains a perfect fit to the training set. We utilize the recently proposed predictive normalized maximum likelihood (pNML) learner which is the min-max regret solution…
Uncertainty Quantification for Regression using Proper Scoring Rules
Department of Machine Learning, MBZUAI（机器学习部门，MBZUAI）；ELLIS Unit Linz and LIT AI Lab, Institute for Machine Learning, Johannes Kepler University Linz（林兹ELLIS单元和LIT AI实验室，机器学习研究所，林兹约翰内斯·开普勒大学）
Accurate quantification of uncertainty is crucial for real-world applications of machine learning. However, modern deep neural networks still produce unreliable predictive uncertainty, often yielding over-confident predictions. In this paper, we are concerned with getting well-calibrated predictions in regression tasks. We propose the calibrated regression method using the maximum mean discrepancy by minimizing the kernel embedding measure. Theoretically, the calibration error of our method asymptotically converges to zero when the sample size is large enough. Experiments on non-trivial real d…
Increasingly high-stakes decisions are made using neural networks in order to make predictions. Specifically, meteorologists and hedge funds apply these techniques to time series data. When it comes to prediction, there are certain limitations for machine learning models (such as lack of expressiveness, vulnerability of domain shifts and overconfidence) which can be solved using uncertainty estimation. There is a set of expectations regarding how uncertainty should ``behave". For instance, a wider prediction horizon should lead to more uncertainty or the model's confidence should be proportion…
Uncertainty Quantification for Regression: A Unified Framework based on kernel scores
LMU Munich（慕尼黑莱茵恩大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；DLR-German Aerospace Center University of Tromsø（德国航空航天中心-特罗姆瑟大学）；German Research Center for Artificial Intelligence (DFKI, DSA)（德国人工智能研究中心）
Among the many ways of quantifying uncertainty in a regression setting, specifying the full quantile function is attractive, as quantiles are amenable to interpretation and evaluation. A model that predicts the true conditional quantiles for each input, at all quantile levels, presents a correct and efficient representation of the underlying uncertainty. To achieve this, many current quantile-based methods focus on optimizing the so-called pinball loss. However, this loss restricts the scope of applicable regression models, limits the ability to target many desirable properties (e.g. calibrati…
Recent years have witnessed an upsurge of interest in employing flexible machine learning models for instrumental variable (IV) regression, but the development of uncertainty quantification methodology is still lacking. In this work we present a novel quasi-Bayesian procedure for IV regression, building upon the recently developed kernelized IV models and the dual/minimax formulation of IV regression. We analyze the frequentist behavior of the proposed method, by establishing minimax optimal contraction rates in $L_2$ and Sobolev norms, and discussing the frequentist validity of credible balls…
This article describes a multivariate polynomial regression method where the uncertainty of the input parameters are approximated with Gaussian distributions, derived from the central limit theorem for large weighted sums, directly from the training sample. The estimated uncertainties can be propagated into the optimal fit function, as an alternative to the statistical bootstrap method. This uncertainty can be propagated further into a loss function like quantity, with which it is possible to calculate the expected loss function, and allows to select the optimal polynomial degree with statisti…
Paris-Saclay University（巴黎萨克雷大学）；ENSTA Paris（巴黎高等先进技术学院）；Institut Polytechnique de Paris（巴黎理工学院）
Monocular depth is important in many tasks, such as 3D reconstruction and autonomous driving. Deep learning based models achieve state-of-the-art performance in this field. A set of novel approaches for estimating monocular depth consists of transforming the regression task into a classification one. However, there is a lack of detailed descriptions and comparisons for Classification Approaches for Regression (CAR) in the community and no in-depth exploration of their potential for uncertainty estimation. To this end, this paper will introduce a taxonomy and summary of CAR approaches, a new un…
University of California, Berkeley（加州大学伯克利分校）；Technion - Israel Institute of Technology（以色列理工学院）
Image-to-image regression is an important learning task, used frequently in biological imaging. Current algorithms, however, do not generally offer statistical guarantees that protect against a model's mistakes and hallucinations. To address this, we develop uncertainty quantification techniques with rigorous statistical guarantees for image-to-image regression problems. In particular, we show how to derive uncertainty intervals around each pixel that are guaranteed to contain the true value with a user-specified confidence probability. Our methods work in conjunction with any base machine lea…
University of Southern California（南加利福尼亚大学）；Amazon Web Services（亚马逊云科技）
Despite impressive state-of-the-art performance on a wide variety of machine learning tasks, deep learning methods can produce over-confident predictions, particularly with limited training data. Therefore, quantifying uncertainty is particularly important in critical applications such as lesion detection and clinical diagnosis, where a realistic assessment of uncertainty is essential in determining surgical margins, disease status and appropriate treatment. In this work, we propose a novel approach that uses quantile regression for quantifying aleatoric uncertainty in both supervised and unsu…
Fully Heteroscedastic Count Regression with Deep Double Poisson Networks
Brigham Young University, Department of Computer Science（Brigham Young 大学计算机科学系）；Arizona State University, School of Computing and Augmented Intelligence（亚利桑那州立大学计算与增强智能学院）；Ohio State University, Department of Marketing（俄亥俄州立大学市场营销系）
Data Intensive Studies Center（数据密集型研究中心）；Tufts University（塔夫茨大学）；University at Buffalo, The State University of New York（纽约州立大学布法罗分校）
In complex physical process characterization, such as the measurement of the regression rate for solid hybrid rocket fuels, where both the observation data and the model used have uncertainties originating from multiple sources, combining these in a systematic way for quantities of interest(QoI) remains a challenge. In this paper, we present a forward propagation uncertainty quantification (UQ) process to produce a probabilistic distribution for the observed regression rate $\dot{r}$. We characterized two input data uncertainty sources from the experiment (the distortion from the camera $U_c$ …
UPLME: Uncertainty-Aware Probabilistic Language Modelling for Robust Empathy Regression
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University（电气工程、计算与数学科学学院，科廷大学）；North South University（北南大学）；BRAC University（布拉克大学）；The Australian National University（澳大利亚国立大学）；University of ÓBuda（奥布达大学）
e:fs TechHub GmbH（e:fs科技中心有限公司）；Ruhr West University of Applied Sciences（鲁尔西区应用科学大学）
Reliable spatial uncertainty evaluation of object detection models is of special interest and has been subject of recent work. In this work, we review the existing definitions for uncertainty calibration of probabilistic regression tasks. We inspect the calibration properties of common detection networks and extend state-of-the-art recalibration methods. Our methods use a Gaussian process (GP) recalibration scheme that yields parametric distributions as output (e.g. Gaussian or Cauchy). The usage of GP recalibration allows for a local (conditional) uncertainty calibration by capturing dependen…
In this paper we consider regression problems subject to arbitrary noise in the operator or design matrix. This characterization appropriately models many physical phenomena with uncertainty in the regressors. Although the problem has been studied extensively for ordinary/total least squares, and via models that implicitly or explicitly assume Gaussianity, less attention has been paid to improving estimation for regression problems under general uncertainty in the design matrix. To address difficulties encountered when dealing with distributions of sums of random variables, we rely on the sadd…
The Alan Turing Institute（艾伦图灵研究所）；Imperial College London（伦敦帝国理工学院）
We introduce a novel rule-based approach for handling regression problems. The new methodology carries elements from two frameworks: (i) it provides information about the uncertainty of the parameters of interest using Bayesian inference, and (ii) it allows the incorporation of expert knowledge through rule-based systems. The blending of those two different frameworks can be particularly beneficial for various domains (e.g. engineering), where, even though the significance of uncertainty quantification motivates a Bayesian approach, there is no simple way to incorporate researcher intuition in…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Fabrication process variations can significantly influence the performance and yield of nano-scale electronic and photonic circuits. Stochastic spectral methods have achieved great success in quantifying the impact of process variations, but they suffer from the curse of dimensionality. Recently, low-rank tensor methods have been developed to mitigate this issue, but two fundamental challenges remain open: how to automatically determine the tensor rank and how to adaptively pick the informative simulation samples. This paper proposes a novel tensor regression method to address these two challe…
Deterministic neural networks (NNs) are increasingly being deployed in safety critical domains, where calibrated, robust, and efficient measures of uncertainty are crucial. In this paper, we propose a novel method for training non-Bayesian NNs to estimate a continuous target as well as its associated evidence in order to learn both aleatoric and epistemic uncertainty. We accomplish this by placing evidential priors over the original Gaussian likelihood function and training the NN to infer the hyperparameters of the evidential distribution. We additionally impose priors during training such th…
We reexamine the classical linear regression model when the model is subject to two types of uncertainty: (i) some of covariates are either missing or completely inaccessible, and (ii) the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician. By following the recent theory of sublinear expectation, we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables, which encompass an infinite family of probability distributions for the response variable in the sense of …
Salesforce Research（赛富时研究院）；University of California, Berkeley（加州大学伯克利分校）
Estimating the data uncertainty in regression tasks is often done by learning a quantile function or a prediction interval of the true label conditioned on the input. It is frequently observed that quantile regression -- a vanilla algorithm for learning quantiles with asymptotic guarantees -- tends to \emph{under-cover} than the desired coverage level in reality. While various fixes have been proposed, a more fundamental understanding of why this under-coverage bias happens in the first place remains elusive. In this paper, we present a rigorous theoretical study on the coverage of uncertainty…
In scientific applications, there often are several competing models that could be fit to the observed data, so quantification of the model uncertainty is of fundamental importance. In this paper, we develop an inferential model (IM) approach for simultaneously valid probabilistic inference over a collection of assertions of interest without requiring any prior input. Our construction guarantees that the approach is optimal in the sense that it is the most efficient among those which are valid. Connections between the IM's simultaneous validity and post-selection inference are also made. We ap…
Bonn-Rhein-Sieg University of Applied Sciences（波恩-莱茵-锡格应用技术大学）；German Aerospace Center (DLR)（德国航空航天中心）；University of Luxembourg（卢森堡大学）
We benchmark the robustness of maximum likelihood based uncertainty estimation methods to outliers in training data for regression tasks. Outliers or noisy labels in training data results in degraded performances as well as incorrect estimation of uncertainty. We propose the use of a heavy-tailed distribution (Laplace distribution) to improve the robustness to outliers. This property is evaluated using standard regression benchmarks and on a high-dimensional regression task of monocular depth estimation, both containing outliers. In particular, heavy-tailed distribution based maximum likelihoo…
