Interval Deep Learning for Uncertainty Quantification in Safety Applications2021-05-13 ${\displaystyle \cong }$ |

Deep neural networks (DNNs) are becoming more prevalent in important safety-critical applications, where reliability in the prediction is paramount. Despite their exceptional prediction capabilities, current DNNs do not have an implicit mechanism to quantify and propagate significant input data uncertainty -- which is common in safety-critical applications. In many cases, this uncertainty is epistemic and can arise from multiple sources, such as lack of knowledge about the data generating process, imprecision, ignorance, and poor understanding of physics phenomena. Recent approaches have focused on quantifying parameter uncertainty, but approaches to end-to-end training of DNNs with epistemic input data uncertainty are more limited and largely problem-specific. In this work, we present a DNN optimized with gradient-based methods capable to quantify input and parameter uncertainty by means of interval analysis, which we call Deep Interval Neural Network (DINN). We perform experiments on an air pollution dataset with sensor uncertainty and show that the DINN can produce accurate bounded estimates from uncertain input data. |

Exploring Uncertainty in Deep Learning for Construction of Prediction Intervals2021-04-26 ${\displaystyle \cong }$ |

Deep learning has achieved impressive performance on many tasks in recent years. However, it has been found that it is still not enough for deep neural networks to provide only point estimates. For high-risk tasks, we need to assess the reliability of the model predictions. This requires us to quantify the uncertainty of model prediction and construct prediction intervals. In this paper, We explore the uncertainty in deep learning to construct the prediction intervals. In general, We comprehensively consider two categories of uncertainties: aleatory uncertainty and epistemic uncertainty. We design a special loss function, which enables us to learn uncertainty without uncertainty label. We only need to supervise the learning of regression task. We learn the aleatory uncertainty implicitly from the loss function. And that epistemic uncertainty is accounted for in ensembled form. Our method correlates the construction of prediction intervals with the uncertainty estimation. Impressive results on some publicly available datasets show that the performance of our method is competitive with other state-of-the-art methods. |

STUaNet: Understanding uncertainty in spatiotemporal collective human mobility2021-02-08 ${\displaystyle \cong }$ |

The high dynamics and heterogeneous interactions in the complicated urban systems have raised the issue of uncertainty quantification in spatiotemporal human mobility, to support critical decision-makings in risk-aware web applications such as urban event prediction where fluctuations are of significant interests. Given the fact that uncertainty quantifies the potential variations around prediction results, traditional learning schemes always lack uncertainty labels, and conventional uncertainty quantification approaches mostly rely upon statistical estimations with Bayesian Neural Networks or ensemble methods. However, they have never involved any spatiotemporal evolution of uncertainties under various contexts, and also have kept suffering from the poor efficiency of statistical uncertainty estimation while training models with multiple times. To provide high-quality uncertainty quantification for spatiotemporal forecasting, we propose an uncertainty learning mechanism to simultaneously estimate internal data quality and quantify external uncertainty regarding various contextual interactions. To address the issue of lacking labels of uncertainty, we propose a hierarchical data turbulence scheme where we can actively inject controllable uncertainty for guidance, and hence provide insights to both uncertainty quantification and weak supervised learning. Finally, we re-calibrate and boost the prediction performance by devising a gated-based bridge to adaptively leverage the learned uncertainty into predictions. Extensive experiments on three real-world spatiotemporal mobility sets have corroborated the superiority of our proposed model in terms of both forecasting and uncertainty quantification. |

Uncertainty Modelling in Deep Neural Networks for Image Data2020-11-17 ${\displaystyle \cong }$ |

Quantifying uncertainty in a model's predictions is important as it enables, for example, the safety of an AI system to be increased by acting on the model's output in an informed manner. We cannot expect a system to be 100% accurate or perfect at its task, however, we can equip the system with some tools to inform us if it is not certain about a prediction. This way, a second check can be performed, or the task can be passed to a human specialist. This is crucial for applications where the cost of an error is high, such as in autonomous vehicle control, medical image analysis, financial estimations or legal fields. Deep Neural Networks are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in DNNs is a challenging and yet on-going problem. Although there have been many efforts to equip NNs with tools to estimate uncertainty, such as Monte Carlo Dropout, most of the previous methods only focus on one of the three types of model, data or distributional uncertainty. In this paper we propose a complete framework to capture and quantify all of these three types of uncertainties in DNNs for image classification. This framework includes an ensemble of CNNs for model uncertainty, a supervised reconstruction auto-encoder to capture distributional uncertainty and using the output of activation functions in the last layer of the network, to capture data uncertainty. Finally we demonstrate the efficiency of our method on popular image datasets for classification. |

DEUP: Direct Epistemic Uncertainty Prediction2021-02-16 ${\displaystyle \cong }$ |

Epistemic uncertainty is the part of out-of-sample prediction error due to the lack of knowledge of the learner. Whereas previous work was focusing on model variance, we propose a principled approach for directly estimating epistemic uncertainty by learning to predict generalization error and subtracting an estimate of aleatoric uncertainty, i.e., intrinsic unpredictability. This estimator of epistemic uncertainty includes the effect of model bias and can be applied in non-stationary learning environments arising in active learning or reinforcement learning. In addition to demonstrating these properties of Direct Epistemic Uncertainty Prediction (DEUP), we illustrate its advantage against existing methods for uncertainty estimation on downstream tasks including sequential model optimization and reinforcement learning. We also evaluate the quality of uncertainty estimates from DEUP for probabilistic classification of images and for estimating uncertainty about synergistic drug combinations. |

Uncertainty Quantification in Deep Learning for Safer Neuroimage Enhancement2019-07-31 ${\displaystyle \cong }$ |

Deep learning (DL) has shown great potential in medical image enhancement problems, such as super-resolution or image synthesis. However, to date, little consideration has been given to uncertainty quantification over the output image. Here we introduce methods to characterise different components of uncertainty in such problems and demonstrate the ideas using diffusion MRI super-resolution. Specifically, we propose to account for $intrinsic$ uncertainty through a heteroscedastic noise model and for $parameter$ uncertainty through approximate Bayesian inference, and integrate the two to quantify $predictive$ uncertainty over the output image. Moreover, we introduce a method to propagate the predictive uncertainty on a multi-channelled image to derived scalar parameters, and separately quantify the effects of intrinsic and parameter uncertainty therein. The methods are evaluated for super-resolution of two different signal representations of diffusion MR images---DTIs and Mean Apparent Propagator MRI---and their derived quantities such as MD and FA, on multiple datasets of both healthy and pathological human brains. Results highlight three key benefits of uncertainty modelling for improving the safety of DL-based image enhancement systems. Firstly, incorporating uncertainty improves the predictive performance even when test data departs from training data. Secondly, the predictive uncertainty highly correlates with errors, and is therefore capable of detecting predictive "failures". Results demonstrate that such an uncertainty measure enables subject-specific and voxel-wise risk assessment of the output images. Thirdly, we show that the method for decomposing predictive uncertainty into its independent sources provides high-level "explanations" for the performance by quantifying how much uncertainty arises from the inherent difficulty of the task or the limited training examples. |

Real-time Uncertainty Decomposition for Online Learning Control2020-10-06 ${\displaystyle \cong }$ |

Safety-critical decisions based on machine learning models require a clear understanding of the involved uncertainties to avoid hazardous or risky situations. While aleatoric uncertainty can be explicitly modeled given a parametric description, epistemic uncertainty rather describes the presence or absence of training data. This paper proposes a novel generic method for modeling epistemic uncertainty and shows its advantages over existing approaches for neural networks on various data sets. It can be directly combined with aleatoric uncertainty estimates and allows for prediction in real-time as the inference is sample-free. We exploit this property in a model-based quadcopter control setting and demonstrate how the controller benefits from a differentiation between aleatoric and epistemic uncertainty in online learning of thermal disturbances. |

Probabilistic Neighbourhood Component Analysis: Sample Efficient Uncertainty Estimation in Deep Learning2020-07-18 ${\displaystyle \cong }$ |

While Deep Neural Networks (DNNs) achieve state-of-the-art accuracy in various applications, they often fall short in accurately estimating their predictive uncertainty and, in turn, fail to recognize when these predictions may be wrong. Several uncertainty-aware models, such as Bayesian Neural Network (BNNs) and Deep Ensembles have been proposed in the literature for quantifying predictive uncertainty. However, research in this area has been largely confined to the big data regime. In this work, we show that the uncertainty estimation capability of state-of-the-art BNNs and Deep Ensemble models degrades significantly when the amount of training data is small. To address the issue of accurate uncertainty estimation in the small-data regime, we propose a probabilistic generalization of the popular sample-efficient non-parametric kNN approach. Our approach enables deep kNN classifier to accurately quantify underlying uncertainties in its prediction. We demonstrate the usefulness of the proposed approach by achieving superior uncertainty quantification as compared to state-of-the-art on a real-world application of COVID-19 diagnosis from chest X-Rays. Our code is available at https://github.com/ankurmallick/sample-efficient-uq |

Objective Evaluation of Deep Uncertainty Predictions for COVID-19 Detection2020-12-22 ${\displaystyle \cong }$ |

Deep neural networks (DNNs) have been widely applied for detecting COVID-19 in medical images. Existing studies mainly apply transfer learning and other data representation strategies to generate accurate point estimates. The generalization power of these networks is always questionable due to being developed using small datasets and failing to report their predictive confidence. Quantifying uncertainties associated with DNN predictions is a prerequisite for their trusted deployment in medical settings. Here we apply and evaluate three uncertainty quantification techniques for COVID-19 detection using chest X-Ray (CXR) images. The novel concept of uncertainty confusion matrix is proposed and new performance metrics for the objective evaluation of uncertainty estimates are introduced. Through comprehensive experiments, it is shown that networks pertained on CXR images outperform networks pretrained on natural image datasets such as ImageNet. Qualitatively and quantitatively evaluations also reveal that the predictive uncertainty estimates are statistically higher for erroneous predictions than correct predictions. Accordingly, uncertainty quantification methods are capable of flagging risky predictions with high uncertainty estimates. We also observe that ensemble methods more reliably capture uncertainties during the inference. |

Fail-Safe Execution of Deep Learning based Systems through Uncertainty Monitoring2021-02-01 ${\displaystyle \cong }$ |

Modern software systems rely on Deep Neural Networks (DNN) when processing complex, unstructured inputs, such as images, videos, natural language texts or audio signals. Provided the intractably large size of such input spaces, the intrinsic limitations of learning algorithms, and the ambiguity about the expected predictions for some of the inputs, not only there is no guarantee that DNN's predictions are always correct, but rather developers must safely assume a low, though not negligible, error probability. A fail-safe Deep Learning based System (DLS) is one equipped to handle DNN faults by means of a supervisor, capable of recognizing predictions that should not be trusted and that should activate a healing procedure bringing the DLS to a safe state. In this paper, we propose an approach to use DNN uncertainty estimators to implement such a supervisor. We first discuss the advantages and disadvantages of existing approaches to measure uncertainty for DNNs and propose novel metrics for the empirical assessment of the supervisor that rely on such approaches. We then describe our publicly available tool UNCERTAINTY-WIZARD, which allows transparent estimation of uncertainty for regular tf.keras DNNs. Lastly, we discuss a large-scale study conducted on four different subjects to empirically validate the approach, reporting the lessons-learned as guidance for software engineers who intend to monitor uncertainty for fail-safe execution of DLS. |

Uncertainty-Aware CNNs for Depth Completion: Uncertainty from Beginning to End2020-06-05 ${\displaystyle \cong }$ |

The focus in deep learning research has been mostly to push the limits of prediction accuracy. However, this was often achieved at the cost of increased complexity, raising concerns about the interpretability and the reliability of deep networks. Recently, an increasing attention has been given to untangling the complexity of deep networks and quantifying their uncertainty for different computer vision tasks. Differently, the task of depth completion has not received enough attention despite the inherent noisy nature of depth sensors. In this work, we thus focus on modeling the uncertainty of depth data in depth completion starting from the sparse noisy input all the way to the final prediction. We propose a novel approach to identify disturbed measurements in the input by learning an input confidence estimator in a self-supervised manner based on the normalized convolutional neural networks (NCNNs). Further, we propose a probabilistic version of NCNNs that produces a statistically meaningful uncertainty measure for the final prediction. When we evaluate our approach on the KITTI dataset for depth completion, we outperform all the existing Bayesian Deep Learning approaches in terms of prediction accuracy, quality of the uncertainty measure, and the computational efficiency. Moreover, our small network with 670k parameters performs on-par with conventional approaches with millions of parameters. These results give strong evidence that separating the network into parallel uncertainty and prediction streams leads to state-of-the-art performance with accurate uncertainty estimates. |

Validating uncertainty in medical image translation2020-02-11 ${\displaystyle \cong }$ |

Medical images are increasingly used as input to deep neural networks to produce quantitative values that aid researchers and clinicians. However, standard deep neural networks do not provide a reliable measure of uncertainty in those quantitative values. Recent work has shown that using dropout during training and testing can provide estimates of uncertainty. In this work, we investigate using dropout to estimate epistemic and aleatoric uncertainty in a CT-to-MR image translation task. We show that both types of uncertainty are captured, as defined, providing confidence in the output uncertainty estimates. |

A Comparison of Uncertainty Estimation Approaches in Deep Learning Components for Autonomous Vehicle Applications2020-07-02 ${\displaystyle \cong }$ |

A key factor for ensuring safety in Autonomous Vehicles (AVs) is to avoid any abnormal behaviors under undesirable and unpredicted circumstances. As AVs increasingly rely on Deep Neural Networks (DNNs) to perform safety-critical tasks, different methods for uncertainty quantification have recently been proposed to measure the inevitable source of errors in data and models. However, uncertainty quantification in DNNs is still a challenging task. These methods require a higher computational load, a higher memory footprint, and introduce extra latency, which can be prohibitive in safety-critical applications. In this paper, we provide a brief and comparative survey of methods for uncertainty quantification in DNNs along with existing metrics to evaluate uncertainty predictions. We are particularly interested in understanding the advantages and downsides of each method for specific AV tasks and types of uncertainty sources. |

Estimating Risk and Uncertainty in Deep Reinforcement Learning2020-02-17 ${\displaystyle \cong }$ |

We propose a method for disentangling epistemic and aleatoric uncertainties in deep reinforcement learning. Aleatoric uncertainty, or risk, which arises from inherently stochastic environments or agents, must be accounted for in the design of risk-sensitive algorithms. Epistemic uncertainty, which stems from limited data, is important both for risk-sensitivity and for efficient exploration. Our method combines elements from distributional reinforcement learning and approximate Bayesian inference techniques with neural networks, allowing us to disentangle both types of uncertainty on the expected return of a policy. Specifically, the learned return distribution provides the aleatoric uncertainty, and the Bayesian posterior yields the epistemic uncertainty. Although our approach in principle requires a large number of samples from the Bayesian posterior to estimate the epistemic uncertainty, we show that two networks already yield a useful approximation. We perform experiments that illustrate our method and some applications. |

Regression with Uncertainty Quantification in Large Scale Complex Data2019-12-04 ${\displaystyle \cong }$ |

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. We also demonstrate this method's efficacy on stochastic, highly volatile time-series data where stock prices are predicted for the next time interval. The resulting uncertainty graph summarizes significant anomalies in the stock price chart. Furthermore, we apply this method to the task of age estimation from the challenging IMDb-Wiki dataset of half a million face images. We successfully predict the uncertainties associated with the prediction and empirically analyze the underlying causes of the uncertainties. This uncertainty quantification can be used to pre-process low quality datasets and further enable learning. |

Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness2020-06-17 ${\displaystyle \cong }$ |

Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer with a Gaussian process. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches. |

SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates2020-08-24 ${\displaystyle \cong }$ |

Uncertainty quantification is a fundamental yet unsolved problem for deep learning. The Bayesian framework provides a principled way of uncertainty estimation but is often not scalable to modern deep neural nets (DNNs) that have a large number of parameters. Non-Bayesian methods are simple to implement but often conflate different sources of uncertainties and require huge computing resources. We propose a new method for quantifying uncertainties of DNNs from a dynamical system perspective. The core of our method is to view DNN transformations as state evolution of a stochastic dynamical system and introduce a Brownian motion term for capturing epistemic uncertainty. Based on this perspective, we propose a neural stochastic differential equation model (SDE-Net) which consists of (1) a drift net that controls the system to fit the predictive function; and (2) a diffusion net that captures epistemic uncertainty. We theoretically analyze the existence and uniqueness of the solution to SDE-Net. Our experiments demonstrate that the SDE-Net model can outperform existing uncertainty estimation methods across a series of tasks where uncertainty plays a fundamental role. |

Computer Model Calibration with Time Series Data using Deep Learning and Quantile Regression2020-09-08 ${\displaystyle \cong }$ |

Computer models play a key role in many scientific and engineering problems. One major source of uncertainty in computer model experiment is input parameter uncertainty. Computer model calibration is a formal statistical procedure to infer input parameters by combining information from model runs and observational data. The existing standard calibration framework suffers from inferential issues when the model output and observational data are high-dimensional dependent data such as large time series due to the difficulty in building an emulator and the non-identifiability between effects from input parameters and data-model discrepancy. To overcome these challenges we propose a new calibration framework based on a deep neural network (DNN) with long-short term memory layers that directly emulates the inverse relationship between the model output and input parameters. Adopting the 'learning with noise' idea we train our DNN model to filter out the effects from data model discrepancy on input parameter inference. We also formulate a new way to construct interval predictions for DNN using quantile regression to quantify the uncertainty in input parameter estimates. Through a simulation study and real data application with WRF-hydro model we show that our approach can yield accurate point estimates and well calibrated interval estimates for input parameters. |

Approaching Neural Network Uncertainty Realism2021-01-08 ${\displaystyle \cong }$ |

Statistical models are inherently uncertain. Quantifying or at least upper-bounding their uncertainties is vital for safety-critical systems such as autonomous vehicles. While standard neural networks do not report this information, several approaches exist to integrate uncertainty estimates into them. Assessing the quality of these uncertainty estimates is not straightforward, as no direct ground truth labels are available. Instead, implicit statistical assessments are required. For regression, we propose to evaluate uncertainty realism -- a strict quality criterion -- with a Mahalanobis distance-based statistical test. An empirical evaluation reveals the need for uncertainty measures that are appropriate to upper-bound heavy-tailed empirical errors. Alongside, we transfer the variational U-Net classification architecture to standard supervised image-to-image tasks. We adopt it to the automotive domain and show that it significantly improves uncertainty realism compared to a plain encoder-decoder model. |

Towards calibrated and scalable uncertainty representations for neural networks2019-12-03 ${\displaystyle \cong }$ |

For many applications it is critical to know the uncertainty of a neural network's predictions. While a variety of neural network parameter estimation methods have been proposed for uncertainty estimation, they have not been rigorously compared across uncertainty measures. We assess four of these parameter estimation methods to calibrate uncertainty estimation using four different uncertainty measures: entropy, mutual information, aleatoric uncertainty and epistemic uncertainty. We evaluate the calibration of these parameter estimation methods using expected calibration error. Additionally, we propose a novel method of neural network parameter estimation called RECAST, which combines cosine annealing with warm restarts with Stochastic Gradient Langevin Dynamics, capturing more diverse parameter distributions. When benchmarked against mutilated image data, we show that RECAST is well-calibrated and when combined with predictive entropy and epistemic uncertainty it offers the best calibrated measure of uncertainty when compared to recent methods. |