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. |

Improving model calibration with accuracy versus uncertainty optimization2020-12-14 ${\displaystyle \cong }$ |

Obtaining reliable and accurate quantification of uncertainty estimates from deep neural networks is important in safety-critical applications. A well-calibrated model should be accurate when it is certain about its prediction and indicate high uncertainty when it is likely to be inaccurate. Uncertainty calibration is a challenging problem as there is no ground truth available for uncertainty estimates. We propose an optimization method that leverages the relationship between accuracy and uncertainty as an anchor for uncertainty calibration. We introduce a differentiable accuracy versus uncertainty calibration (AvUC) loss function that allows a model to learn to provide well-calibrated uncertainties, in addition to improved accuracy. We also demonstrate the same methodology can be extended to post-hoc uncertainty calibration on pretrained models. We illustrate our approach with mean-field stochastic variational inference and compare with state-of-the-art methods. Extensive experiments demonstrate our approach yields better model calibration than existing methods on large-scale image classification tasks under distributional shift. |

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. |

Real-time parameter inference in reduced-order flame models with heteroscedastic Bayesian neural network ensembles2020-10-11 ${\displaystyle \cong }$ |

The estimation of model parameters with uncertainties from observed data is a ubiquitous inverse problem in science and engineering. In this paper, we suggest an inexpensive and easy to implement parameter estimation technique that uses a heteroscedastic Bayesian Neural Network trained using anchored ensembling. The heteroscedastic aleatoric error of the network models the irreducible uncertainty due to parameter degeneracies in our inverse problem, while the epistemic uncertainty of the Bayesian model captures uncertainties which may arise from an input observation's out-of-distribution nature. We use this tool to perform real-time parameter inference in a 6 parameter G-equation model of a ducted, premixed flame from observations of acoustically excited flames. We train our networks on a library of 2.1 million simulated flame videos. Results on the test dataset of simulated flames show that the network recovers flame model parameters, with the correlation coefficient between predicted and true parameters ranging from 0.97 to 0.99, and well-calibrated uncertainty estimates. The trained neural networks are then used to infer model parameters from real videos of a premixed Bunsen flame captured using a high-speed camera in our lab. Re-simulation using inferred parameters shows excellent agreement between the real and simulated flames. Compared to Ensemble Kalman Filter-based tools that have been proposed for this problem in the combustion literature, our neural network ensemble achieves better data-efficiency and our sub-millisecond inference times represent a savings on computational costs by several orders of magnitude. This allows us to calibrate our reduced-order flame model in real-time and predict the thermoacoustic instability behaviour of the flame more accurately. |

Variational Calibration of Computer Models2018-10-29 ${\displaystyle \cong }$ |

Bayesian calibration of black-box computer models offers an established framework to obtain a posterior distribution over model parameters. Traditional Bayesian calibration involves the emulation of the computer model and an additive model discrepancy term using Gaussian processes; inference is then carried out using MCMC. These choices pose computational and statistical challenges and limitations, which we overcome by proposing the use of approximate Deep Gaussian processes and variational inference techniques. The result is a practical and scalable framework for calibration, which obtains competitive performance compared to the state-of-the-art. |

Improving Classifier Confidence using Lossy Label-Invariant Transformations2020-11-08 ${\displaystyle \cong }$ |

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 when applied to models with large input spaces (e.g., ImageNet). In this paper, we present the recursive lossy label-invariant calibration (ReCal) technique that leverages label-invariant transformations of the input that induce a loss of discriminatory information to recursively group (and calibrate) inputs - without requiring model retraining. We show that ReCal outperforms other calibration methods on multiple datasets, especially, on large-scale datasets such as ImageNet. |

Calibrating Deep Neural Network Classifiers on Out-of-Distribution Datasets2020-06-16 ${\displaystyle \cong }$ |

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 propose a new post-hoc confidence calibration method, called CCAC (Confidence Calibration with an Auxiliary Class), for DNN classifiers on OOD datasets. The key novelty of CCAC is an auxiliary class in the calibration model which separates mis-classified samples from correctly classified ones, thus effectively mitigating the target DNN's being confidently wrong. We also propose a simplified version of CCAC to reduce free parameters and facilitate transfer to a new unseen dataset. Our experiments on different DNN models, datasets and applications show that CCAC can consistently outperform the prior post-hoc calibration methods. |

Ensemble-Compression: A New Method for Parallel Training of Deep Neural Networks2017-07-18 ${\displaystyle \cong }$ |

Parallelization framework has become a necessity to speed up the training of deep neural networks (DNN) recently. Such framework typically employs the Model Average approach, denoted as MA-DNN, in which parallel workers conduct respective training based on their own local data while the parameters of local models are periodically communicated and averaged to obtain a global model which serves as the new start of local models. However, since DNN is a highly non-convex model, averaging parameters cannot ensure that such global model can perform better than those local models. To tackle this problem, we introduce a new parallel training framework called Ensemble-Compression, denoted as EC-DNN. In this framework, we propose to aggregate the local models by ensemble, i.e., averaging the outputs of local models instead of the parameters. As most of prevalent loss functions are convex to the output of DNN, the performance of ensemble-based global model is guaranteed to be at least as good as the average performance of local models. However, a big challenge lies in the explosion of model size since each round of ensemble can give rise to multiple times size increment. Thus, we carry out model compression after each ensemble, specialized by a distillation based method in this paper, to reduce the size of the global model to be the same as the local ones. Our experimental results demonstrate the prominent advantage of EC-DNN over MA-DNN in terms of both accuracy and speedup. |

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. |

CRUDE: Calibrating Regression Uncertainty Distributions Empirically2020-07-03 ${\displaystyle \cong }$ |

The importance of calibrated uncertainty estimates in machine learning is growing apparent across 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: Calibrating Regression Uncertainty Distributions Empirically (CRUDE). CRUDE makes the weaker assumption that error distributions have a constant arbitrary shape across the output space, shifted by predicted mean and scaled by predicted standard deviation. CRUDE requires no training of the calibration estimator, aside from a parameter to account for fixed bias in the predicted mean. Across an extensive set of regression tasks, CRUDE demonstrates consistently sharper, better calibrated, and more accurate uncertainty estimates than state-of-the-art techniques. |

Scalable Bayesian neural networks by layer-wise input augmentation2020-10-26 ${\displaystyle \cong }$ |

We introduce implicit Bayesian neural networks, a simple and scalable approach for uncertainty representation in deep learning. Standard Bayesian approach to deep learning requires the impractical inference of the posterior distribution over millions of parameters. Instead, we propose to induce a distribution that captures the uncertainty over neural networks by augmenting each layer's inputs with latent variables. We present appropriate input distributions and demonstrate state-of-the-art performance in terms of calibration, robustness and uncertainty characterisation over large-scale, multi-million parameter image classification tasks. |

Reliable Uncertainties for Bayesian Neural Networks using Alpha-divergences2020-08-15 ${\displaystyle \cong }$ |

Bayesian Neural Networks (BNNs) often result uncalibrated after training, usually tending towards overconfidence. Devising effective calibration methods with low impact in terms of computational complexity is thus of central interest. In this paper we present calibration methods for BNNs based on the alpha divergences from Information Geometry. We compare the use of alpha divergence in training and in calibration, and we show how the use in calibration provides better calibrated uncertainty estimates for specific choices of alpha and is more efficient especially for complex network architectures. We empirically demonstrate the advantages of alpha calibration in regression problems involving parameter estimation and inferred correlations between output uncertainties. |

Amortized Conditional Normalized Maximum Likelihood2020-11-05 ${\displaystyle \cong }$ |

While deep neural networks provide good performance for a range of challenging tasks, calibration and uncertainty estimation remain major challenges. In this paper, we propose the amortized conditional normalized maximum likelihood (ACNML) method as a scalable general-purpose approach for uncertainty estimation, calibration, and out-of-distribution robustness with deep networks. Our algorithm builds on the conditional normalized maximum likelihood (CNML) coding scheme, which has minimax optimal properties according to the minimum description length principle, but is computationally intractable to evaluate exactly for all but the simplest of model classes. We propose to use approximate Bayesian inference technqiues to produce a tractable approximation to the CNML distribution. Our approach can be combined with any approximate inference algorithm that provides tractable posterior densities over model parameters. We demonstrate that ACNML compares favorably to a number of prior techniques for uncertainty estimation in terms of calibration on out-of-distribution inputs. |

A Causal Lens for Peeking into Black Box Predictive Models: Predictive Model Interpretation via Causal Attribution2020-08-01 ${\displaystyle \cong }$ |

With the increasing adoption of predictive models trained using machine learning across a wide range of high-stakes applications, e.g., health care, security, criminal justice, finance, and education, there is a growing need for effective techniques for explaining such models and their predictions. We aim to address this problem in settings where the predictive model is a black box; That is, we can only observe the response of the model to various inputs, but have no knowledge about the internal structure of the predictive model, its parameters, the objective function, and the algorithm used to optimize the model. We reduce the problem of interpreting a black box predictive model to that of estimating the causal effects of each of the model inputs on the model output, from observations of the model inputs and the corresponding outputs. We estimate the causal effects of model inputs on model output using variants of the Rubin Neyman potential outcomes framework for estimating causal effects from observational data. We show how the resulting causal attribution of responsibility for model output to the different model inputs can be used to interpret the predictive model and to explain its predictions. We present results of experiments that demonstrate the effectiveness of our approach to the interpretation of black box predictive models via causal attribution in the case of deep neural network models trained on one synthetic data set (where the input variables that impact the output variable are known by design) and two real-world data sets: Handwritten digit classification, and Parkinson's disease severity prediction. Because our approach does not require knowledge about the predictive model algorithm and is free of assumptions regarding the black box predictive model except that its input-output responses be observable, it can be applied, in principle, to any black box predictive model. |

Uncertainty Propagation in Deep Neural Network Using Active Subspace2020-01-11 ${\displaystyle \cong }$ |

The inputs of deep neural network (DNN) from real-world data usually come with uncertainties. Yet, it is challenging to propagate the uncertainty in the input features to the DNN predictions at a low computational cost. This work employs a gradient-based subspace method and response surface technique to accelerate the uncertainty propagation in DNN. Specifically, the active subspace method is employed to identify the most important subspace in the input features using the gradient of the DNN output to the inputs. Then the response surface within that low-dimensional subspace can be efficiently built, and the uncertainty of the prediction can be acquired by evaluating the computationally cheap response surface instead of the DNN models. In addition, the subspace can help explain the adversarial examples. The approach is demonstrated in MNIST datasets with a convolutional neural network. Code is available at: https://github.com/jiweiqi/nnsubspace. |

Integration of AI and mechanistic modeling in generative adversarial networks for stochastic inverse problems2020-09-17 ${\displaystyle \cong }$ |

The problem of finding distributions of input parameters for deterministic mechanistic models to match distributions of model outputs to stochastic observations, i.e., the "Stochastic Inverse Problem" (SIP), encompasses a range of common tasks across a variety of scientific disciplines. Here, we demonstrate that SIP could be reformulated as a constrained optimization problem and adapted for applications in intervention studies to simultaneously infer model input parameters for two sets of observations, under control conditions and under an intervention. In the constrained optimization problem, the solution of SIP is enforced to accommodate the prior knowledge on the model input parameters and to produce outputs consistent with given observations by minimizing the divergence between the inferred distribution of input parameters and the prior. Unlike in standard SIP, the prior incorporates not only knowledge about model input parameters for objects in each set, but also information on the joint distribution or the deterministic map between the model input parameters in two sets of observations. To solve standard and intervention SIP, we employed conditional generative adversarial networks (GANs) and designed novel GANs that incorporate multiple generators and discriminators and have structures that reflect the underlying constrained optimization problems. This reformulation allows us to build computationally scalable solutions to tackle complex model input parameter inference scenarios, which appear routinely in physics, biophysics, economics and other areas, and which currently could not be handled with existing methods. |

From parameter calibration to parameter learning: Revolutionizing large-scale geoscientific modeling with big data2020-07-30 ${\displaystyle \cong }$ |

The behaviors and skills of models in many geoscientific domains strongly depend on spatially varying parameters that lack direct observations and must be determined by calibration. Calibration, which solves inverse problems, is a classical but inefficient and stochasticity-ridden approach to reconcile models and observations. Using a widely applied hydrologic model and soil moisture observations as a case study, here we propose a novel, forward-mapping parameter learning (fPL) framework. Whereas evolutionary algorithm (EA)-based calibration solves inversion problems one by one, fPL solves a pattern recognition problem and learns a more robust, universal mapping. fPL can save orders-of-magnitude computational time compared to EA-based calibration, while, surprisingly, producing equivalent ending skill metrics. With more training data, fPL learned across sites and showed super-convergence, scaling much more favorably. Moreover, a more important benefit emerged: fPL produced spatially-coherent parameters in better agreement with physical processes. As a result, it demonstrated better results for out-of-training-set locations and uncalibrated variables. Compared to purely data-driven models, fPL can output unobserved variables, in this case simulated evapotranspiration, which agrees better with satellite-based estimates than the comparison EA. The deep-learning-powered fPL frameworks can be uniformly applied to myriad other geoscientific models. We contend that a paradigm shift from inverse parameter calibration to parameter learning will greatly propel various geoscientific domains. |

Automatic Calibration of Dynamic and Heterogeneous Parameters in Agent-based Model2019-08-09 ${\displaystyle \cong }$ |

While simulations have been utilized in diverse domains, such as urban growth modeling, market dynamics modeling, etc; some of these applications may require validations based upon some real-world observations modeled in the simulation, as well. This validation has been categorized into either qualitative face-validation or quantitative empirical validation, but as the importance and the accumulation of data grows, the importance of the quantitative validation has been highlighted in the recent studies, i.e. digital twin. The key component of quantitative validation is finding a calibrated set of parameters to regenerate the real-world observations with simulation models. While this parameter calibration has been fixed throughout a simulation execution, this paper expands the static parameter calibration in two dimensions: dynamic calibration and heterogeneous calibration. First, dynamic calibration changes the parameter values over the simulation period by reflecting the simulation output trend. Second, heterogeneous calibration changes the parameter values per simulated entity clusters by considering the similarities of entity states. We experimented the suggested calibrations on one hypothetical case and another real-world case. As a hypothetical scenario, we use the Wealth Distribution Model to illustrate how our calibration works. As a real-world scenario, we selected Real Estate Market Model because of three reasons. First, the models have heterogeneous entities as being agent-based models; second, they are economic models with real-world trends over time; and third, they are applicable to the real-world scenarios where we can gather validation data. |

Distribution Calibration for Regression2019-05-15 ${\displaystyle \cong }$ |

We are concerned with obtaining well-calibrated output distributions from regression models. Such distributions allow us to quantify the uncertainty that the model has regarding the predicted target value. We introduce the novel concept of distribution calibration, and demonstrate its advantages over the existing definition of quantile calibration. We further propose a post-hoc approach to improving the predictions from previously trained regression models, using multi-output Gaussian Processes with a novel Beta link function. The proposed method is experimentally verified on a set of common regression models and shows improvements for both distribution-level and quantile-level calibration. |

X-CAL: Explicit Calibration for Survival Analysis2021-01-13 ${\displaystyle \cong }$ |

Survival analysis models the distribution of time until an event of interest, such as discharge from the hospital or admission to the ICU. When a model's predicted number of events within any time interval is similar to the observed number, it is called well-calibrated. A survival model's calibration can be measured using, for instance, distributional calibration (D-CALIBRATION) [Haider et al., 2020] which computes the squared difference between the observed and predicted number of events within different time intervals. Classically, calibration is addressed in post-training analysis. We develop explicit calibration (X-CAL), which turns D-CALIBRATION into a differentiable objective that can be used in survival modeling alongside maximum likelihood estimation and other objectives. X-CAL allows practitioners to directly optimize calibration and strike a desired balance between predictive power and calibration. In our experiments, we fit a variety of shallow and deep models on simulated data, a survival dataset based on MNIST, on length-of-stay prediction using MIMIC-III data, and on brain cancer data from The Cancer Genome Atlas. We show that the models we study can be miscalibrated. We give experimental evidence on these datasets that X-CAL improves D-CALIBRATION without a large decrease in concordance or likelihood. |