We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.
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Understanding to what extent neural networks memorize training data is an intriguing question with practical and theoretical implications. In this paper we show that in some cases a significant fraction of the training data can in fact be reconstructed from the parameters of a trained neural network classifier. We propose a novel reconstruction scheme that stems from recent theoretical results about the implicit bias in training neural networks with gradient-based methods. To the best of our knowledge, our results are the first to show that reconstructing a large portion of the actual training samples from a trained neural network classifier is generally possible. This has negative implications on privacy, as it can be used as an attack for revealing sensitive training data. We demonstrate our method for binary MLP classifiers on a few standard computer vision datasets.
We investigate statistical uncertainty quantification for reinforcement learning (RL) and its implications in exploration policy. Despite ever-growing literature on RL applications, fundamental questions about inference and error quantification, such as large-sample behaviors, appear to remain quite open. In this paper, we fill in the literature gap by studying the central limit theorem behaviors of estimated Q-values and value functions under various RL settings. In particular, we explicitly identify closed-form expressions of the asymptotic variances, which allow us to efficiently construct asymptotically valid confidence regions for key RL quantities. Furthermore, we utilize these asymptotic expressions to design an effective exploration strategy, which we call Q-value-based Optimal Computing Budget Allocation (Q-OCBA). The policy relies on maximizing the relative discrepancies among the Q-value estimates. Numerical experiments show superior performances of our exploration strategy than other benchmark policies.
In this paper we propose the adaptive lasso for predictive quantile regression (ALQR). Reflecting empirical findings, we allow predictors to have various degrees of persistence and exhibit different signal strengths. The number of predictors is allowed to grow with the sample size. We study regularity conditions under which stationary, local unit root, and cointegrated predictors are present simultaneously. We next show the convergence rates, model selection consistency, and asymptotic distributions of ALQR. We apply the proposed method to the out-of-sample quantile prediction problem of stock returns and find that it outperforms the existing alternatives. We also provide numerical evidence from additional Monte Carlo experiments, supporting the theoretical results.
We are interested in estimating the uncertainties of deep neural networks, which play an important role in many scientific and engineering problems. In this paper, we present a striking new finding that an ensemble of neural networks with the same weight initialization, trained on datasets that are shifted by a constant bias gives rise to slightly inconsistent trained models, where the differences in predictions are a strong indicator of epistemic uncertainties. Using the neural tangent kernel (NTK), we demonstrate that this phenomena occurs in part because the NTK is not shift-invariant. Since this is achieved via a trivial input transformation, we show that this behavior can therefore be approximated by training a single neural network -- using a technique that we call $\Delta-$UQ -- that estimates uncertainty around prediction by marginalizing out the effect of the biases during inference. We show that $\Delta-$UQ's uncertainty estimates are superior to many of the current methods on a variety of benchmarks -- outlier rejection, calibration under distribution shift, and sequential design optimization of black box functions. Code for $\Delta-$UQ can be accessed at //github.com/LLNL/DeltaUQ
Uncertainty estimation of the trained deep learning network provides important information for improving the learning efficiency or evaluating the reliability of the network prediction. In this paper, we propose a method for the uncertainty estimation for multi-class image classification using test-time mixup augmentation (TTMA). To improve the discrimination ability between the correct and incorrect prediction of the existing aleatoric uncertainty, we propose the data uncertainty by applying the mixup augmentation on the test data and measuring the entropy of the histogram of predicted labels. In addition to the data uncertainty, we propose a class-specific uncertainty presenting the aleatoric uncertainty associated with the specific class, which can provide information on the class confusion and class similarity of the trained network. The proposed methods are validated on two public datasets, the ISIC-18 skin lesion diagnosis dataset, and the CIFAR-100 real-world image classification dataset. The experiments demonstrate that (1) the proposed data uncertainty better separates the correct and incorrect prediction than the existing uncertainty measures thanks to the mixup perturbation, and (2) the proposed class-specific uncertainty provides information on the class confusion and class similarity of the trained network for both datasets.
An in-depth understanding of uncertainty is the first step to making effective decisions under uncertainty. Deep/machine learning (ML/DL) has been hugely leveraged to solve complex problems involved with processing high-dimensional data. However, reasoning and quantifying different types of uncertainties to achieve effective decision-making have been much less explored in ML/DL than in other Artificial Intelligence (AI) domains. In particular, belief/evidence theories have been studied in KRR since the 1960s to reason and measure uncertainties to enhance decision-making effectiveness. We found that only a few studies have leveraged the mature uncertainty research in belief/evidence theories in ML/DL to tackle complex problems under different types of uncertainty. In this survey paper, we discuss several popular belief theories and their core ideas dealing with uncertainty causes and types and quantifying them, along with the discussions of their applicability in ML/DL. In addition, we discuss three main approaches that leverage belief theories in Deep Neural Networks (DNNs), including Evidential DNNs, Fuzzy DNNs, and Rough DNNs, in terms of their uncertainty causes, types, and quantification methods along with their applicability in diverse problem domains. Based on our in-depth survey, we discuss insights, lessons learned, limitations of the current state-of-the-art bridging belief theories and ML/DL, and finally, future research directions.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.
Artificial neural networks thrive in solving the classification problem for a particular rigid task, acquiring knowledge through generalized learning behaviour from a distinct training phase. The resulting network resembles a static entity of knowledge, with endeavours to extend this knowledge without targeting the original task resulting in a catastrophic forgetting. Continual learning shifts this paradigm towards networks that can continually accumulate knowledge over different tasks without the need to retrain from scratch. We focus on task incremental classification, where tasks arrive sequentially and are delineated by clear boundaries. Our main contributions concern 1) a taxonomy and extensive overview of the state-of-the-art, 2) a novel framework to continually determine the stability-plasticity trade-off of the continual learner, 3) a comprehensive experimental comparison of 11 state-of-the-art continual learning methods and 4 baselines. We empirically scrutinize method strengths and weaknesses on three benchmarks, considering Tiny Imagenet and large-scale unbalanced iNaturalist and a sequence of recognition datasets. We study the influence of model capacity, weight decay and dropout regularization, and the order in which the tasks are presented, and qualitatively compare methods in terms of required memory, computation time, and storage.
Ensembles over neural network weights trained from different random initialization, known as deep ensembles, achieve state-of-the-art accuracy and calibration. The recently introduced batch ensembles provide a drop-in replacement that is more parameter efficient. In this paper, we design ensembles not only over weights, but over hyperparameters to improve the state of the art in both settings. For best performance independent of budget, we propose hyper-deep ensembles, a simple procedure that involves a random search over different hyperparameters, themselves stratified across multiple random initializations. Its strong performance highlights the benefit of combining models with both weight and hyperparameter diversity. We further propose a parameter efficient version, hyper-batch ensembles, which builds on the layer structure of batch ensembles and self-tuning networks. The computational and memory costs of our method are notably lower than typical ensembles. On image classification tasks, with MLP, LeNet, and Wide ResNet 28-10 architectures, our methodology improves upon both deep and batch ensembles.
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