Gaussian processes scale prohibitively with the size of the dataset. In response, many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. Therefore in practice, GP models are often as much about the approximation method as they are about the data. Here, we develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended. The most common GP approximations map to an instance in this class, such as methods based on the Cholesky factorization, conjugate gradients, and inducing points. For any method in this class, we prove (i) convergence of its posterior mean in the associated RKHS, (ii) decomposability of its combined posterior covariance into mathematical and computational covariances, and (iii) that the combined variance is a tight worst-case bound for the squared error between the method's posterior mean and the latent function. Finally, we empirically demonstrate the consequences of ignoring computational uncertainty and show how implicitly modeling it improves generalization performance on benchmark datasets.
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Inverse Uncertainty Quantification (IUQ) method has been widely used to quantify the uncertainty of Physical Model Parameters (PMPs) in nuclear Thermal Hydraulics (TH) systems. This paper introduces a novel hierarchical Bayesian model which aims to mitigate two existing challenges in IUQ: the high variability of PMPs under varying experimental conditions, and unknown model discrepancies or outliers causing over-fitting issues. The proposed hierarchical model is compared with the conventional single-level Bayesian model using TRACE code and the measured void fraction data in the BFBT benchmark. A Hamiltonian Monte Carlo Method - No U-Turn Sampler (NUTS) is used for posterior sampling. The results demonstrate the effectiveness of the proposed hierarchical model in providing better estimates of the posterior distributions of PMPs and being less prone to over-fitting. The proposed method also demonstrates a promising approach for generalizing IUQ to larger databases with broad ranges of experimental conditions.
Representation learning has significantly driven the field to develop pretrained models that can act as a valuable starting point when transferring to new datasets. With the rising demand for reliable machine learning and uncertainty quantification, there is a need for pretrained models that not only provide embeddings but also transferable uncertainty estimates. To guide the development of such models, we propose the Uncertainty-aware Representation Learning (URL) benchmark. Besides the transferability of the representations, it also measures the zero-shot transferability of the uncertainty estimate using a novel metric. We apply URL to evaluate eleven uncertainty quantifiers that are pretrained on ImageNet and transferred to eight downstream datasets. We find that approaches that focus on the uncertainty of the representation itself or estimate the prediction risk directly outperform those that are based on the probabilities of upstream classes. Yet, achieving transferable uncertainty quantification remains an open challenge. Our findings indicate that it is not necessarily in conflict with traditional representation learning goals. Code is provided under //github.com/mkirchhof/url .
Conformal inference is a popular tool for constructing prediction intervals (PI). We consider here the scenario of post-selection/selective conformal inference, that is PIs are reported only for individuals selected from an unlabeled test data. To account for multiplicity, we develop a general split conformal framework to construct selective PIs with the false coverage-statement rate (FCR) control. We first investigate the Benjamini and Yekutieli (2005)'s FCR-adjusted method in the present setting, and show that it is able to achieve FCR control but yields uniformly inflated PIs. We then propose a novel solution to the problem, named as Selective COnditional conformal Predictions (SCOP), which entails performing selection procedures on both calibration set and test set and construct marginal conformal PIs on the selected sets by the aid of conditional empirical distribution obtained by the calibration set. Under a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the FCR. More importantly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond exchangeablity and discuss the conditions under which the SCOP is able to control the FCR. As special cases, the SCOP with quantile-based selection or conformal p-values-based multiple testing procedures enjoys valid coverage guarantee under mild conditions. Numerical results confirm the effectiveness and robustness of SCOP in FCR control and show that it achieves more narrowed PIs over existing methods in many settings.
Objective: Quantitative $T_1\rho$ imaging has potential for assessment of biochemical alterations of liver pathologies. Deep learning methods have been employed to accelerate quantitative $T_1\rho$ imaging. To employ artificial intelligence-based quantitative imaging methods in complicated clinical environment, it is valuable to estimate the uncertainty of the predicated $T_1\rho$ values to provide the confidence level of the quantification results. The uncertainty should also be utilized to aid the post-hoc quantitative analysis and model learning tasks. Approach: To address this need, we propose a parametric map refinement approach for learning-based $T_1\rho$ mapping and train the model in a probabilistic way to model the uncertainty. We also propose to utilize the uncertainty map to spatially weight the training of an improved $T_1\rho$ mapping network to further improve the mapping performance and to remove pixels with unreliable $T_1\rho$ values in the region of interest. The framework was tested on a dataset of 51 patients with different liver fibrosis stages. Main results: Our results indicate that the learning-based map refinement method leads to a relative mapping error of less than 3% and provides uncertainty estimation simultaneously. The estimated uncertainty reflects the actual error level, and it can be used to further reduce relative $T_1\rho$ mapping error to 2.60% as well as removing unreliable pixels in the region of interest effectively. Significance: Our studies demonstrate the proposed approach has potential to provide a learning-based quantitative MRI system for trustworthy $T_1\rho$ mapping of the liver.
We employ pressure point analysis and roofline modeling to identify performance bottlenecks and determine an upper bound on the performance of the Canonical Polyadic Alternating Poisson Regression Multiplicative Update (CP-APR MU) algorithm in the SparTen software library. Our analyses reveal that a particular matrix computation, $\Phi^{(n)}$, is the critical performance bottleneck in the SparTen CP-APR MU implementation. Moreover, we find that atomic operations are not a critical bottleneck while higher cache reuse can provide a non-trivial performance improvement. We also utilize grid search on the Kokkos library parallel policy parameters to achieve 2.25x average speedup over the SparTen default for $\Phi^{(n)}$ computation on CPU and 1.70x on GPU. We conclude our investigations by comparing Kokkos implementations of the STREAM benchmark and the matricized tensor times Khatri-Rao product (MTTKRP) benchmark from the Parallel Sparse Tensor Algorithm (PASTA) benchmark suite to implementations using vendor libraries. We show that with a single implementation Kokkos achieves performance comparable to hand-tuned code for fundamental operations that make up tensor decomposition kernels on a wide range of CPU and GPU systems. Overall, we conclude that Kokkos demonstrates good performance portability for simple data-intensive operations but requires tuning for algorithms with more complex dependencies and data access patterns.
Quantifying uncertainty is important for actionable predictions in real-world applications. A crucial part of predictive uncertainty quantification is the estimation of epistemic uncertainty, which is defined as an integral of the product between a divergence function and the posterior. Current methods such as Deep Ensembles or MC dropout underperform at estimating the epistemic uncertainty, since they primarily consider the posterior when sampling models. We suggest Quantification of Uncertainty with Adversarial Models (QUAM) to better estimate the epistemic uncertainty. QUAM identifies regions where the whole product under the integral is large, not just the posterior. Consequently, QUAM has lower approximation error of the epistemic uncertainty compared to previous methods. Models for which the product is large correspond to adversarial models (not adversarial examples!). Adversarial models have both a high posterior as well as a high divergence between their predictions and that of a reference model. Our experiments show that QUAM excels in capturing epistemic uncertainty for deep learning models and outperforms previous methods on challenging tasks in the vision domain.
Uncertainty sampling is a prevalent active learning algorithm that queries sequentially the annotations of data samples which the current prediction model is uncertain about. However, the usage of uncertainty sampling has been largely heuristic: (i) There is no consensus on the proper definition of "uncertainty" for a specific task under a specific loss; (ii) There is no theoretical guarantee that prescribes a standard protocol to implement the algorithm, for example, how to handle the sequentially arrived annotated data under the framework of optimization algorithms such as stochastic gradient descent. In this work, we systematically examine uncertainty sampling algorithms under both stream-based and pool-based active learning. We propose a notion of equivalent loss which depends on the used uncertainty measure and the original loss function and establish that an uncertainty sampling algorithm essentially optimizes against such an equivalent loss. The perspective verifies the properness of existing uncertainty measures from two aspects: surrogate property and loss convexity. Furthermore, we propose a new notion for designing uncertainty measures called \textit{loss as uncertainty}. The idea is to use the conditional expected loss given the features as the uncertainty measure. Such an uncertainty measure has nice analytical properties and generality to cover both classification and regression problems, which enable us to provide the first generalization bound for uncertainty sampling algorithms under both stream-based and pool-based settings, in the full generality of the underlying model and problem. Lastly, we establish connections between certain variants of the uncertainty sampling algorithms with risk-sensitive objectives and distributional robustness, which can partly explain the advantage of uncertainty sampling algorithms when the sample size is small.
Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems.
Gaussian variational inference and the Laplace approximation are popular alternatives to Markov chain Monte Carlo that formulate Bayesian posterior inference as an optimization problem, enabling the use of simple and scalable stochastic optimization algorithms. However, a key limitation of both methods is that the solution to the optimization problem is typically not tractable to compute; even in simple settings the problem is nonconvex. Thus, recently developed statistical guarantees -- which all involve the (data) asymptotic properties of the global optimum -- are not reliably obtained in practice. In this work, we provide two major contributions: a theoretical analysis of the asymptotic convexity properties of variational inference with a Gaussian family and the maximum a posteriori (MAP) problem required by the Laplace approximation; and two algorithms -- consistent Laplace approximation (CLA) and consistent stochastic variational inference (CSVI) -- that exploit these properties to find the optimal approximation in the asymptotic regime. Both CLA and CSVI involve a tractable initialization procedure that finds the local basin of the optimum, and CSVI further includes a scaled gradient descent algorithm that provably stays locally confined to that basin. Experiments on nonconvex synthetic and real-data examples show that compared with standard variational and Laplace approximations, both CSVI and CLA improve the likelihood of obtaining the global optimum of their respective optimization problems.
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.
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