We analyze the epistemic uncertainty (EU) of supervised learning in Bayesian inference by focusing on the excess risk. Existing analysis is limited to the Bayesian setting, which assumes a correct model and exact Bayesian posterior distribution. Thus we cannot apply the existing theory to modern Bayesian algorithms, such as variational inference. To address this, we present a novel EU analysis in the frequentist setting, where data is generated from an unknown distribution. We show a relation between the generalization ability and the widely used EU measurements, such as the variance and entropy of the predictive distribution. Then we show their convergence behaviors theoretically. Finally, we propose new variational inference that directly controls the prediction and EU evaluation performances based on the PAC-Bayesian theory. Numerical experiments show that our algorithm significantly improves the EU evaluation over the existing methods.
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The value of uncertainty quantification on predictions for trained neural networks (NNs) on quantum chemical reference data is quantitatively explored. For this, the architecture of the PhysNet NN was suitably modified and the resulting model was evaluated with different metrics to quantify calibration, quality of predictions, and whether prediction error and the predicted uncertainty can be correlated. The results from training on the QM9 database and evaluating data from the test set within and outside the distribution indicate that error and uncertainty are not linearly related. The results clarify that noise and redundancy complicate property prediction for molecules even in cases for which changes - e.g. double bond migration in two otherwise identical molecules - are small. The model was then applied to a real database of tautomerization reactions. Analysis of the distance between members in feature space combined with other parameters shows that redundant information in the training dataset can lead to large variances and small errors whereas the presence of similar but unspecific information returns large errors but small variances. This was, e.g., observed for nitro-containing aliphatic chains for which predictions were difficult although the training set contained several examples for nitro groups bound to aromatic molecules. This underlines the importance of the composition of the training data and provides chemical insight into how this affects the prediction capabilities of a ML model. Finally, the approach put forward can be used for information-based improvement of chemical databases for target applications through active learning optimization.
Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.
Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical settings hinge on the assumption that the kernel hyperparameters are known, which does not apply in general. To mitigate this, we introduce robust Gaussian process uniform error bounds in settings with unknown hyperparameters. Our approach computes a confidence region in the space of hyperparameters, which enables us to obtain a probabilistic upper bound for the model error of a Gaussian process with arbitrary hyperparameters. We do not require to know any bounds for the hyperparameters a priori, which is an assumption commonly found in related work. Instead, we are able to derive bounds from data in an intuitive fashion. We additionally employ the proposed technique to derive performance guarantees for a class of learning-based control problems. Experiments show that the bound performs significantly better than vanilla and fully Bayesian Gaussian processes.
We study generalizations of online bipartite matching in which each arriving vertex (customer) views a ranked list of offline vertices (products) and matches to (purchases) the first one they deem acceptable. The number of products that the customer has patience to view can be stochastic and dependent on the products seen. We develop a framework that views the interaction with each customer as an abstract resource consumption process, and derive new results for these online matching problems under the adversarial, non-stationary, and IID arrival models, assuming we can (approximately) solve the product ranking problem for each single customer. To that end, we show new results for product ranking under two cascade-click models: an optimal algorithm for item-dependent hazard rates, and a 1/2-approximate algorithm for general item-independent patience distributions. We also present a constant-factor 0.027-approximate algorithm in a new model where items are not initially available and arrive over time. Finally, we present three negative results of interest: one formalizing the notion of a stochasticity gap exhibited by existing approaches to this problem, an example showing the analysis of SimpleGreedy in existing work to be tight, and another one for the single-customer problem in which any constant-factor approximation is impossible when compared to a benchmark that knows the realization of the patience in advance. A corollary of this last result is that for general single-item online accept/reject problems with IID arrivals, any constant-factor approximation is impossible if the number of arrivals is unknown.
Using a novel toy nautical navigation environment, we show that dynamic programming can be used when only incomplete information about a partially observed Markov decision process (POMDP) is known. By incorporating uncertainty into our model, we show that navigation policies can be constructed that maintain safety, outperforming the baseline performance of traditional dynamic programming for Markov decision processes (MDPs). Adding in controlled sensing methods, we show that these policies can also lower measurement costs at the same time.
Uncertainty control and scalability to large datasets are the two main issues for the deployment of Gaussian process models into the autonomous material and chemical space exploration pipelines. One way to address both of these issues is by introducing the latent inducing variables and choosing the right approximation for the marginal log-likelihood objective. Here, we show that variational learning of the inducing points in the high-dimensional molecular descriptor space significantly improves both the prediction quality and uncertainty estimates on test configurations from a sample molecular dynamics dataset. Additionally, we show that inducing points can learn to represent the configurations of the molecules of different types that were not present within the initialization set of inducing points. Among several evaluated approximate marginal log-likelihood objectives, we show that the predictive log-likelihood provides both the predictive quality comparable to the exact Gaussian process model and excellent uncertainty control. Finally, we comment on whether a machine learning model makes predictions by interpolating the molecular configurations in high-dimensional descriptor space. We show that despite our intuition, and even for densely sampled molecular dynamics datasets, most of the predictions are done in the extrapolation regime.
We propose a novel framework for multitask reinforcement learning based on the minimum description length (MDL) principle. In this approach, which we term MDL-control (MDL-C), the agent learns the common structure among the tasks with which it is faced and then distills it into a simpler representation which facilitates faster convergence and generalization to new tasks. In doing so, MDL-C naturally balances adaptation to each task with epistemic uncertainty about the task distribution. We motivate MDL-C via formal connections between the MDL principle and Bayesian inference, derive theoretical performance guarantees, and demonstrate MDL-C's empirical effectiveness on both discrete and high-dimensional continuous control tasks. %Empirically, this framework is used to modify existing policy optimization approaches and improves their multitask performance in both discrete and high-dimensional continuous control problems.
A fundamental class of inferential problems are those characterised by there having been a substantial degree of pre-data (or prior) belief that the value of a model parameter $\theta_j$ was equal or lay close to a specified value $\theta^{*}_j$, which may, for example, be the value that indicates the absence of a treatment effect or the lack of correlation between two variables. This paper puts forward a generally applicable 'push-button' solution to problems of this type that circumvents the severe difficulties that arise when attempting to apply standard methods of inference, including the Bayesian method, to such problems. Usually the only input of major note that is required from the user in implementing this solution is the assignment of a pre-data or prior probability to the hypothesis that the parameter $\theta_j$ lies in a narrow interval $[\theta_{j0},\theta_{j1}]$ that is assumed to contain the value of interest $\theta^{*}_j$. On the other hand, the end result that is achieved by applying this method is, conveniently, a joint post-data distribution over all the parameters $\theta_1,\theta_2,\ldots,\theta_k$ of the model concerned. The proposed method is constructed by naturally combining a simple Bayesian argument with an approach to inference called organic fiducial inference that was developed in a number of earlier papers. To begin with, the main theoretical arguments underlying this combined Bayesian and fiducial method are presented and discussed in detail. Various applications and useful extensions of this methodology are then outlined in the latter part of the paper. The examples that are considered are made relevant to the analysis of clinical trial data where appropriate.
The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often refereed to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of hitherto attempts at handling uncertainty in general and formalizing this distinction in particular.
We propose a new method for event extraction (EE) task based on an imitation learning framework, specifically, inverse reinforcement learning (IRL) via generative adversarial network (GAN). The GAN estimates proper rewards according to the difference between the actions committed by the expert (or ground truth) and the agent among complicated states in the environment. EE task benefits from these dynamic rewards because instances and labels yield to various extents of difficulty and the gains are expected to be diverse -- e.g., an ambiguous but correctly detected trigger or argument should receive high gains -- while the traditional RL models usually neglect such differences and pay equal attention on all instances. Moreover, our experiments also demonstrate that the proposed framework outperforms state-of-the-art methods, without explicit feature engineering.
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