Uncertainty Quantification with Statistical Guarantees in End-to-End Autonomous Driving Control Rhiannon Michelmore 1, Matthew Wicker 1, Luca Laurenti 1, Luca Cardelli 1, Y arin Gal 1, Marta Kwiatkowska 1 Abstract -- Deep neural network controllers for autonomous driving have recently benefited from significant performance improvements, and have begun deployment in the real world. Prior to their widespread adoption, safety guarantees are needed on the controller behaviour that properly take account of the uncertainty within the model as well as sensor noise. Bayesian neural networks, which assume a prior over the weights, have been shown capable of producing such uncertainty measures, but properties surrounding their safety have not yet been quantified for use in autonomous driving scenarios. In this paper, we develop a framework based on a state-of-the-art simulator for evaluating end-to-end Bayesian controllers. In addition to computing pointwise uncertainty measures that can be computed in real time and with statistical guarantees, we also provide a method for estimating the probability that, given a scenario, the controller keeps the car safe within a finite horizon. We experimentally evaluate the quality of uncertainty computation by three Bayesian inference methods in different scenarios and show how the uncertainty measures can be combined and calibrated for use in collision avoidance. Our results suggest that uncertainty estimates can greatly aid decision making in autonomous driving. I NTRODUCTION Deep Neural Networks (DNNs) have seen a surge in popularity over the past decade, and their use has become widespread in many fields including safety-critical systems such as medical diagnosis and, in particular, autonomous cars. The latter have driven millions of miles without human intervention [1], [2], but offer few safety guarantees.

One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffusion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this paper is bringing together techniques from stochastic control with methods from statistics for stochastic processes to find a way to both learn the dynamics of the underlying process and control in a reasonable way at the same time. More precisely, we study a long-term average impulse control problem, a stochastic version of the classical Faustmann timber harvesting problem. One of the problems that immediately arises is an exploration vs. exploitation-behavior as is well known for problems in machine learning. We propose a way to deal with this issue by combining exploration- and exploitation periods in a suitable way. Our main finding is that this construction can be based on the rates of convergence of estimators for the invariant density. Using this, we obtain that the average cumulated regret is of uniform order $O({T^{-1/3}})$.

We provide high probability sample complexity guarantees for non-parametric structure learning of tree-shaped graphical models whose nodes are discrete random variables with a finite or countable alphabet, both in the noiseless and noisy regimes. First, we introduce a new, fundamental quantity called the (noisy) information threshold, which arises naturally from the error analysis of the Chow-Liu algorithm and characterizes not only the sample complexity, but also the inherent impact of the noise on the structure learning task, without explicit assumptions on the distribution of the model. This allows us to present the first non-parametric, high-probability finite sample complexity bounds on tree-structure learning from potentially noise-corrupted data. In particular, for number of nodes $p$, success rate $1-\delta$, and a fixed value of the information threshold, our sample complexity bounds for exact structure recovery are of the order of $\mathcal{O}\big(\log^{1+\zeta} (p/\delta)\big) $, for all $\zeta>0$, for both noiseless and noisy settings. Subsequently, we apply our results on two classes of hidden models, namely, the $M$-ary erasure channel and the generalized symmetric channel, illustrating the usefulness and importance of our framework. As a byproduct of our analysis, this paper resolves the open problem of tree structure learning in the presence of non-identically distributed observation noise, providing explicit conditions on the convergence of the Chow-Liu algorithm under this setting as well.

Practical reinforcement learning problems are often formulated as constrained Markov decision process (CMDP) problems, in which the agent has to maximize the expected return while satisfying a set of prescribed safety constraints. In this study, we propose a novel simulator-based method to approximately solve a CMDP problem without making any compromise on the safety constraints. We achieve this by decomposing the CMDP into a pair of MDPs; reconnaissance MDP and planning MDP. The purpose of reconnaissance MDP is to evaluate the set of actions that are safe, and the purpose of planning MDP is to maximize the return while using the actions authorized by reconnaissance MDP. RMDP can define a set of safe policies for any given set of safety constraint, and this set of safe policies can be used to solve another CMDP problem with different reward. Our method is not only computationally less demanding than the previous simulator-based approaches to CMDP, but also capable of finding a competitive reward-seeking policy in a high dimensional environment, including those involving multiple moving obstacles.

The Mind is Flat: The Illusion of Mental Depth and the Improvised Mind.

A continuing challenge for machine learning is providing methods to perform computation on data while ensuring the data remains private. In this paper we build on the provable privacy guarantees of differential privacy which has been combined with Gaussian processes through the previously published \emph{cloaking method}. In this paper we solve several shortcomings of this method, starting with the problem of predictions in regions with low data density. We experiment with the use of inducing points to provide a sparse approximation and show that these can provide robust differential privacy in outlier areas and at higher dimensions. We then look at classification, and modify the Laplace approximation approach to provide differentially private predictions. We then combine this with the sparse approximation and demonstrate the capability to perform classification in high dimensions. We finally explore the issue of hyperparameter selection and develop a method for their private selection. This paper and associated libraries provide a robust toolkit for combining differential privacy and GPs in a practical manner.

Whenever a social media user decides to share a story, she is typically pleased to receive likes, comments, shares, or, more generally, feedback from her followers. As a result, she may feel compelled to use the feedback she receives to (re-)estimate her followers' preferences and decides which stories to share next to receive more (positive) feedback. Under which conditions can she succeed? In this work, we first look into this problem from a theoretical perspective and then provide a set of practical algorithms to identify and characterize such behavior in social media. More specifically, we address the above problem from the viewpoint of sequential decision making and utility maximization. For a wide variety of utility functions, we first show that, to succeed, a user needs to actively trade off exploitation-- sharing stories which lead to more (positive) feedback--and exploration-- sharing stories to learn about her followers' preferences. However, exploration is not necessary if a user utilizes the feedback her followers provide to other users in addition to the feedback she receives. Then, we develop a utility estimation framework for observation data, which relies on statistical hypothesis testing to determine whether a user utilizes the feedback she receives from each of her followers to decide what to post next. Experiments on synthetic data illustrate our theoretical findings and show that our estimation framework is able to accurately recover users' underlying utility functions. Experiments on several real datasets gathered from Twitter and Reddit reveal that up to 82% (43%) of the Twitter (Reddit) users in our datasets do use the feedback they receive to decide what to post next.

Domain-specific knowledge graphs constructed from natural language text are ubiquitous in today's world. In many such scenarios the base text, from which the knowledge graph is constructed, concerns itself with practical, on-hand, actual or ground-reality information about the domain. Product documentation in software engineering domain are one example of such base texts. Other examples include blogs and texts related to digital artifacts, reports on emerging markets and business models, patient medical records, etc. Though the above sources contain a wealth of knowledge about their respective domains, the conceptual knowledge on which they are based is often missing or unclear. Access to this conceptual knowledge can enormously increase the utility of available data and assist in several tasks such as knowledge graph completion, grounding, querying, etc. Our contributions in this paper are twofold. First, we propose a novel Markovian stochastic model for document generation from conceptual knowledge. The uniqueness of our approach lies in the fact that the conceptual knowledge in the writer's mind forms a component of the parameter set of our stochastic model. Secondly, we solve the inverse problem of learning the best conceptual knowledge from a given document, by finding model parameters which maximize the likelihood of generating the specific document over all possible parameter values. This likelihood maximization is done using an application of Baum-Welch algorithm, which is a known special case of Expectation-Maximization (EM) algorithm. We run our conceptualization algorithm on several well-known natural language sources and obtain very encouraging results. The results of our extensive experiments concur with the hypothesis that the information contained in these sources has a well-defined and rigorous underlying conceptual structure, which can be discovered using our method.

The ECG remains the most widely used diagnostic test for characterization of cardiac structure and electrical activity. We hypothesized that parallel advances in computing power, machine learning algorithms, and availability of large-scale data could substantially expand the clinical inferences derived from the ECG while at the same time preserving interpretability for medical decision-making. We identified 36 186 ECGs from the University of California, San Francisco database that would enable training of models for estimation of cardiac structure or function or detection of disease. We segmented the ECG into standard component waveforms and intervals using a novel combination of convolutional neural networks and hidden Markov models and evaluated this segmentation by comparing resulting electrical intervals against 141 864 measurements produced during the clinical workflow. We then built a patient-level ECG profile, a 725-element feature vector and used this profile to train and interpret machine learning models for examples of cardiac structure (left ventricular mass, left atrial volume, and mitral annulus e-prime) and disease (pulmonary arterial hypertension, hypertrophic cardiomyopathy, cardiac amyloid, and mitral valve prolapse).

Neural networks are popular models for regression. They are often trained via back-propagation to find a value of the weights that correctly predicts the observed data. Although back-propagation has shown good performance in many applications, it cannot easily output an estimate of the uncertainty in the predictions made. Measuring this uncertainty in the predictions of machine learning models is a critical aspect with important applications. Uncertainty estimates can be obtained by following a Bayesian approach in which a posterior distribution of the model parameters is computed. The posterior distribution summarizes which parameter values are compatible with the data. Typically,this posterior distribution is intractable and has to be approximated. Several approaches have been considered for solving this problem. We propose here a general method for approximate Bayesian inference based on minimizing{\alpha}-divergences which allows for flexible approximate distributions. The method is evaluated in the context of Bayesian neural networks for regression on extensive experiments. The results show that it often gives better performance in terms of the test log-likelihood and sometimes in terms of the squared error.