End-to-end learning has recently emerged as a promising technique to tackle the problem of autonomous driving. Existing works show that learning a navigation policy from raw sensor data may reduce the system's reliance on external sensing systems, (e.g. GPS), and or outperform traditional methods based on state estimation and planning. However, existing end-to-end methods generally trade off performance for safety, hindering their diffusion to real-life applications. For example, when confronted with an input which is radically different from the training data, end-to-end autonomous driving systems are likely to fail, compromising the safety of the vehicle. To detect such failure cases, this work proposes a general framework for uncertainty estimation which enables a policy trained end-to-end to predict not only action commands, but also a confidence about its own predictions. In contrast to previous works, our framework can be applied to any existing neural network and task, without the need to change the network's architecture or loss, or to train the network. In order to do so, we generate confidence levels by forward propagation of input and model uncertainties using Bayesian inference. We test our framework on the task of steering angle regression for an autonomous car, and compare our approach to existing methods with both qualitative and quantitative results on a real dataset. Finally, we show an interesting by-product of our framework: robustness against adversarial attacks.
To account for model uncertainty in deep learning, a distribution is placed over neural network (NN) weights @math , defining a Bayesian neural network @cite_0 @cite_8 @cite_13 . The work of @cite_17 provides a mathematically grounded framework to capture model uncertainty leveraging dropout at test-time @cite_5 . Specifically, they propose to approximate the intractable posterior distribution over network weights @math given a specific training set @math by collecting multiple predictions for a single input, each with a different realization of weights due to dropout. This method is often referred to as (MC) .
@cite_8 @cite_0 @cite_5 @cite_13 @cite_17
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A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible (1) objective comparisons between solutions using alternative network architectures, (2) objective stopping rules for network pruning or growing procedures, (3) objective choice of magnitude and type of weight decay terms or additive regularizers (for penalizing large weights, etc.), (4) a measure of the effective number of well-determined parameters in a model, (5) quantified estimates of the error bars on network parameters and on network output, and (6) objective comparisons with alternative learning and interpolation models such as splines and radial basis functions. The Bayesian "evidence" automatically embodies "Occam's razor," penalizing overflexible and overcomplex models. The Bayesian approach helps detect poor underlying assumptions in learning models. For learning models well matched to a problem, a good correlation between generalization ability and the Bayesian evidence is obtained. (1) The outputs of a typical multi-output classification network do not satisfy the axioms of probability; probabilities should be positive and sum to one. This problem can be solved by treating the trained network as a preprocessor that produces a feature vector that can be further processed, for instance by classical statistical estimation techniques. (2) We present a method for computing the first two moments of the probability distribution indicating the range of outputs that are consistent with the input and the training data. It is particularly useful to combine these two ideas: we implement the ideas of section 1 using Parzen windows, where the shape and relative size of each window is computed using the ideas of section 2. This allows us to make contact between important theoretical ideas (e.g. the ensemble formalism) and practical techniques (e.g. back-prop). Our results also shed new light on and generalize the well-known "softmax" scheme. Deep neural nets with a large number of parameters are very powerful machine learning systems. However, overfitting is a serious problem in such networks. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. Dropout is a technique for addressing this problem. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much. During training, dropout samples from an exponential number of different "thinned" networks. At test time, it is easy to approximate the effect of averaging the predictions of all these thinned networks by simply using a single unthinned network that has smaller weights. This significantly reduces overfitting and gives major improvements over other regularization methods. We show that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets. From the Publisher: Artificial "neural networks" are now widely used as flexible models for regression classification applications, but questions remain regarding what these models mean, and how they can safely be used when training data is limited. Bayesian Learning for Neural Networks shows that Bayesian methods allow complex neural network models to be used without fear of the "overfitting" that can occur with traditional neural network learning methods. Insight into the nature of these complex Bayesian models is provided by a theoretical investigation of the priors over functions that underlie them. Use of these models in practice is made possible using Markov chain Monte Carlo techniques. Both the theoretical and computational aspects of this work are of wider statistical interest, as they contribute to a better understanding of how Bayesian methods can be applied to complex problems. Presupposing only the basic knowledge of probability and statistics, this book should be of interest to many researchers in statistics, engineering, and artificial intelligence. Software for Unix systems that implements the methods described is freely available over the Internet.