Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Fixed Point Theory

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Geostatistics For Environmental Scientists

We present an approach for learning spatial traversability maps for driving in complex, urban environments based on an extensive dataset demonstrating the driving behaviour of human experts. The direct end-to-end mapping from raw input data to cost bypasses the effort of manually designing parts of the pipeline, exploits a large number of data samples, and can be framed additionally to refine handcrafted cost maps produced based on manual hand-engineered features. To achieve this, we introduce a maximum-entropy-based, non-linear inverse reinforcement learning IRL framework which exploits the capacity of fully convolutional neural networks FCNs to represent the cost model underlying driving behaviours. The application of a high-capacity, deep, parametric approach successfully scales to more complex environments and driving behaviours, while at deployment being run-time independent of training dataset size. After benchmarking against state-of-the-art IRL approaches, we focus on demonstrating scalability and performance on an ambitious dataset collected over the course of 1 year including more than 25,000 demonstration trajectories extracted from over 120 km of urban driving. We evaluate the resulting cost representations by showing the advantages over a carefully, manually designed cost map and furthermore demonstrate its robustness towards systematic errors by learning accurate representations even in the presence of calibration perturbations. Importantly, we demonstrate that a manually designed cost map can be refined to more accurately handle corner cases that are scarcely seen in the environment, such as stairs, slopes and underpasses, by further incorporating human priors into the training framework.

Large-scale cost function learning for path planning using deep inverse reinforcement learning

In this work, we present an approach to learn cost maps for driving in complex urban environments from a large number of demonstrations of human driving behaviour. The learned cost maps are constructed directly from raw sensor measurements, bypassing the effort of manually designing cost maps as well as features. When deploying the cost maps, the trajectories generated not only replicate human-like driving behaviour but are also demonstrably robust against systematic errors in putative robot configuration. To achieve this we deploy a Maximum Entropy based, non-linear IRL framework which uses Fully Convolutional Neural Networks (FCNs) to represent the cost model underlying expert driving behaviour. Using a deep, parametric approach enables us to scale efficiently to large datasets and complex behaviours while being run-time independent of dataset extent during deployment. We demonstrate scalability and performance on an ambitious dataset collected over the course of one year including more than 25k demonstration trajectories extracted from over 120km of driving and 13 different drivers. We evaluate against a carefully designed cost map and, in addition, demonstrate robustness to systematic errors by learning precise cost-maps even in the presence of system calibration perturbations.

Watch this: Scalable cost-function learning for path planning in urban environments

The EU Proposal for a General Data Protection Regulation has caused a wide debate between lawyers and legal scholars and many opinions have been voiced on the issue of the right to be forgotten. In order to analyse the relevance of the new rule provided by Article 17 of the Proposal, this paper considers the original idea of the right to be forgotten, pre-existing in both European and U.S. legal frameworks. This article focuses on the new provisions of Article 17 of the EU Proposal for a General Data Protection Regulation and evaluates its effects on court decisions. The author assumes that the new provisions do not seem to represent a revolutionary change to the existing rules with regard to the right granted to the individual, but instead have an impact on the extension of the protection of the information disseminated on-line.

The EU Proposal for a General Data Protection Regulation and the Roots of the ‘Right to Be Forgotten’

This paper looks into the problem of detecting network anomalies by analyzing NetFlow records. While many previous works have used statistical models and machine learning techniques in a supervised way, such solutions have the limitations that they require large amount of labeled data for training and are unlikely to detect zero-day attacks. Existing anomaly detection solutions also do not provide an easy way to explain or identify attacks in the anomalous traffic. To address these limitations, we develop and present GEE, a framework for detecting and explaining anomalies in network traffic. GEE comprises of two components: (i)Variational Autoencoder (VAE)- an unsupervised deep-learning technique for detecting anomalies, and (ii)a gradient-based fingerprinting technique for explaining anomalies. Evaluation of GEE on the recent UGR dataset demonstrates that our approach is effective in detecting different anomalies as well as identifying fingerprints that are good representations of these various attacks.

GEE: A Gradient-based Explainable Variational Autoencoder for Network Anomaly Detection

Existing inverse reinforcement learning (IRL) algorithms have assumed each expert's demonstrated trajectory to be produced by only a single reward function. This paper presents a novel generalization of the IRL problem that allows each trajectory to be generated by multiple locally consistent reward functions, hence catering to more realistic and complex experts' behaviors. Solving our generalized IRL problem thus involves not only learning these reward functions but also the stochastic transitions between them at any state (including unvisited states). By representing our IRL problem with a probabilistic graphical model, an expectation-maximization (EM) algorithm can be devised to iteratively learn the different reward functions and the stochastic transitions between them in order to jointly improve the likelihood of the expert's demonstrated trajectories. As a result, the most likely partition of a trajectory into segments that are generated from different locally consistent reward functions selected by EM can be derived. Empirical evaluation on synthetic and real-world datasets shows that our IRL algorithm outperforms the state-of-the-art EM clustering with maximum likelihood IRL, which is, interestingly, a reduced variant of our approach.

/pdf/inverse-reinforcement-learning-with-locally-consistent-12h5dcq9zy.pdf

Inverse reinforcement learning with locally consistent reward functions

This paper studies the problem of approximately unlearning a Bayesian model from a small subset of the training data to be erased. We frame this problem as one of minimizing the Kullback-Leibler divergence between the approximate posterior belief of model parameters after directly unlearning from erased data vs. the exact posterior belief from retraining with remaining data. Using the variational inference (VI) framework, we show that it is equivalent to minimizing an evidence upper bound which trades off between fully unlearning from erased data vs. not entirely forgetting the posterior belief given the full data (i.e., including the remaining data); the latter prevents catastrophic unlearning that can render the model useless. In model training with VI, only an approximate (instead of exact) posterior belief given the full data can be obtained, which makes unlearning even more challenging. We propose two novel tricks to tackle this challenge. We empirically demonstrate our unlearning methods on Bayesian models such as sparse Gaussian process and logistic regression using synthetic and real-world datasets.

Variational Bayesian Unlearning.

This paper looks into the problem of detecting network anomalies by analyzing NetFlow records. While many previous works have used statistical models and machine learning techniques in a supervised way, such solutions have the limitations that they require large amount of labeled data for training and are unlikely to detect zero-day attacks. Existing anomaly detection solutions also do not provide an easy way to explain or identify attacks in the anomalous traffic. To address these limitations, we develop and present GEE, a framework for detecting and explaining anomalies in network traffic. GEE comprises of two components: (i) Variational Autoencoder (VAE) - an unsupervised deep-learning technique for detecting anomalies, and (ii) a gradient-based fingerprinting technique for explaining anomalies. Evaluation of GEE on the recent UGR dataset demonstrates that our approach is effective in detecting different anomalies as well as identifying fingerprints that are good representations of these various attacks.

The problem of inverse reinforcement learning (IRL) is relevant to a variety of tasks including value alignment and robot learning from demonstration. Despite significant algorithmic contributions in recent years, IRL remains an ill-posed problem at its core; multiple reward functions coincide with the observed behavior and the actual reward function is not identifiable without prior knowledge or supplementary information. This paper presents an IRL framework called Bayesian optimization-IRL (BO-IRL) which identifies multiple solutions that are consistent with the expert demonstrations by efficiently exploring the reward function space. BO-IRL achieves this by utilizing Bayesian Optimization along with our newly proposed kernel that (a) projects the parameters of policy invariant reward functions to a single point in a latent space and (b) ensures nearby points in the latent space correspond to reward functions yielding similar likelihoods. This projection allows the use of standard stationary kernels in the latent space to capture the correlations present across the reward function space. Empirical results on synthetic and real-world environments (model-free and model-based) show that BO-IRL discovers multiple reward functions while minimizing the number of expensive exact policy optimizations.

/pdf/efficient-exploration-of-reward-functions-in-inverse-17v38d2cjo.pdf

Quoc Phong Nguyen

Papers

GEE: A Gradient-based Explainable Variational Autoencoder for Network Anomaly Detection

Inverse reinforcement learning with locally consistent reward functions

Variational Bayesian Unlearning.

GEE: A Gradient-based Explainable Variational Autoencoder for Network Anomaly Detection

Efficient Exploration of Reward Functions in Inverse Reinforcement Learning via Bayesian Optimization