This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2 -Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)Hölder continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent Hölder-stability results for optimal transport maps. 1. Introduction Numerous problems involve the comparison of point clouds, i.e. sets of points that lie in a metric space and for which the spatial distribution is of interest. Seeing the point clouds as discrete probability measures in a metric space, it is natural to compare them using Wasserstein distances defined by the optimal transport theory [37]. These distances have indeed been successfully used in a variety of applications in machine learning [11, 3, 25, 23, 19, 1] and in statistics [39, 12, 8, 35]. In the discrete setting, many efficient algorithms have been proposed to compute or approximate the Wasserstein distances, such as Sinkhorn-Knopp and auction algorithms - see [34] and references therein.

Emergent properties of the local geometry of neural loss landscapesStanislav Fort Surya Ganguli Stanford University Stanford, CA, USA Stanford University Stanford, CA, USA Abstract The local geometry of high dimensional neural network loss landscapes can both challenge our cherished theoretical intuitions as well as dramatically impact the practical success of neural network training. Indeed recent works have observed 4 striking local properties of neural loss landscapes on classification tasks: (1) the landscape exhibits exactly C directions of high positive curvature, where C is the number of classes; (2) gradient directions are largely confined to this extremely low dimensional subspace of positive Hessian curvature, leaving the vast majority of directions in weight space unexplored; (3) gradient descent transiently explores intermediate regions of higher positive curvature before eventually finding flatter minima; (4) training can be successful even when confined to low dimensional random affine hy-perplanes, as long as these hyperplanes intersect a Goldilocks zone of higher than average curvature. We develop a simple theoretical model of gradients and Hessians, justified by numerical experiments on architectures and datasets used in practice, that simultaneously accounts for all 4 of these surprising and seemingly unrelated properties. Our unified model provides conceptual insights into the emergence of these properties and makes connections with diverse topics in neural networks, random matrix theory, and spin glasses, including the neural tangent kernel, BBP phase transitions, and Derrida's random energy model. 1 Introduction The geometry of neural network loss landscapes and the implications of this geometry for both optimization and generalization have been subjects of intense interest in many works, ranging from studies on the lack of local minima at significantly higher loss than that of the global minimum [1, 2] to studies debating relations between the curvature of local minima and their generalization properties [3, 4, 5, 6]. Fundamentally, the neural network loss landscape is a scalar loss function over a very high D dimensional parameter space that could depend a priori in highly nontrivial ways on the very structure of real-world data itself as well as intricate properties of the neural network architecture. Moreover, the regions of this loss landscape explored by gradient descent could themselves have highly atypical geometric properties relative to randomly chosen points in the landscape.

Contextual bandits are online learners that, given an input, select an arm and receive a reward for that arm. They use the reward as a learning signal and aim to maximize the total reward over the inputs. Contextual bandits are commonly used to solve recommendation or ranking problems. This paper considers a learning setting in which multiple parties aim to train a contextual bandit together in a private way: the parties aim to maximize the total reward but do not want to share any of the relevant information they possess with the other parties. Specifically, multiple parties have access to (different) features that may benefit the learner but that cannot be shared with other parties. One of the parties pulls the arm but other parties may not learn which arm was pulled. One party receives the reward but the other parties may not learn the reward value. This paper develops a privacy-preserving contextual bandit algorithm that combines secure multi-party computation with a differential private mechanism based on epsilon-greedy exploration in contextual bandits.

Choosing the most adequate kernel is crucial in many Machine Learning applications. Gaussian Process is a state-of-the-art technique for regression and classification that heavily relies on a kernel function. However, in the Gaussian Process literature, kernels have usually been either ad hoc designed, selected from a predefined set, or searched for in a space of compositions of kernels which have been defined a priori. In this paper, we propose a Genetic-Programming algorithm that represents a kernel function as a tree of elementary mathematical expressions. By means of this representation, a wider set of kernels can be modeled, where potentially better solutions can be found, although new challenges also arise. The proposed algorithm is able to overcome these difficulties and find kernels that accurately model the characteristics of the data. This method has been tested in several real-world time-series extrapolation problems, improving the state-of-the-art results while reducing the complexity of the kernels.

Mathematical modeling with Ordinary Differential Equations (ODEs) has proven to be extremely successful in a variety of fields, including biology. However, these models are completely deterministic given a certain set of initial conditions. We convert mathematical ODE models of three benchmark biological systems to Dynamic Bayesian Networks (DBNs). The DBN model can handle model uncertainty and data uncertainty in a principled manner. They can be used for temporal data mining for noisy and missing variables. We apply Particle Filtering algorithm to infer the model variables by re-estimating the models parameters of various biological ODE models. The model parameters are automatically re-estimated using temporal evidence in the form of data streams. The results show that DBNs are capable of inferring the model variables of the ODE model with high accuracy in situations where data is missing, incomplete, sparse and irregular and true values of model parameters are not known.

Causal inference is a fundemental problem in statistics and has wide applications in different fields. Transfer Entropy (TE) is a important notion defined for measuring causality, which is essentially conditional Mutual Information (MI). Copula Entropy (CE) is a theory on measurement of statistical independence and is equivalent to MI. In this paper, we prove that TE can be represented with only CE and then propose a non-parametric method for estimating TE via CE. The proposed method was applied to analyze the Beijing PM2.5 data in the experiments. Experimental results show that the proposed method can infer causality relationships from data effectively and hence help to understand the data better.

We present multi-point optimization: an optimization technique that allows to train several models simultaneously without the need to keep the parameters of each one individually. The proposed method is used for a thorough empirical analysis of the loss landscape of neural networks. By extensive experiments on FashionMNIST and CIFAR10 datasets we demonstrate two things: 1) loss surface is surprisingly diverse and intricate in terms of landscape patterns it contains, and 2) adding batch normalization makes it more smooth. Source code to reproduce all the reported results is available on GitHub: https://github.com/universome/loss-patterns.

Satellite imaging is a critical technology for monitoring and responding to natural disasters such as flooding. Despite the capabilities of modern satellites, there is still much to be desired from the perspective of first response organisations like UNICEF. Two main challenges are rapid access to data, and the ability to automatically identify flooded regions in images. We describe a prototypical flood segmentation system, identifying cloud, water and land, that could be deployed on a constellation of small satellites, performing processing on board to reduce downlink bandwidth by 2 orders of magnitude. We target PhiSat-1, part of the FSSCAT mission, which is planned to be launched by the European Space Agency (ESA) near the start of 2020 as a proof of concept for this new technology.

Multi-step greedy policies have been extensively used in model-based Reinforcement Learning (RL) and in the case when a model of the environment is available (e.g., in the game of Go). In this work, we explore the benefits of multi-step greedy policies in model-free RL when employed in the framework of multi-step Dynamic Programming (DP): multi-step Policy and Value Iteration. These algorithms iteratively solve short-horizon decision problems and converge to the optimal solution of the original one. By using model-free algorithms as solvers of the short-horizon problems we derive fully model-free algorithms which are instances of the multi-step DP framework. As model-free algorithms are prone to instabilities w.r.t. the decision problem horizon, this simple approach can help in mitigating these instabilities and results in an improved model-free algorithms. We test this approach and show results on both discrete and continuous control problems.

Magnetoencephalography and electroencephalography (M/EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity. Estimating the location and magnitude of the current sources that generated these electromagnetic fields is a challenging ill-posed regression problem known as \emph{source imaging}. When considering a group study, a common approach consists in carrying out the regression tasks independently for each subject. An alternative is to jointly localize sources for all subjects taken together, while enforcing some similarity between them. By pooling all measurements in a single multi-task regression, one makes the problem better posed, offering the ability to identify more sources and with greater precision. The Minimum Wasserstein Estimates (MWE) promotes focal activations that do not perfectly overlap for all subjects, thanks to a regularizer based on Optimal Transport (OT) metrics. MWE promotes spatial proximity on the cortical mantel while coping with the varying noise levels across subjects. On realistic simulations, MWE decreases the localization error by up to 4 mm per source compared to individual solutions. Experiments on the Cam-CAN dataset show a considerable improvement in spatial specificity in population imaging. Our analysis of a multimodal dataset shows how multi-subject source localization closes the gap between MEG and fMRI for brain mapping.