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Echo State Networks (ESN) are a type of Recurrent Neural Networks that yields promising results in representing time series and nonlinear dynamic systems. Although they are equipped with a very efficient training procedure, Reservoir Computing strategies, such as the ESN, require the use of high order networks, i.e. large number of layers, resulting in number of states that is magnitudes higher than the number of model inputs and outputs. This not only makes the computation of a time step more costly, but also may pose robustness issues when applying ESNs to problems such as Model Predictive Control (MPC) and other optimal control problems. One such way to circumvent this is through Model Order Reduction strategies such as the Proper Orthogonal Decomposition (POD) and its variants (POD-DEIM), whereby we find an equivalent lower order representation to an already trained high dimension ESN. The objective of this work is to investigate and analyze the performance of POD methods in Echo State Networks, evaluating their effectiveness. To this end, we evaluate the Memory Capacity (MC) of the POD-reduced network in comparison to the original (full order) ENS. We also perform experiments on two different numerical case studies: a NARMA10 difference equation and an oil platform containing two wells and one riser. The results show that there is little loss of performance comparing the original ESN to a POD-reduced counterpart, and also that the performance of a POD-reduced ESN tend to be superior to a normal ESN of the same size. Also we attain speedups of around $80\%$ in comparison to the original ESN.

The notion of concept drift refers to the phenomenon that the distribution generating the observed data changes over time. If drift is present, machine learning models may become inaccurate and need adjustment. Many technologies for learning with drift rely on the interleaved test-train error (ITTE) as a quantity which approximates the model generalization error and triggers drift detection and model updates. In this work, we investigate in how far this procedure is mathematically justified. More precisely, we relate a change of the ITTE to the presence of real drift, i.e., a changed posterior, and to a change of the training result under the assumption of optimality. We support our theoretical findings by empirical evidence for several learning algorithms, models, and datasets.

We study the bias of classical quantile regression and instrumental variable quantile regression estimators. While being asymptotically first-order unbiased, these estimators can have non-negligible second-order biases. We derive a higher-order stochastic expansion of these estimators using empirical process theory. Based on this expansion, we derive an explicit formula for the second-order bias and propose a feasible bias correction procedure that uses finite-difference estimators of the bias components. The proposed bias correction method performs well in simulations. We provide an empirical illustration using Engel's classical data on household expenditure.

In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a $hp$-version discontinuous Galerkin finite element approximation in the time direction with an $H^1(\Omega)$-conforming finite element approximation in the spatial variables. Error bounds at the temporal nodal points are derived under a weak restriction on the temporal step size in terms of the spatial mesh size. Numerical experiments are presented to verify the theoretical results.

This paper studies the high-dimensional quantile regression problem under the transfer learning framework, where possibly related source datasets are available to make improvements on the estimation or prediction based solely on the target data. In the oracle case with known transferable sources, a smoothed two-step transfer learning algorithm based on convolution smoothing is proposed and the L1/L2 estimation error bounds of the corresponding estimator are also established. To avoid including non-informative sources, we propose a clustering-based algorithm to select the transferable sources adaptively and establish its selection consistency under regular conditions; we also provide an alternative model averaging procedure, of which the optimality of the excess risk is proved. Monte Carlo simulations as well as an empirical analysis of gene expression data demonstrate the effectiveness of the proposed procedure.

This paper proposes Distributed Model Predictive Covariance Steering (DMPCS), a novel method for safe multi-robot control under uncertainty. The scope of our approach is to blend covariance steering theory, distributed optimization and model predictive control (MPC) into a single methodology that is safe, scalable and decentralized. Initially, we pose a problem formulation that uses the Wasserstein distance to steer the state distributions of a multi-robot team to desired targets, and probabilistic constraints to ensure safety. We then transform this problem into a finite-dimensional optimization one by utilizing a disturbance feedback policy parametrization for covariance steering and a tractable approximation of the safety constraints. To solve the latter problem, we derive a decentralized consensus-based algorithm using the Alternating Direction Method of Multipliers (ADMM). This method is then extended to a receding horizon form, which yields the proposed DMPCS algorithm. Simulation experiments on large-scale problems with up to hundreds of robots successfully demonstrate the effectiveness and scalability of DMPCS. Its superior capability in achieving safety is also highlighted through a comparison against a standard stochastic MPC approach. A video with all simulation experiments is available in //youtu.be/Hks-0BRozxA.

Many scientific problems require identifying a small set of covariates that are associated with a target response and estimating their effects. Often, these effects are nonlinear and include interactions, so linear and additive methods can lead to poor estimation and variable selection. Unfortunately, methods that simultaneously express sparsity, nonlinearity, and interactions are computationally intractable -- with runtime at least quadratic in the number of covariates, and often worse. In the present work, we solve this computational bottleneck. We show that suitable interaction models have a kernel representation, namely there exists a "kernel trick" to perform variable selection and estimation in $O$(# covariates) time. Our resulting fit corresponds to a sparse orthogonal decomposition of the regression function in a Hilbert space (i.e., a functional ANOVA decomposition), where interaction effects represent all variation that cannot be explained by lower-order effects. On a variety of synthetic and real data sets, our approach outperforms existing methods used for large, high-dimensional data sets while remaining competitive (or being orders of magnitude faster) in runtime.

The present study is an extension of the work done in [16] and [10], where a two-level Parareal method with averaging was examined. The method proposed in this paper is a multi-level Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for multi-scale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level.

The Gaussian mechanism is one differential privacy mechanism commonly used to protect numerical data. However, it may be ill-suited to some applications because it has unbounded support and thus can produce invalid numerical answers to queries, such as negative ages or human heights in the tens of meters. One can project such private values onto valid ranges of data, though such projections lead to the accumulation of private query responses at the boundaries of such ranges, thereby harming accuracy. Motivated by the need for both privacy and accuracy over bounded domains, we present a bounded Gaussian mechanism for differential privacy, which has support only on a given region. We present both univariate and multivariate versions of this mechanism and illustrate a significant reduction in variance relative to comparable existing work.

Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they require the knowledge of the values of the expectations to estimate. Motivated by the form of these optimal estimators and some interesting properties, we therefore propose an adaptive algorithm. The general idea is to adaptively update the parameters of the estimators for approaching the optimal ones. We suggest then a quantitative stopping criterion that exploits the trade-off between approaching these optimal parameters and having a sufficient budget left. This left budget is then used to draw a new independent sample from the final sampling distribution, allowing to get unbiased estimators of the expectations. We show how to apply our procedure to sensitivity analysis, by estimating Sobol' indices and quantifying the impact of the input distributions. Finally, realistic test cases show the practical interest of the proposed algorithm, and its significant improvement over estimating the expectations separately.