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Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic (PD) model is used in regions of fracture and smooth PUM is used in the surrounding linear elastic media. The method is a so-called global-local enrichment strategy. The elastic fields of the undamaged media provide appropriate boundary data for the localized PD simulations. The first steps for a combined PD/PUM simulator are presented. In part I of this series, we show that the local PD approximation can be utilized to enrich the global PUM approximation to capture the true material response with high accuracy efficiently. Test problems are provided demonstrating the validity and potential of this numerical approach.

In this paper, we provide bounds in Wasserstein and total variation distances between the distributions of the successive iterates of two functional autoregressive processes with isotropic Gaussian noise of the form $Y_{k+1} = \mathrm{T}_\gamma(Y_k) + \sqrt{\gamma\sigma^2} Z_{k+1}$ and $\tilde{Y}_{k+1} = \tilde{\mathrm{T}}_\gamma(\tilde{Y}_k) + \sqrt{\gamma\sigma^2} \tilde{Z}_{k+1}$. More precisely, we give non-asymptotic bounds on $\rho(\mathcal{L}(Y_{k}),\mathcal{L}(\tilde{Y}_k))$, where $\rho$ is an appropriate weighted Wasserstein distance or a $V$-distance, uniformly in the parameter $\gamma$, and on $\rho(\pi_{\gamma},\tilde{\pi}_{\gamma})$, where $\pi_{\gamma}$ and $\tilde{\pi}_{\gamma}$ are the respective stationary measures of the two processes. The class of considered processes encompasses the Euler-Maruyama discretization of Langevin diffusions and its variants. The bounds we derive are of order $\gamma$ as $\gamma \to 0$. To obtain our results, we rely on the construction of a discrete sticky Markov chain $(W_k^{(\gamma)})_{k \in \mathbb{N}}$ which bounds the distance between an appropriate coupling of the two processes. We then establish stability and quantitative convergence results for this process uniformly on $\gamma$. In addition, we show that it converges in distribution to the continuous sticky process studied in previous work. Finally, we apply our result to Bayesian inference of ODE parameters and numerically illustrate them on two particular problems.

Monte Carlo simulations of physics processes at particle colliders like the Large Hadron Collider at CERN take up a major fraction of the computational budget. For some simulations, a single data point takes seconds, minutes, or even hours to compute from first principles. Since the necessary number of data points per simulation is on the order of $10^9$ - $10^{12}$, machine learning regressors can be used in place of physics simulators to significantly reduce this computational burden. However, this task requires high-precision regressors that can deliver data with relative errors of less than $1\%$ or even $0.1\%$ over the entire domain of the function. In this paper, we develop optimal training strategies and tune various machine learning regressors to satisfy the high-precision requirement. We leverage symmetry arguments from particle physics to optimize the performance of the regressors. Inspired by ResNets, we design a Deep Neural Network with skip connections that outperform fully connected Deep Neural Networks. We find that at lower dimensions, boosted decision trees far outperform neural networks while at higher dimensions neural networks perform significantly better. We show that these regressors can speed up simulations by a factor of $10^3$ - $10^6$ over the first-principles computations currently used in Monte Carlo simulations. Additionally, using symmetry arguments derived from particle physics, we reduce the number of regressors necessary for each simulation by an order of magnitude. Our work can significantly reduce the training and storage burden of Monte Carlo simulations at current and future collider experiments.

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.

We study the dynamics and implicit bias of gradient flow (GF) on univariate ReLU neural networks with a single hidden layer in a binary classification setting. We show that when the labels are determined by the sign of a target network with $r$ neurons, with high probability over the initialization of the network and the sampling of the dataset, GF converges in direction (suitably defined) to a network achieving perfect training accuracy and having at most $\mathcal{O}(r)$ linear regions, implying a generalization bound. Unlike many other results in the literature, under an additional assumption on the distribution of the data, our result holds even for mild over-parameterization, where the width is $\tilde{\mathcal{O}}(r)$ and independent of the sample size.

Overparametrization often helps improve the generalization performance. This paper proposes a dual view of overparametrization suggesting that downsampling may also help generalize. Motivated by this dual view, we characterize two out-of-sample prediction risks of the sketched ridgeless least square estimator in the proportional regime $m\asymp n \asymp p$, where $m$ is the sketching size, $n$ the sample size, and $p$ the feature dimensionality. Our results reveal the statistical role of downsampling. Specifically, downsampling does not always hurt the generalization performance, and may actually help improve it in some cases. We identify the optimal sketching sizes that minimize the out-of-sample prediction risks, and find that the optimally sketched estimator has stabler risk curves that eliminates the peaks of those for the full-sample estimator. We then propose a practical procedure to empirically identify the optimal sketching size. Finally, we extend our results to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

Modern reinforcement learning (RL) often faces an enormous state-action space. Existing analytical results are typically for settings with a small number of state-actions, or simple models such as linearly modeled Q-functions. To derive statistically efficient RL policies handling large state-action spaces, with more general Q-functions, some recent works have considered nonlinear function approximation using kernel ridge regression. In this work, we derive sample complexities for kernel based Q-learning when a generative model exists. We propose a nonparametric Q-learning algorithm which finds an $\epsilon$-optimal policy in an arbitrarily large scale discounted MDP. The sample complexity of the proposed algorithm is order optimal with respect to $\epsilon$ and the complexity of the kernel (in terms of its information gain). To the best of our knowledge, this is the first result showing a finite sample complexity under such a general model.

The proliferation of automated data collection schemes and the advances in sensorics are increasing the amount of data we are able to monitor in real-time. However, given the high annotation costs and the time required by quality inspections, data is often available in an unlabeled form. This is fostering the use of active learning for the development of soft sensors and predictive models. In production, instead of performing random inspections to obtain product information, labels are collected by evaluating the information content of the unlabeled data. Several query strategy frameworks for regression have been proposed in the literature but most of the focus has been dedicated to the static pool-based scenario. In this work, we propose a new strategy for the stream-based scenario, where instances are sequentially offered to the learner, which must instantaneously decide whether to perform the quality check to obtain the label or discard the instance. The approach is inspired by the optimal experimental design theory and the iterative aspect of the decision-making process is tackled by setting a threshold on the informativeness of the unlabeled data points. The proposed approach is evaluated using numerical simulations and the Tennessee Eastman Process simulator. The results confirm that selecting the examples suggested by the proposed algorithm allows for a faster reduction in the prediction error.

Predictive models -- as with machine learning -- can underpin causal inference, to estimate the effects of an intervention at the population or individual level. This opens the door to a plethora of models, useful to match the increasing complexity of health data, but also the Pandora box of model selection: which of these models yield the most valid causal estimates? Classic machine-learning cross-validation procedures are not directly applicable. Indeed, an appropriate selection procedure for causal inference should equally weight both outcome errors for each individual, treated or not treated, whereas one outcome may be seldom observed for a sub-population. We study how more elaborate risks benefit causal model selection. We show theoretically that simple risks are brittle to weak overlap between treated and non-treated individuals as well as to heterogeneous errors between populations. Rather a more elaborate metric, the R-risk appears as a proxy of the oracle error on causal estimates, observable at the cost of an overlap re-weighting. As the R-risk is defined not only from model predictions but also by using the conditional mean outcome and the treatment probability, using it for model selection requires adapting cross validation. Extensive experiments show that the resulting procedure gives the best causal model selection.

We propose a paradigm for interpretable Manifold Learning for scientific data analysis, whereby we parametrize a manifold with $d$ smooth functions from a scientist-provided dictionary of meaningful, domain-related functions. When such a parametrization exists, we provide an algorithm for finding it based on sparse non-linear regression in the manifold tangent bundle, bypassing more standard manifold learning algorithms. We also discuss conditions for the existence of such parameterizations in function space and for successful recovery from finite samples. We demonstrate our method with experimental results from a real scientific domain.