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We propose a novel method for sampling and optimization tasks based on a stochastic interacting particle system. We explain how this method can be used for the following two goals: (i) generating approximate samples from a given target distribution; (ii) optimizing a given objective function. The approach is derivative-free and affine invariant, and is therefore well-suited for solving inverse problems defined by complex forward models: (i) allows generation of samples from the Bayesian posterior and (ii) allows determination of the maximum a posteriori estimator. We investigate the properties of the proposed family of methods in terms of various parameter choices, both analytically and by means of numerical simulations. The analysis and numerical simulation establish that the method has potential for general purpose optimization tasks over Euclidean space; contraction properties of the algorithm are established under suitable conditions, and computational experiments demonstrate wide basins of attraction for various specific problems. The analysis and experiments also demonstrate the potential for the sampling methodology in regimes in which the target distribution is unimodal and close to Gaussian; indeed we prove that the method recovers a Laplace approximation to the measure in certain parametric regimes and provide numerical evidence that this Laplace approximation attracts a large set of initial conditions in a number of examples.

In clinical research, the effect of a treatment or intervention is widely assessed through clinical importance, instead of statistical significance. In this paper, we propose a principled statistical inference framework to learning the minimal clinically important difference (MCID), a vital concept in assessing clinical importance. We formulate the scientific question into a novel statistical learning problem, develop an efficient algorithm for parameter estimation, and establish the asymptotic theory for the proposed estimator. We conduct comprehensive simulation studies to examine the finite sample performance of the proposed method. We also re-analyze the ChAMP (Chondral Lesions And Meniscus Procedures) trial, where the primary outcome is the patient-reported pain score and the ultimate goal is to determine whether there exists a significant difference in post-operative knee pain between patients undergoing debridement versus observation of chondral lesions during the surgery. Some previous analysis of this trial exhibited that the effect of debriding the chondral lesions does not reach a statistical significance. Our analysis reinforces this conclusion that the effect of debriding the chondral lesions is not only statistically non-significant, but also clinically un-important.

This paper examines the performance trade-offs between an introduced linear flexibility market model for congestion management and a benchmark second-order cone programming (SOCP) formulation. The linear market model incorporates voltage magnitudes and reactive powers, while providing a simpler formulation than the SOCP model, which enables its practical implementation. The paper provides a structured comparison of the two formulations relying on developed deterministic and statistical Monte Carlo case analyses using two distribution test systems (the Matpower 69-bus and 141-bus systems). The case analyses show that with the increasing spread of offered flexibility throughout the system, the linear formulation increasingly preserves the reliability of the computed system variables as compared to the SOCP formulation, while more lenient imposed voltage limits can improve the approximation of prices and power flows at the expense of a less accurate computation of voltage magnitudes.

Gaussian process (GP) models that combine both categorical and continuous input variables have found use e.g. in longitudinal data analysis and computer experiments. However, standard inference for these models has the typical cubic scaling, and common scalable approximation schemes for GPs cannot be applied since the covariance function is non-continuous. In this work, we derive a basis function approximation scheme for mixed-domain covariance functions, which scales linearly with respect to the number of observations and total number of basis functions. The proposed approach is naturally applicable to Bayesian GP regression with arbitrary observation models. We demonstrate the approach in a longitudinal data modelling context and show that it approximates the exact GP model accurately, requiring only a fraction of the runtime compared to fitting the corresponding exact model.

We introduce a flexible and scalable class of Bayesian geostatistical models for discrete data, based on the class of nearest neighbor mixture transition distribution processes (NNMP), referred to as discrete NNMP. The proposed class characterizes spatial variability by a weighted combination of first-order conditional probability mass functions (pmfs) for each one of a given number of neighbors. The approach supports flexible modeling for multivariate dependence through specification of general bivariate discrete distributions that define the conditional pmfs. Moreover, the discrete NNMP allows for construction of models given a pre-specified family of marginal distributions that can vary in space, facilitating covariate inclusion. In particular, we develop a modeling and inferential framework for copula-based NNMPs that can attain flexible dependence structures, motivating the use of bivariate copula families for spatial processes. Compared to the traditional class of spatial generalized linear mixed models, where spatial dependence is introduced through a transformation of response means, our process-based modeling approach provides both computational and inferential advantages. We illustrate the benefits with synthetic data examples and an analysis of North American Breeding Bird Survey data.

The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.

This paper aims to explore models based on the extreme gradient boosting (XGBoost) approach for business risk classification. Feature selection (FS) algorithms and hyper-parameter optimizations are simultaneously considered during model training. The five most commonly used FS methods including weight by Gini, weight by Chi-square, hierarchical variable clustering, weight by correlation, and weight by information are applied to alleviate the effect of redundant features. Two hyper-parameter optimization approaches, random search (RS) and Bayesian tree-structured Parzen Estimator (TPE), are applied in XGBoost. The effect of different FS and hyper-parameter optimization methods on the model performance are investigated by the Wilcoxon Signed Rank Test. The performance of XGBoost is compared to the traditionally utilized logistic regression (LR) model in terms of classification accuracy, area under the curve (AUC), recall, and F1 score obtained from the 10-fold cross validation. Results show that hierarchical clustering is the optimal FS method for LR while weight by Chi-square achieves the best performance in XG-Boost. Both TPE and RS optimization in XGBoost outperform LR significantly. TPE optimization shows a superiority over RS since it results in a significantly higher accuracy and a marginally higher AUC, recall and F1 score. Furthermore, XGBoost with TPE tuning shows a lower variability than the RS method. Finally, the ranking of feature importance based on XGBoost enhances the model interpretation. Therefore, XGBoost with Bayesian TPE hyper-parameter optimization serves as an operative while powerful approach for business risk modeling.

Combining Bayesian nonparametrics and a forward model selection strategy, we construct parsimonious Bayesian deep networks (PBDNs) that infer capacity-regularized network architectures from the data and require neither cross-validation nor fine-tuning when training the model. One of the two essential components of a PBDN is the development of a special infinite-wide single-hidden-layer neural network, whose number of active hidden units can be inferred from the data. The other one is the construction of a greedy layer-wise learning algorithm that uses a forward model selection criterion to determine when to stop adding another hidden layer. We develop both Gibbs sampling and stochastic gradient descent based maximum a posteriori inference for PBDNs, providing state-of-the-art classification accuracy and interpretable data subtypes near the decision boundaries, while maintaining low computational complexity for out-of-sample prediction.

Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.