Improving the resilience of a network protects the system from natural disasters and malicious attacks. This is typically achieved by introducing new edges, which however may reach beyond the maximum number of connections a node could sustain. Many studies then resort to the degree-preserving operation of rewiring, which swaps existing edges $AC, BD$ to new edges $AB, CD$. A significant line of studies focuses on this technique for theoretical and practical results while leaving three limitations: network utility loss, local optimality, and transductivity. In this paper, we propose ResiNet, a reinforcement learning (RL)-based framework to discover resilient network topologies against various disasters and attacks. ResiNet is objective agnostic which allows the utility to be balanced by incorporating it into the objective function. The local optimality, typically seen in greedy algorithms, is addressed by casting the cumulative resilience gain into a sequential decision process of step-wise rewiring. The transductivity, which refers to the necessity to run a computationally intensive optimization for each input graph, is lifted by our variant of RL with auto-regressive permutation-invariant variable action space. ResiNet is armed by our technical innovation, Filtration enhanced GNN (FireGNN), which distinguishes graphs with minor differences. It is thus possible for ResiNet to capture local structure changes and adapt its decision among consecutive graphs, which is known to be infeasible for GNN. Extensive experiments demonstrate that with a small number of rewiring operations, ResiNet achieves a near-optimal resilience gain on multiple graphs while balancing the utility, with a large margin compared to existing approaches.

We introduce a new library named abess that implements a unified framework of best-subset selection for solving diverse machine learning problems, e.g., linear regression, classification, and principal component analysis. Particularly, the abess certifiably gets the optimal solution within polynomial times under the linear model. Our efficient implementation allows abess to attain the solution of best-subset selection problems as fast as or even 100x faster than existing competing variable (model) selection toolboxes. Furthermore, it supports common variants like best group subset selection and $\ell_2$ regularized best-subset selection. The core of the library is programmed in C++. For ease of use, a Python library is designed for conveniently integrating with scikit-learn, and it can be installed from the Python library Index. In addition, a user-friendly R library is available at the Comprehensive R Archive Network. The source code is available at: https://github.com/abess-team/abess.

Exploration and analysis of massive datasets has recently generated increasing interest in the research and development communities. It has long been a recognized problem that many datasets contain significant levels of missing numerical data. We introduce a mathematically principled stochastic optimization imputation method based on the theory of Kriging. This is shown to be a powerful method for imputation. However, its computational effort and potential numerical instabilities produce costly and/or unreliable predictions, potentially limiting its use on large scale datasets. In this paper, we apply a recently developed multi-level stochastic optimization approach to the problem of imputation in massive medical records. The approach is based on computational applied mathematics techniques and is highly accurate. In particular, for the Best Linear Unbiased Predictor (BLUP) this multi-level formulation is exact, and is also significantly faster and more numerically stable. This permits practical application of Kriging methods to data imputation problems for massive datasets. We test this approach on data from the National Inpatient Sample (NIS) data records, Healthcare Cost and Utilization Project (HCUP), Agency for Healthcare Research and Quality. Numerical results show the multi-level method significantly outperforms current approaches and is numerically robust. In particular, it has superior accuracy as compared with methods recommended in the recent report from HCUP on the important problem of missing data, which could lead to sub-optimal and poorly based funding policy decisions. In comparative benchmark tests it is shown that the multilevel stochastic method is significantly superior to recommended methods in the report, including Predictive Mean Matching (PMM) and Predicted Posterior Distribution (PPD), with up to 75% reductions in error.

Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory.

This is a tutorial and survey paper on various methods for Sufficient Dimension Reduction (SDR). We cover these methods with both statistical high-dimensional regression perspective and machine learning approach for dimensionality reduction. We start with introducing inverse regression methods including Sliced Inverse Regression (SIR), Sliced Average Variance Estimation (SAVE), contour regression, directional regression, Principal Fitted Components (PFC), Likelihood Acquired Direction (LAD), and graphical regression. Then, we introduce forward regression methods including Principal Hessian Directions (pHd), Minimum Average Variance Estimation (MAVE), Conditional Variance Estimation (CVE), and deep SDR methods. Finally, we explain Kernel Dimension Reduction (KDR) both for supervised and unsupervised learning. We also show that supervised KDR and supervised PCA are equivalent.

At the core of statistical learning lies the problem of understanding the generalization performance (e.g., out-of-sample errors) of the learning algorithms in use. Conventional wisdom in statistics held that including too many covariates when training statistical models can hurt generalization (despite improving training accuracy), due to the undesired over-fit. This leads to the classical conclusion that: proper regularization -- through either adding certain penalty functions to the loss function or algorithmic self-regularization -- seems to be critical in achieving the desired accuracy (e.g., Friedman et al. (2001); Wei et al. (2019)). However, an evolving line of works in machine learning observes empirical evidence that suggests, to the surprise of many statisticians, over-parameterization is not necessarily harmful. Indeed, many machine learning models (such as random forests or deep neural networks) are trained until the training error vanishes to zero -- meaning that they are able to perfectly interpolate the data -- while still generalizing well (e.g., Zhang et al. (2021); Wyner et al. (2017); Belkin et al. (2019)). As a key observation to explain this phenomenon, many models when trained by gradient type methods (e.g., gradient descent, stochastic gradient descent, AdaBoost) converge to certain minimum norm interpolators, which implicitly favor models with smaller model complexity. These empirical mysteries inspire a recent flurry of activity towards understanding the generalization properties of various interpolators.

Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions on the form of the component distributions. Additionally, we establish a key equivalence between kernel-based data-clustering and kernel density-based clustering. This enables us to provide consistency guarantees for kernel-based estimators of non-parametric mixture models. Along with theoretical implications, this connection could have practical implications, including in the systematic choice of the bandwidth of the Gaussian kernel in the context of clustering.

We propose a novel class of graph neural networks based on the discretised Beltrami flow, a non-Euclidean diffusion PDE. In our model, node features are supplemented with positional encodings derived from the graph topology and jointly evolved by the Beltrami flow, producing simultaneously continuous feature learning and topology evolution. The resulting model generalises many popular graph neural networks and achieves state-of-the-art results on several benchmarks.

Random forests and recursive trees are widely used in applied statistics and computer science. The popularity of recursive trees relies on several factors: their easy interpretability, the fact that they can be used for both regression and classification tasks, the small number of hyper-parameters to be tuned and finally, their non-parametric nature that allows their use to infer arbitrarily complex relations between the input and the output space. A random forest combines several randomized trees, improving the prediction accuracy at a cost of a slight lost in interpretation. This technique is easily parallelizable which has made it one of the most popular tools for handling high dimensional data sets. It has been successfully involved in various practical problems, including chemioinformatics, ecology, 3D object recognition, bioinformatics and econometrics. Biau and Scornet (2016) present a detailed list of applications as well as a review on random forests. In the present work we have focused on the ability of random forests to deal with missing values.

Self-supervised representation learning methods aim to provide powerful deep feature learning without the requirement of large annotated datasets, thus alleviating the annotation bottleneck that is one of the main barriers to practical deployment of deep learning today. These methods have advanced rapidly in recent years, with their efficacy approaching and sometimes surpassing fully supervised pre-training alternatives across a variety of data modalities including image, video, sound, text and graphs. This article introduces this vibrant area including key concepts, the four main families of approach and associated state of the art, and how self-supervised methods are applied to diverse modalities of data. We further discuss practical considerations including workflows, representation transferability, and compute cost. Finally, we survey the major open challenges in the field that provide fertile ground for future work.