Sparse parametric models are of great interest in statistical learning and are often analyzed by means of regularized estimators. Pathwise methods allow to efficiently compute the full solution path for penalized estimators, for any possible value of the penalization parameter $\lambda$. In this paper we deal with the pathwise optimization for bridge-type problems; i.e. we are interested in the minimization of a loss function, such as negative log-likelihood or residual sum of squares, plus the sum of $\ell^q$ norms with $q\in(0,1]$ involving adpative coefficients. For some loss functions this regularization achieves asymptotically the oracle properties (such as the selection consistency). Nevertheless, since the objective function involves nonconvex and nondifferentiable terms, the minimization problem is computationally challenging. The aim of this paper is to apply some general algorithms, arising from nonconvex optimization theory, to compute efficiently the path solutions for the adaptive bridge estimator with multiple penalties. In particular, we take into account two different approaches: accelerated proximal gradient descent and blockwise alternating optimization. The convergence and the path consistency of these algorithms are discussed. In order to assess our methods, we apply these algorithms to the penalized estimation of diffusion processes observed at discrete times. This latter represents a recent research topic in the field of statistics for time-dependent data.

Training data attribution (TDA) is the task of attributing model behavior to elements in the training data. This paper draws attention to the common setting where one has access only to the final trained model, and not the training algorithm or intermediate information from training. To serve as a gold standard for TDA in this "final-model-only" setting, we propose further training, with appropriate adjustment and averaging, to measure the sensitivity of the given model to training instances. We then unify existing gradient-based methods for TDA by showing that they all approximate the further training gold standard in different ways. We investigate empirically the quality of these gradient-based approximations to further training, for tabular, image, and text datasets and models. We find that the approximation quality of first-order methods is sometimes high but decays with the amount of further training. In contrast, the approximations given by influence function methods are more stable but surprisingly lower in quality.

The weak-to-strong generalization phenomenon is the driver for important machine learning applications including highly data-efficient learning and, most recently, performing superalignment. While decades of research have resulted in numerous algorithms that produce strong empirical performance, understanding what aspects of data enable weak-to-strong generalization has been understudied. We propose a simple data-centric mechanism that characterizes weak-to-strong generalization: the overlap density. Intuitively, generalization tracks the number of points that contain overlaps, i.e., both easy patterns (learnable by a weak model) and challenging patterns (only learnable by a stronger model), as with such points, weak predictions can be used to learn challenging patterns by stronger models. We provide a practical overlap detection algorithm to find such points in datasets and leverage them to learn, among multiple sources of data, which to query when seeking to maximize overlap density and thereby enhance weak-to-strong generalization. We present a theoretical result showing that the generalization benefit is a function of the overlap density and a regret bound for our data selection algorithm. Empirically, we validate the mechanism and the overlap detection algorithm on a wide array of settings.

This paper presents a novel approach for optimal control of nonlinear stochastic systems using infinitesimal generator learning within infinite-dimensional reproducing kernel Hilbert spaces. Our learning framework leverages data samples of system dynamics and stage cost functions, with only control penalties and constraints provided. The proposed method directly learns the diffusion operator of a controlled Fokker-Planck-Kolmogorov equation in an infinite-dimensional hypothesis space. This operator models the continuous-time evolution of the probability measure of the control system's state. We demonstrate that this approach seamlessly integrates with modern convex operator-theoretic Hamilton-Jacobi-Bellman recursions, enabling a data-driven solution to the optimal control problem. Furthermore, our statistical learning framework includes nonparametric estimators for uncontrolled forward infinitesimal generators as a special case. Numerical experiments, ranging from synthetic differential equations to simulated robotic systems, showcase the advantages of our approach compared to both modern data-driven and classical nonlinear programming methods for optimal control.

The elastic net penalty is frequently employed in high-dimensional statistics for parameter regression and variable selection. It is particularly beneficial compared to lasso when the number of predictors greatly surpasses the number of observations. However, empirical evidence has shown that the $\ell_q$-norm penalty (where $0 < q < 1$) often provides better regression compared to the $\ell_1$-norm penalty, demonstrating enhanced robustness in various scenarios. In this paper, we explore a generalized elastic net model that employs a $\ell_r$-norm (where $r \geq 1$) in loss function to accommodate various types of noise, and employs a $\ell_q$-norm (where $0 < q < 1$) to replace the $\ell_1$-norm in elastic net penalty. Theoretically, we establish the computable lower bounds for the nonzero entries of the generalized first-order stationary points of the proposed generalized elastic net model. For implementation, we develop two efficient algorithms based on the locally Lipschitz continuous $\epsilon$-approximation to $\ell_q$-norm. The first algorithm employs an alternating direction method of multipliers (ADMM), while the second utilizes a proximal majorization-minimization method (PMM), where the subproblems are addressed using the semismooth Newton method (SNN). We also perform extensive numerical experiments with both simulated and real data, showing that both algorithms demonstrate superior performance. Notably, the PMM-SSN is efficient than ADMM, even though the latter provides a simpler implementation.

Logistic regression is a classical model for describing the probabilistic dependence of binary responses to multivariate covariates. We consider the predictive performance of the maximum likelihood estimator (MLE) for logistic regression, assessed in terms of logistic risk. We consider two questions: first, that of the existence of the MLE (which occurs when the dataset is not linearly separated), and second that of its accuracy when it exists. These properties depend on both the dimension of covariates and on the signal strength. In the case of Gaussian covariates and a well-specified logistic model, we obtain sharp non-asymptotic guarantees for the existence and excess logistic risk of the MLE. We then generalize these results in two ways: first, to non-Gaussian covariates satisfying a certain two-dimensional margin condition, and second to the general case of statistical learning with a possibly misspecified logistic model. Finally, we consider the case of a Bernoulli design, where the behavior of the MLE is highly sensitive to the parameter direction.

We investigate active data collection strategies for operator learning when the target operator is linear and the input functions are drawn from a mean-zero stochastic process with continuous covariance kernels. With an active data collection strategy, we establish an error convergence rate in terms of the decay rate of the eigenvalues of the covariance kernel. Thus, with sufficiently rapid eigenvalue decay of the covariance kernels, arbitrarily fast error convergence rates can be achieved. This contrasts with the passive (i.i.d.) data collection strategies, where the convergence rate is never faster than $\sim n^{-1}$. In fact, for our setting, we establish a \emph{non-vanishing} lower bound for any passive data collection strategy, regardless of the eigenvalues decay rate of the covariance kernel. Overall, our results show the benefit of active over passive data collection strategies in operator learning.

Stochastic gradient descent is a classic algorithm that has gained great popularity especially in the last decades as the most common approach for training models in machine learning. While the algorithm has been well-studied when stochastic gradients are assumed to have a finite variance, there is significantly less research addressing its theoretical properties in the case of infinite variance gradients. In this paper, we establish the asymptotic behavior of stochastic gradient descent in the context of infinite variance stochastic gradients, assuming that the stochastic gradient is regular varying with index $\alpha\in(1,2)$. The closest result in this context was established in 1969 , in the one-dimensional case and assuming that stochastic gradients belong to a more restrictive class of distributions. We extend it to the multidimensional case, covering a broader class of infinite variance distributions. As we show, the asymptotic distribution of the stochastic gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-Uhlenbeck process driven by an appropriate stable L\'evy process. Additionally, we explore the applications of these results in linear regression and logistic regression models.

Rolling element bearings are critical components of rotating machinery, with their performance directly influencing the efficiency and reliability of industrial systems. At the same time, bearing faults are a leading cause of machinery failures, often resulting in costly downtime, reduced productivity, and, in extreme cases, catastrophic damage. This study presents a methodology that utilizes Kolmogorov-Arnold Networks to address these challenges through automatic feature selection, hyperparameter tuning and interpretable fault analysis within a unified framework. By training shallow network architectures and minimizing the number of selected features, the framework produces lightweight models that deliver explainable results through feature attribution and symbolic representations of their activation functions. Validated on two widely recognized datasets for bearing fault diagnosis, the framework achieved perfect F1-Scores for fault detection and high performance in fault and severity classification tasks, including 100% F1-Scores in most cases. Notably, it demonstrated adaptability by handling diverse fault types, such as imbalance and misalignment, within the same dataset. The symbolic representations enhanced model interpretability, while feature attribution offered insights into the optimal feature types or signals for each studied task. These results highlight the framework's potential for practical applications, such as real-time machinery monitoring, and for scientific research requiring efficient and explainable models.

Estimates of seismic wave speeds in the Earth (seismic velocity models) are key input parameters to earthquake simulations for ground motion prediction. Owing to the non-uniqueness of the seismic inverse problem, typically many velocity models exist for any given region. The arbitrary choice of which velocity model to use in earthquake simulations impacts ground motion predictions. However, current hazard analysis methods do not account for this source of uncertainty. We present a proof-of-concept ground motion prediction workflow for incorporating uncertainties arising from inconsistencies between existing seismic velocity models. Our analysis is based on the probabilistic fusion of overlapping seismic velocity models using scalable Gaussian process (GP) regression. Specifically, we fit a GP to two synthetic 1-D velocity profiles simultaneously, and show that the predictive uncertainty accounts for the differences between the models. We subsequently draw velocity model samples from the predictive distribution and estimate peak ground displacement using acoustic wave propagation through the velocity models. The resulting distribution of possible ground motion amplitudes is much wider than would be predicted by simulating shaking using only the two input velocity models. This proof-of-concept illustrates the importance of probabilistic methods for physics-based seismic hazard analysis.