The spectacular success ofdeep generativemodels calls forquantitativetools to measure their statistical performance. Divergence frontiers have recently been proposed as an evaluation framework for generative models, due to their ability to measure the quality-diversity trade-off inherent to deep generative modeling. We establish non-asymptotic bounds on the sample complexity of divergence frontiers.

Forlinear regression, we give a polynomial-time algorithm based on Celis-Dennis-Tapia optimization algorithms. For binary classification, we show how to efficiently implement itusing aproper agnostic learner (i.e., anEmpirical Risk Minimizer) for the class of interest.

Many machine learning applications require outputs that satisfy complex, dynamic constraints. This task is particularly challenging in Graph Neural Network models due to the variable output sizes of graph-structured data. In this paper, we introduce ProjNet, a Graph Neural Network framework which satisfies input-dependant constraints. ProjNet combines a sparse vector clipping method with the Component-Averaged Dykstra (CAD) algorithm, an iterative scheme for solving the best-approximation problem. We establish a convergence result for CAD and develop a GPU-accelerated implementation capable of handling large-scale inputs efficiently. To enable end-to-end training, we introduce a surrogate gradient for CAD that is both computationally efficient and better suited for optimization than the exact gradient. We validate ProjNet on four classes of constrained optimisation problems: linear programming, two classes of non-convex quadratic programs, and radio transmit power optimization, demonstrating its effectiveness across diverse problem settings.

We develop a novel semigroup contraction analysis based on Lyapunov techniques to prove the exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schr\"odinger bridges with respect to general classes of $\phi$-divergences as well as in weighted Banach spaces. To the best of our knowledge, these are the first results of this type in the literature on entropic transport and the Sinkhorn algorithm. We also illustrate the impact of these results in the context of multivariate linear Gaussian models as well as statistical finite mixture models including Gaussian-kernel density estimation of complex data distributions arising in generative models.

Precision matrix estimation is essential in various fields, yet it is challenging when samples for the target study are limited. Transfer learning can enhance estimation accuracy by leveraging data from related source studies. We propose Trans-Glasso, a two-step transfer learning method for precision matrix estimation. First, we obtain initial estimators using a multi-task learning objective that captures shared and unique features across studies. Then, we refine these estimators through differential network estimation to adjust for structural differences between the target and source precision matrices. Under the assumption that most entries of the target precision matrix are shared with source matrices, we derive non-asymptotic error bounds and show that Trans-Glasso achieves minimax optimality under certain conditions. Extensive simulations demonstrate Trans Glasso's superior performance compared to baseline methods, particularly in small-sample settings. We further validate Trans-Glasso in applications to gene networks across brain tissues and protein networks for various cancer subtypes, showcasing its effectiveness in biological contexts. Additionally, we derive the minimax optimal rate for differential network estimation, representing the first such guarantee in this area.

In this work, we explore the intersection of sparse coding theory and deep learning to enhance our understanding of feature extraction capabilities in advanced neural network architectures. We begin by introducing a novel class of Deep Sparse Coding (DSC) models and establish a thorough theoretical analysis of their uniqueness and stability properties. By applying iterative algorithms to these DSC models, we derive convergence rates for convolutional neural networks (CNNs) in their ability to extract sparse features. This provides a strong theoretical foundation for the use of CNNs in sparse feature learning tasks. We additionally extend this convergence analysis to more general neural network architectures, including those with diverse activation functions, as well as self-attention and transformer-based models. This broadens the applicability of our findings to a wide range of deep learning methods for deep sparse feature extraction. Inspired by the strong connection between sparse coding and CNNs, we also explore training strategies to encourage neural networks to learn more sparse features. Through numerical experiments, we demonstrate the effectiveness of these approaches, providing valuable insights for the design of efficient and interpretable deep learning models.

We present Collaborative Trees, a novel tree model designed for regression prediction, along with its bagging version, which aims to analyze complex statistical associations between features and uncover potential patterns inherent in the data. We decompose the mean decrease in impurity from the proposed tree model to analyze the additive and interaction effects of features on the response variable. Additionally, we introduce network diagrams to visually depict how each feature contributes additively to the response and how pairs of features contribute interaction effects. Through a detailed demonstration using an embryo growth dataset, we illustrate how the new statistical tools aid data analysis, both visually and numerically. Moreover, we delve into critical aspects of tree modeling, such as prediction performance, inference stability, and bias in feature importance measures, leveraging real datasets and simulation experiments for comprehensive discussions. On the theory side, we show that Collaborative Trees, built upon a ``sum of trees'' approach with our own innovative tree model regularization, exhibit characteristics akin to matching pursuit, under the assumption of high-dimensional independent binary input features (or one-hot feature groups). This newfound link sheds light on the superior capability of our tree model in estimating additive effects of features, a crucial factor for accurate interaction effect estimation.

Multi-distribution learning (MDL), which seeks to learn a shared model that minimizes the worst-case risk across $k$ distinct data distributions, has emerged as a unified framework in response to the evolving demand for robustness, fairness, multi-group collaboration, etc. Achieving data-efficient MDL necessitates adaptive sampling, also called on-demand sampling, throughout the learning process. However, there exist substantial gaps between the state-of-the-art upper and lower bounds on the optimal sample complexity. Focusing on a hypothesis class of Vapnik-Chervonenkis (VC) dimension $d$, we propose a novel algorithm that yields an $varepsilon$-optimal randomized hypothesis with a sample complexity on the order of $(d+k)/\varepsilon^2$ (modulo some logarithmic factor), matching the best-known lower bound. Our algorithmic ideas and theory have been further extended to accommodate Rademacher classes. The proposed algorithms are oracle-efficient, which access the hypothesis class solely through an empirical risk minimization oracle. Additionally, we establish the necessity of randomization, unveiling a large sample size barrier when only deterministic hypotheses are permitted. These findings successfully resolve three open problems presented in COLT 2023 (i.e., Awasthi et al., (2023, Problem 1, 3 and 4)).