Predictions from statistical physics postulate that recovery of the communities in Stochastic Block Model (SBM) is possible in polynomial time above, and only above, the Kesten-Stigum (KS) threshold. This conjecture has given rise to a rich literature, proving that non-trivial community recovery is indeed possible in SBM above the KS threshold, as long as the number $K$ of communities remains smaller than $\sqrt{n}$, where $n$ is the number of nodes in the observed graph. Failure of low-degree polynomials below the KS threshold was also proven when $K=o(\sqrt{n})$. When $K\geq \sqrt{n}$, Chin et al.(2025) recently prove that, in a sparse regime, community recovery in polynomial time is possible below the KS threshold by counting non-backtracking paths. This breakthrough result lead them to postulate a new threshold for the many communities regime $K\geq \sqrt{n}$. In this work, we provide evidences that confirm their conjecture for $K\geq \sqrt{n}$: 1- We prove that, for any density of the graph, low-degree polynomials fail to recover communities below the threshold postulated by Chin et al.(2025); 2- We prove that community recovery is possible in polynomial time above the postulated threshold, not only in the sparse regime of~Chin et al., but also in some (but not all) moderately sparse regimes by essentially counting clique occurence in the observed graph.

Latent space model plays a crucial role in network analysis, and accurate estimation of latent variables is essential for downstream tasks such as link prediction. However, the large number of parameters to be estimated presents a challenge, especially when the latent space dimension is not exceptionally small. In this paper, we propose a transfer learning method that leverages information from networks with latent variables similar to those in the target network, thereby improving the estimation accuracy for the target. Given transferable source networks, we introduce a two-stage transfer learning algorithm that accommodates differences in node numbers between source and target networks. In each stage, we derive sufficient identification conditions and design tailored projected gradient descent algorithms for estimation. Theoretical properties of the resulting estimators are established. When the transferable networks are unknown, a detection algorithm is introduced to identify suitable source networks. Simulation studies and analyses of two real datasets demonstrate the effectiveness of the proposed methods.

The Lookahead optimizer enhances deep learning models by employing a dual-weight update mechanism, which has been shown to improve the performance of underlying optimizers such as SGD. However, most theoretical studies focus on its convergence on training data, leaving its generalization capabilities less understood. Existing generalization analyses are often limited by restrictive assumptions, such as requiring the loss function to be globally Lipschitz continuous, and their bounds do not fully capture the relationship between optimization and generalization. In this paper, we address these issues by conducting a rigorous stability and generalization analysis of the Lookahead optimizer with minibatch SGD. We leverage on-average model stability to derive generalization bounds for both convex and strongly convex problems without the restrictive Lipschitzness assumption. Our analysis demonstrates a linear speedup with respect to the batch size in the convex setting.

We highlight a fundamental connection between information geometry and variational Bayes (VB) and discuss its consequences for machine learning. Under certain conditions, a VB solution always requires estimation or computation of natural gradients. We show several consequences of this fact by using the natural-gradient descent algorithm of Khan and Rue (2023) called the Bayesian Learning Rule (BLR). These include (i) a simplification of Bayes' rule as addition of natural gradients, (ii) a generalization of quadratic surrogates used in gradient-based methods, and (iii) a large-scale implementation of VB algorithms for large language models. Neither the connection nor its consequences are new but we further emphasize the common origins of the two fields of information geometry and Bayes with a hope to facilitate more work at the intersection of the two fields.

Machine learning algorithms often assume that training samples are independent. When data points are connected by a network, the induced dependency between samples is both a challenge, reducing effective sample size, and an opportunity to improve prediction by leveraging information from network neighbors. Multiple methods taking advantage of this opportunity are now available, but many, including graph neural networks, are not easily interpretable, limiting their usefulness for understanding how a model makes its predictions. Others, such as network-assisted linear regression, are interpretable but often yield substantially worse prediction performance. We bridge this gap by proposing a family of flexible network-assisted models built upon a generalization of random forests (RF+), which achieves highly-competitive prediction accuracy and can be interpreted through feature importance measures. In particular, we develop a suite of interpretation tools that enable practitioners to not only identify important features that drive model predictions, but also quantify the importance of the network contribution to prediction. Importantly, we provide both global and local importance measures as well as sample influence measures to assess the impact of a given observation. This suite of tools broadens the scope and applicability of network-assisted machine learning for high-impact problems where interpretability and transparency are essential.

We study the estimation of distributional treatment effects in randomized experiments with imperfect compliance. When participants do not adhere to their assigned treatments, we leverage treatment assignment as an instrumental variable to identify the local distributional treatment effect-the difference in outcome distributions between treatment and control groups for the subpopulation of compliers. We propose a regression-adjusted estimator based on a distribution regression framework with Neyman-orthogonal moment conditions, enabling robustness and flexibility with high-dimensional covariates. Our approach accommodates continuous, discrete, and mixed discrete-continuous outcomes, and applies under a broad class of covariate-adaptive randomization schemes, including stratified block designs and simple random sampling. We derive the estimator's asymptotic distribution and show that it achieves the semiparametric efficiency bound. Simulation results demonstrate favorable finite-sample performance, and we demonstrate the method's practical relevance in an application to the Oregon Health Insurance Experiment.

In transfer learning, a source domain often carries diverse knowledge, and different domains usually emphasize different types of knowledge. Different from handling only a single type of knowledge from all domains in traditional transfer learning methods, we introduce an ensemble learning framework with a weak mode of convergence in the form of Statistical Invariant (SI) for multi-source transfer learning, formulated as Stochastic Ensemble Multi-Source Transfer Learning Using Statistical Invariant (SETrLUSI). The proposed SI extracts and integrates various types of knowledge from both source and target domains, which not only effectively utilizes diverse knowledge but also accelerates the convergence process. Further, SETrLUSI incorporates stochastic SI selection, proportional source domain sampling, and target domain bootstrapping, which improves training efficiency while enhancing model stability. Experiments show that SETrLUSI has good convergence and outperforms related methods with a lower time cost.

Online controlled experiments, also known as A/B testing, are the digital equivalent of randomized controlled trials for estimating the impact of marketing campaigns on website visitors. Stratified sampling is a traditional technique for variance reduction to improve the sensitivity (or statistical power) of controlled experiments; this technique first divides the population into strata (homogeneous subgroups) based on stratification variables and then draws samples from each stratum to avoid sampling bias. To enhance the estimation accuracy of stratified sampling, we focus on the problem of selecting a subset of stratification variables that are effective in variance reduction. We design an efficient algorithm that selects stratification variables one by one by simulating a series of stratified sampling processes. We also estimate the computational complexity of our subset selection algorithm. Computational experiments using synthetic and real-world datasets demonstrate that our method can outperform other variance reduction techniques especially when multiple variables have a certain correlation with the outcome variable. Our subset selection method for stratified sampling can improve the sensitivity of online controlled experiments, thus enabling more reliable marketing decisions.

Transformers and their attention mechanism have been revolutionary in the field of Machine Learning. While originally proposed for the language data, they quickly found their way to the image, video, graph, etc. data modalities with various signal geometries. Despite this versatility, generalizing the attention mechanism to scenarios where data is presented at different scales from potentially different modalities is not straightforward. The attempts to incorporate hierarchy and multi-modality within transformers are largely based on ad hoc heuristics, which are not seamlessly generalizable to similar problems with potentially different structures. To address this problem, in this paper, we take a fundamentally different approach: we first propose a mathematical construct to represent multi-modal, multi-scale data. We then mathematically derive the neural attention mechanics for the proposed construct from the first principle of entropy minimization. We show that the derived formulation is optimal in the sense of being the closest to the standard Softmax attention while incorporating the inductive biases originating from the hierarchical/geometric information of the problem. We further propose an efficient algorithm based on dynamic programming to compute our derived attention mechanism. By incorporating it within transformers, we show that the proposed hierarchical attention mechanism not only can be employed to train transformer models in hierarchical/multi-modal settings from scratch, but it can also be used to inject hierarchical information into classical, pre-trained transformer models post training, resulting in more efficient models in zero-shot manner.

Accurate ranking of important features is a fundamental challenge in interpretable machine learning with critical applications in scientific discovery and decision-making. Unlike feature selection and feature importance, the specific problem of ranking important features has received considerably less attention. We introduce RAMPART (Ranked Attributions with MiniPatches And Recursive Trimming), a framework that utilizes any existing feature importance measure in a novel algorithm specifically tailored for ranking the top-$k$ features. Our approach combines an adaptive sequential halving strategy that progressively focuses computational resources on promising features with an efficient ensembling technique using both observation and feature subsampling. Unlike existing methods that convert importance scores to ranks as post-processing, our framework explicitly optimizes for ranking accuracy. We provide theoretical guarantees showing that RAMPART achieves the correct top-$k$ ranking with high probability under mild conditions, and demonstrate through extensive simulation studies that RAMPART consistently outperforms popular feature importance methods, concluding with a high-dimensional genomics case study.