We study the algorithmic task of learning Boolean disjunctions in the distribution-free agnostic PAC model. The best known agnostic learner for the class of disjunctions over $\{0, 1\}^n$ is the $L_1$-polynomial regression algorithm, achieving complexity $2^{\tilde{O}(n^{1/2})}$. This complexity bound is known to be nearly best possible within the class of Correlational Statistical Query (CSQ) algorithms. In this work, we develop an agnostic learner for this concept class with complexity $2^{\tilde{O}(n^{1/3})}$. Our algorithm can be implemented in the Statistical Query (SQ) model, providing the first separation between the SQ and CSQ models in distribution-free agnostic learning.

We propose a scalable, approximate inference hypernetwork framework for a general model of history-dependent processes. The flexible data model is based on a neural ordinary differential equation (NODE) representing the evolution of internal states together with a trainable observation model subcomponent. The posterior distribution corresponding to the data model parameters (weights and biases) follows a stochastic differential equation with a drift term related to the score of the posterior that is learned jointly with the data model parameters. This Langevin sampling approach offers flexibility in balancing the computational budget between the evaluation cost of the data model and the approximation of the posterior density of its parameters. We demonstrate performance of the hypernetwork on chemical reaction and material physics data and compare it to mean-field variational inference.

In order to reduce the cost of experimental evaluation for models, we introduce a computational theory of evaluation for prediction and decision models: build evaluation model to accelerate the evaluation procedures. We prove upper bounds of generalized error and generalized causal effect error of given evaluation models. We also prove efficiency, and consistency to estimated causal effect from deployed subject to evaluation metric by prediction. To learn evaluation models, we propose a meta-learner to handle heterogeneous evaluation subjects space problem. Comparing with existed evaluation approaches, our (conditional) evaluation model reduced 24.1\%-99.0\% evaluation errors across 12 scenes, including individual medicine, scientific simulation, social experiment, business activity, and quantum trade. The evaluation time is reduced 3-7 order of magnitude comparing with experiments or simulations.

Spectral algorithms leverage spectral regularization techniques to analyze and process data, providing a flexible framework for addressing supervised learning problems. To deepen our understanding of their performance in real-world scenarios where the distributions of training and test data may differ, we conduct a rigorous investigation into the convergence behavior of spectral algorithms under distribution shifts, specifically within the framework of reproducing kernel Hilbert spaces. Our study focuses on the case of covariate shift. In this scenario, the marginal distributions of the input data differ between the training and test datasets, while the conditional distribution of the output given the input remains unchanged. Under this setting, we analyze the generalization error of spectral algorithms and show that they achieve minimax optimality when the density ratios between the training and test distributions are uniformly bounded. However, we also identify a critical limitation: when the density ratios are unbounded, the spectral algorithms may become suboptimal. To address this limitation, we propose a weighted spectral algorithm that incorporates density ratio information into the learning process. Our theoretical analysis shows that this weighted approach achieves optimal capacity-independent convergence rates. Furthermore, by introducing a weight clipping technique, we demonstrate that the convergence rates of the weighted spectral algorithm can approach the optimal capacity-dependent convergence rates arbitrarily closely. This improvement resolves the suboptimality issue in unbounded density ratio scenarios and advances the state-of-the-art by refining existing theoretical results.

We study the approximation gap between the dynamics of a polynomial-width neural network and its infinite-width counterpart, both trained using projected gradient descent in the mean-field scaling regime. We demonstrate how to tightly bound this approximation gap through a differential equation governed by the mean-field dynamics. A key factor influencing the growth of this ODE is the local Hessian of each particle, defined as the derivative of the particle's velocity in the mean-field dynamics with respect to its position. We apply our results to the canonical feature learning problem of estimating a well-specified single-index model; we permit the information exponent to be arbitrarily large, leading to convergence times that grow polynomially in the ambient dimension $d$. We show that, due to a certain ``self-concordance'' property in these problems -- where the local Hessian of a particle is bounded by a constant times the particle's velocity -- polynomially many neurons are sufficient to closely approximate the mean-field dynamics throughout training.

We develop two classes of variance-reduced fast operator splitting methods to approximate solutions of both finite-sum and stochastic generalized equations. Our approach integrates recent advances in accelerated fixed-point methods, co-hypomonotonicity, and variance reduction. First, we introduce a class of variance-reduced estimators and establish their variance-reduction bounds. This class covers both unbiased and biased instances and comprises common estimators as special cases, including SVRG, SAGA, SARAH, and Hybrid-SGD. Next, we design a novel accelerated variance-reduced forward-backward splitting (FBS) algorithm using these estimators to solve finite-sum and stochastic generalized equations. Our method achieves both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ convergence rates on the expected squared norm $\mathbb{E}[ \| G_{\lambda}x^k\|^2]$ of the FBS residual $G_{\lambda}$, where $k$ is the iteration counter. Additionally, we establish, for the first time, almost sure convergence rates and almost sure convergence of iterates to a solution in stochastic accelerated methods. Unlike existing stochastic fixed-point algorithms, our methods accommodate co-hypomonotone operators, which potentially include nonmonotone problems arising from recent applications. We further specify our method to derive an appropriate variant for each stochastic estimator -- SVRG, SAGA, SARAH, and Hybrid-SGD -- demonstrating that they achieve the best-known complexity for each without relying on enhancement techniques. Alternatively, we propose an accelerated variance-reduced backward-forward splitting (BFS) method, which attains similar convergence rates and oracle complexity as our FBS method. Finally, we validate our results through several numerical experiments and compare their performance.

Self-Supervised Learning (SSL) powers many current AI systems. As research interest and investment grow, the SSL design space continues to expand. The Platonic view of SSL, following the Platonic Representation Hypothesis (PRH), suggests that despite different methods and engineering approaches, all representations converge to the same Platonic ideal. However, this phenomenon lacks precise theoretical explanation. By synthesizing evidence from Identifiability Theory (IT), we show that the PRH can emerge in SSL. However, current IT cannot explain SSL's empirical success. To bridge the gap between theory and practice, we propose expanding IT into what we term Singular Identifiability Theory (SITh), a broader theoretical framework encompassing the entire SSL pipeline. SITh would allow deeper insights into the implicit data assumptions in SSL and advance the field towards learning more interpretable and generalizable representations. We highlight three critical directions for future research: 1) training dynamics and convergence properties of SSL; 2) the impact of finite samples, batch size, and data diversity; and 3) the role of inductive biases in architecture, augmentations, initialization schemes, and optimizers.

Cluster weighted models with multivariate skewed distributions for functional data Cristina Anton, 1 Roy Shivam Ram Shreshtth 2 1 Department of Mathematics and Statistics, MacEwan University, 103C, 10700-104 Ave., Edmonton, AB T5J 4S2, Canada, email: popescuc@macewan.ca 2 Department of Mathematics and Statistics, Indian Institute of Technology Kanpur Abstract We propose a clustering method, funWeightClustSkew, based on mixtures of functional linear regression models and three skewed multivariate distributions: the variance-gamma distribution, the skew-t distribution, and the normal-inverse Gaussian distribution. Our approach follows the framework of the functional high dimensional data clustering (funHDDC) method, and we extend to functional data the cluster weighted models based on skewed distributions used for finite dimensional multivariate data. We consider several parsimonious models, and to estimate the parameters we construct an expectation maximization (EM) algorithm. We illustrate the performance of funWeightClustSkew for simulated data and for the Air Quality dataset. Keywords: Cluster weighted models, Functional linear regression, EM algorithm, Skewed distributions, Multivariate functional principal component analysis 1 Introduction Smart devices and other modern technologies record huge amounts of data measured continuously in time. These data are better represented as curves instead of finite-dimensional vectors, and they are analyzed using statistical methods specific to functional data (Ramsay and Silverman, 2006; Ferraty and Vieu, 2006; Horv ath and Kokoszka, 2012). Many times more than one curve is collected for one individual, e.g.

We introduce ALT, an open-source Python package created for efficient and accurate time series classification (TSC). The package implements the adaptive law-based transformation (ALT) algorithm, which transforms raw time series data into a linearly separable feature space using variable-length shifted time windows. This adaptive approach enhances its predecessor, the linear law-based transformation (LLT), by effectively capturing patterns of varying temporal scales. The software is implemented for scalability, interpretability, and ease of use, achieving state-of-the-art performance with minimal computational overhead. Extensive benchmarking on real-world datasets demonstrates the utility of ALT for diverse TSC tasks in physics and related domains.

The International Prize in Statistics Foundation has awarded Grace Wahba the 2025 prize for "her groundbreaking work on smoothing splines, which has transformed data analysis and machine learning". Professor Wahba was among the earliest to pioneer the use of nonparametric regression modeling. Recent advances in computing and availability of large data sets have further popularized these models, especially under the guise of machine learning algorithms such as gradient boosting and neural networks. Nevertheless, the use of smoothing splines remains a mainstay of nonparametric regression. In seminal research that began in the early 1970s, Wahba developed theoretical foundations and computational algorithms for fitting smoothing splines to noisy data.