We present NeuralOperator, an open-source Python library for operator learning. Neural operators generalize neural networks to maps between function spaces instead of finite-dimensional Euclidean spaces. They can be trained and inferenced on input and output functions given at various discretizations, satisfying a discretization convergence properties. Part of the official PyTorch Ecosystem, NeuralOperator provides all the tools for training and deploying neural operator models, as well as developing new ones, in a high-quality, tested, open-source package. It combines cutting-edge models and customizability with a gentle learning curve and simple user interface for newcomers and researchers.

We introduce MarkDiffusion, an open-source Python toolkit for generative watermarking of latent diffusion models. It comprises three key components: a unified implementation framework for streamlined watermarking algorithm integration and user-friendly interfaces; a mechanism visualization suite that intuitively presents embedded and extracted watermark patterns to aid public understanding; and a comprehensive evaluation module offering standard implementations of 24 tools for assessing detectability, robustness, and output quality, plus 8 automated evaluation pipelines. Counts reflect the initial release; see the repository for the latest version. Through MarkDiffusion, we seek to assist researchers, enhance public awareness of and engagement with generative watermarking, help build consensus, and advance research and applications. Code is available at https://github.com/THU-BPM/MarkDiffusion.

Black-box optimization (BBO) underpins advances in domains such as AutoML and Materials Informatics, yet implementations of algorithms and benchmarks remain fragmented across research communities. We introduce OptunaHub (https://hub.optuna.org/), a community-oriented, decentralized platform for distributing BBO components under a unified Optuna-compatible interface. OptunaHub enables independent publication, discovery, and reuse of optimization algorithms and benchmark problems through a lightweight Python module, a contributor-driven registry, and a searchable web interface. The source code is publicly available in the optunahub, optunahub-registry, and optunahub-web repositories under the Optuna organization on GitHub (https://github.com/optuna/).

Chain-of-Thought (CoT) prompting and its variants have gained significant attention as effective methods for solving multi-step reasoning tasks with pretrained large language models (LLMs). However, their theoretical underpinnings remain insufficiently explored. We analyze CoT prompting from a statistical perspective, offering insights into why “pretrained LLMs + CoT prompting” performs well. Additionally, we examine the role of the transformer architecture and the inclusion of intermediate reasoning steps in enhancing performance. We introduce a multi-step latent variable model to capture the reasoning process. In this model, we show that the estimator induced by CoT prompting approximates a Bayesian estimator that solves the reasoning task by inferring the posterior distribution from examples in the prompt. We prove that the statistical error of the CoT estimator consists of (i) a prompting error, which is incurred in inferring the desired task from the prompt, and (ii) a pretraining error, which is the statistical error of the pretrained LLM. We further prove that the prompting error decreases exponentially as the number of examples in the prompt increases. For the pretrained LLM, we construct a transformer model class that explicitly approximates the target distribution and establish the generalization error under the Pac-Bayes framework.

We introduce Prob-GParareal, a probabilistic extension of the GParareal algorithm designed to provide uncertainty quantification for the Parallel-in-Time (PinT) solution of (ordinary and partial) differential equations (ODEs, PDEs). The method employs Gaussian processes (GPs) to model the Parareal correction function, in line with GParareal, further enabling the propagation of numerical uncertainty across time and yielding probabilistic forecasts of the system's evolution. Furthermore, Prob-GParareal accommodates probabilistic initial conditions and maintains compatibility with classical numerical solvers, ensuring its straightforward integration into existing Parareal frameworks. Here, we first conduct a theoretical analysis of the computational complexity and derive error bounds of Prob-GParareal. Then, we numerically demonstrate the accuracy and robustness of the proposed algorithm on five benchmark ODE systems, including chaotic, stiff, and bifurcation problems. To showcase the flexibility and potential scalability of the proposed algorithm, we also consider Prob-nnGParareal, a variant obtained by replacing the GPs in Parareal with the nearest-neighbors GPs, illustrating its improved computational performance on an additional PDE example. This work bridges a critical gap in the development of probabilistic counterparts to established PinT methods.

We consider the relationship between learnability of a "base class" of functions on a set $X$, and learnability of a class of statistical functions derived from the base class. For example, we refine results showing that learnability of a family $h_p: p \in \Theta$ of functions implies learnability of the family of functions $h_\mu(p) = \mathbb{E}_\mu[h_p]$, where $\mathbb{E}_\mu$ is the expectation with respect to $\mu$, and $\mu$ ranges over probability distributions on $X$. We will look at both Probably Approximately Correct (PAC) learning, where example inputs and outputs are chosen at random, and online learning, where the examples are chosen adversarially. For agnostic learning, we establish improved bounds on the sample complexity of learning for statistical classes, stated in terms of combinatorial dimensions of the base class. We connect these problems to techniques introduced in model theory for "randomizing a structure". We also provide counterexamples for realizable learning, in both the PAC and online settings.

Recently, Masked Pretraining (MPT) based on reconstruction pretraining tasks has risen to a promising self-supervised learning paradigm across various domains and achieves remarkable performance in multiple downstream tasks. However, the theoretical understanding of the working mechanism behind MPT is still limited. In this paper, we introduce a new theoretical framework to analyze MPT and understand the crucial role of masking in extracting meaningful representations. We establish theoretical connections between MPT and another popular self-supervised paradigm: contrastive learning. We prove that the masking technique implicitly creates positive pairs that are semantically similar and the reconstruction loss pulls them together in the feature space. Besides, as a result of the implicit alignment, we point out the dimensional collapse issue of MPT and propose a Uniformity-enhanced MPT (U-MPT) loss that can effectively address this issue and bring significant improvements in downstream tasks including linear evaluation, cross-dataset fine-tuning and out-of-distribution generalization on real-world data sets. Furthermore, we establish downstream guarantees of U-MPT and theoretically analyze the influence of masking strategies. Based on the theoretical analysis, we propose a new masking strategy which enhances the downstream performance of MPT and explains current improvements of masking strategies with our theoretical perspective.

We discuss causal inference for observational studies with possibly invalid instrumental variables. We propose a novel methodology called two-stage curvature identification (\texttt{TSCI}) by exploring the nonlinear treatment model with machine learning. The first-stage machine learning enables improving the instrumental variable's strength and adjusting for different forms of violating the instrumental variable assumptions. The success of \texttt{TSCI} requires the instrumental variable's effect on treatment to differ from its violation form. A novel bias correction step is implemented to remove bias resulting from the potentially high complexity of machine learning. Our proposed \texttt{TSCI} estimator is shown to be asymptotically unbiased and Gaussian even if the machine learning algorithm does not consistently estimate the treatment model. Furthermore, we design a data-dependent method to choose the best among several candidate violation forms. We apply \texttt{TSCI} to study the effect of education on earnings.

Multi-Objective Markov Decision Processes (MOMDPs) are among the most prevalent formal frameworks for addressing sequential decision-making problems involving multiple, potentially conflicting objectives. In most MOMDP approaches, a utility function is employed to aggregate these objectives into a single scalar criterion that encodes user preferences. Despite its widespread adoption, the theoretical foundations of MOMDPs remain incomplete in two main respects: first, there is no general characterisation of the classes of utility functions that guarantee the existence of an optimal policy; second, we do not know which preference relations can be represented by utility functions. This work advances both lines of research through a theoretical analysis of MOMDPs. Specifically, we examine each problem under the two principal formulations of utility functions for MOMDPs: the Scalarised Expected Returns (SER) criterion and the Expected Scalarised Returns (ESR) criterion. Our formal findings allow us to derive formal conditions that describe the families of utility functions and preference relations for which MOMDP algorithms should focus. These analyses can guide the development of new MOMDP algorithms explicitly grounded in our formal results.

Building artificially intelligent geospatial systems requires rapid delivery of spatial data analysis on massive scales with minimal human intervention. Depending on their intended use, learning about underlying spatial processes can also involve model assessment and uncertainty quantification. We devise transfer learning frameworks for deployment in artificially intelligent systems, where a massive data set is split into smaller data sets that stream into the analytical framework to propagate learning and assimilate learning for the entire data set. Specifically, we develop Bayesian predictive stacking for multivariate spatial data and demonstrate rapid automated probabilistic learning from massive spatial data sets. We illustrate the effectiveness of our approach through extensive simulation experiments and through the analysis of a massive dataset on vegetation index that are indistinguishable from traditional (and more expensive) statistical approaches.

This paper proposes a novel class of block quasi-Newton methods for convex optimization which we call symmetric rank-$k$ (SR-$k$) methods. Each iteration of SR-$k$ incorporates the curvature information with $k$ Hessian-vector products achieved from the greedy or random strategy. We prove that SR-$k$ methods have the local superlinear convergence rate of $\mathcal{O}\big((1-k/d)^{t(t-1)/2}\big)$ for minimizing smooth and strongly convex functions, where $d$ is the problem dimension and $t$ is the iteration counter. This is the first explicit superlinear convergence rate for block quasi-Newton methods, and it successfully explains why block quasi-Newton methods converge faster than ordinary quasi-Newton methods in practice. We also leverage the idea of SR-$k$ methods to study the block BFGS and block DFP methods, showing their superior convergence rates.

We inspect the deductive connection between the neural scaling law and Zipf's law--two statements discussed in machine learning and quantitative linguistics. The neural scaling law describes how the cross entropy rate of a foundation model--such as a large language model--changes with respect to the amount of training tokens, parameters, and compute. By contrast, Zipf's law posits that the distribution of tokens exhibits a power law tail. Whereas similar claims have been made in more specific settings, we show that the neural scaling law is a consequence of Zipf's law under certain broad assumptions that we reveal systematically. The derivation steps are as follows: We derive Heaps' law on the vocabulary growth from Zipf's law, Hilberg's hypothesis on the entropy scaling from Heaps' law, and the neural scaling from Hilberg's hypothesis. We illustrate these inference steps by a toy example of the Santa Fe process that satisfies all four statistical laws.

In many real-world applications, researchers aim to deploy models trained in a source domain to a target domain, where obtaining labeled data is often expensive, time-consuming, or even infeasible. While most existing literature assumes that the source and target data follow the same joint distribution, distribution shifts are common in practice. This paper considers a particular type of distribution shift, label shift, and develops an efficient inference procedure for general parameters characterizing the unlabeled target population. A central idea is to model the outcome density ratio between the labeled source data and unlabeled target data. To this end, we propose a progressive estimation strategy that unfolds in three stages: an initial heuristic guess, a consistent estimation, and ultimately, an efficient estimation. This self-evolving process is novel in the statistical literature and of independent interest. We also highlight the connection between our approach and prediction-powered inference (PPI), which uses machine learning models to improve statistical inference in related settings. We rigorously establish the asymptotic properties of the proposed estimators and demonstrate their superior performance compared to existing methods. Through simulation studies and multiple real-world applications, we illustrate both the theoretical contributions and practical benefits of our approach.

We consider a bandit problem where the budget is smaller than the number of arms, which may be infinite. In this regime, the usual objective in the literature is to minimize simple regret. To analyze broad classes of distributions with potentially unbounded support, where simple regret may not be well-defined, we take a slightly different approach and seek to maximize the expected simple reward of the recommended arm, providing anytime guarantees. To that end, we introduce a distribution-free algorithm, OSE, that adapts to the distribution of arm means and achieves near-optimal rates for several distribution classes. We characterize the sample complexity through the rank-corrected inverse squared gap function. In particular, we recover known upper bounds and transition regimes for $\alpha$ less or greater than $1/2$ when the quantile function is $\lambda_\eta = 1-\eta^{\alpha}$. We additionally identify new transition regimes depending on the noise level relative to $\alpha$, which we conjecture to be nearly optimal. Additionally, we introduce an enhanced practical version, PROSE, that achieves state-of-the-art empirical performance for the main distribution classes considered in the literature.

Mixture distributions are expressive variational families for black-box VI, but their discrete component choices complicate gradient estimation. We systematize reparameterization-based estimators for mixtures in a common notation, giving self-contained derivations and extending several to new settings. In particular, we provide an elementary derivation of a single-sample post-stratified estimator---previously derived via transport equations---and prove a variance reduction relative to simple random sampling. We also broaden the applicability of implicit reparameterization and reduce its computational complexity. Across different benchmarks, we find that stratified estimators are consistently robust when feasible; among single-sample methods, the post-stratified estimator frequently perform best, while implicit reparameterization is the most computationally demanding. Our analysis clarifies when each method should be used and provides efficient algorithms that make mixture-based variational inference practical.

This paper introduces torchsom, an open-source Python library that provides a reference implementation of the Self-Organizing Map (SOM) in PyTorch. This package offers three main features: (i) dimensionality reduction, (ii) clustering, and (iii) friendly data visualization. It relies on a PyTorch backend, enabling (i) fast and efficient training of SOMs through GPU acceleration, and (ii) easy and scalable integration with the PyTorch ecosystem. torchsom also follows the scikit-learn API for ease of use and extensibility. The library is released under the Apache 2.0 license with 90% test coverage, and its source code and documentation are available at https://github.com/michelin/TorchSOM.

We consider a high-probability non-asymptotic confidence estimation in the $\ell^2$-regularized non-linear least-squares setting with fixed design. In particular, we study confidence estimation for local minimizers of the regularized training loss. We show a pointwise confidence bound, meaning that it holds for the prediction on any given fixed test input $x$. Importantly, the proposed confidence bound scales with similarity of the test input to the training data in the implicit feature space of the predictor (for instance, becoming very large when the test input lies far outside of the training data). This desirable last feature is captured by the weighted norm involving the inverse-Hessian matrix of the objective function, which is a generalized version of its counterpart in the linear setting, $x^{\top} \text{Cov}^{-1} x$. Our generalized result can be regarded as a non-asymptotic counterpart of the classical confidence interval based on asymptotic normality of the MLE estimator. We propose an efficient method for computing the weighted norm, which only mildly exceeds the cost of a gradient computation of the loss function. Finally, we complement our analysis with empirical evidence showing that the proposed confidence bound provides better coverage/width trade-off compared to a confidence estimation by bootstrapping, which is a gold-standard method in many applications involving non-linear predictors such as neural networks.

Most learning algorithms with formal regret guarantees essentially rely on trying all possible behaviors, which is problematic when some errors cannot be recovered from. Instead, we allow the learning agent to ask for help from a mentor and to transfer knowledge between similar states. We show that this combination enables the agent to learn both safely and effectively. Under standard online learning assumptions, we provide an algorithm whose regret and number of mentor queries are both sublinear in the time horizon for Markov decision processes with irreversible dynamics and infinite state spaces. Our proof involves a sequence of three reductions, making our result more general than a single algorithm. Conceptually, our result may be the first formal proof that it is possible for an agent to obtain high reward while becoming self-sufficient in an unknown, unbounded, and high-stakes environment without resets.

The importance of explainability in stock prediction is increasingly recognized, especially for audit and regulatory purposes. Meanwhile, financial news corpora are often key drivers behind stock price fluctuations. However, the raw news data obtained is usually highly noisy, has a highly variable scope of influence in time and space, and is not precisely synchronized with stock price data. In this paper, we propose a prediction-explanation network called AgentPEN, which can provide clear explanations for complex temporal price patterns. Specifically, AgentPEN jointly aligns text and price streams by an LLM-based Representation Fusion Agent and then adopts a Deep Recurrent Generation module to explore the distribution of stock movements. The LLM-based Representation Fusion Agent is designed in a Selection-Memory-Fusion manner: the Text Selection Module picks up useful information from massive text data; the Text Memory Module evaluates and writes the text memory from a two-view perspective, including Temporal Memory and Spatial Memory; the Information Fusion Module models the interaction between text and price data. Next, the fused representation is sent to the Deep Recurrent Generation module to convert insights into stock movement predictions. Experiments on multiple real-world datasets have shown that AgentPEN surpasses the state-of-the-art baselines both in prediction accuracy and explainability.

This work studies online multiple testing with feedback, where decisions are made sequentially, and the true state of the hypothesis is revealed after decisions are made, either instantly or with a delay, and under either full or bandit feedback. We propose Generalized alpha-investing with feedback (GAIF) along with its adaptive variants, a feedback-enhanced framework that dynamically adjusts thresholds using revealed outcomes, ensuring finite-sample false discovery rate (FDR)/marginal FDR (mFDR) control. We further extend GAIF to online conformal testing by constructing valid conformal $p$-values and developing feedback-enhanced testing rules with finite-sample mFDR control. We also propose a feedback-driven score selection criterion to adaptively choose the candidate score that is most effective for the testing procedure, together with a theoretical analysis of its optimality. Numerical simulations and real-data applications demonstrate the effectiveness of our methods.