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.

The notion of rank of a Boolean function has been a cornerstone in PAC learning, enabling quasipolynomial-time learning algorithms for polynomial-size decision trees. We present a novel characterization of rank, grounded in the well-known Transformer architecture. We show that the rank of a function $f$ corresponds to the minimum number of Chain of Thought (CoT) iterations required by a single-layer Transformer with hard attention to compute $f$. Based on this characterization, we establish tight bounds on the number of CoT iterations required for specific problems, showing that \(\ell\)-fold function composition necessitates exactly \(\ell\) CoT iterations. Furthermore, we analyze the problem of identifying the position of the \(k\)-th occurrence of 1 in a Boolean sequence, proving that it requires \(k\) CoT iterations. Finally, we introduce the notion of the multi-head rank that captures multi-head single-layer transformers, and perform the analysis of PAC-learnability of the classes of functions with bounded multi-head rank.

scikit-activeml is a user-friendly open-source Python library for active learning on top of scikit-learn. Included are implementations of a large collection of query strategies, models, and visualization tools in pool- and stream-based active learning for classification or regression tasks with single or multiple annotators. The flexible design of the active learning cycle enables individual adaptations to a variety of learning scenarios. Our source code with comprehensive documentation is available at https://scikit-activeml.github.io.

We initiate the study of locally differentially private (LDP) learning with public features. We define semi-feature LDP, where some features are publicly available while the remaining ones, along with the label, require protection under local differential privacy. Under semi-feature LDP, we consider three fundamental estimation problems: non-parametric density estimation, classification, and regression. Given the smoothness assumption, we show that the minimax convergence rate is significantly improved compared to classical LDP. Then, we propose HistOfTree, an estimator that fully leverages the information contained in both public and private features. Theoretically, HistOfTree reaches the minimax optimal convergence rate. Empirically, HistOfTree achieves superior performance on both synthetic and real data. We also explore scenarios where users have the flexibility to select features for protection manually. In such cases, we propose an estimator and a data-driven parameter tuning strategy, leading to analogous theoretical and empirical results.

We study the derivative-informed learning of nonlinear operators between infinite-dimensional Hilbert spaces. Such operators can arise as solution maps of partial differential equations, and their approximation by accurate surrogate models can accelerate simulation-intensive tasks of scientific and engineering interest, including inference, control, and uncertainty quantification. Since efficiently performing such tasks often requires an accurate representation of the operator's derivatives, we analyze the approximation capabilities of neural operators measured by Sobolev norms over infinite-dimensional Gaussian input measures. We focus on the reduced basis neural operator, which employs linear encoders/decoders defined on dominant input/output subspaces. We study two methods for generating the subspaces: principal component analysis (PCA) and derivative-informed subspaces (DIS). These use the dominant eigenvectors of the covariance of the data or the derivatives as bases for the subspaces, respectively. We derive bounds for errors arising from both dimension reduction and latent neural network approximations, including sampling errors associated with the empirical estimation of the PCA/DIS subspaces. Our analysis is validated through numerical experiments, which demonstrate that subspaces informed by the underlying operator (DIS or output PCA) yield smaller generalization errors in the Sobolev norm, while input PCA may underperform unless ranks and training sample sizes are sufficiently large.

Domain adaptation enables generalizable and efficient data-driven research. However, existing work has largely focused on domain adaptation for some intrinsically homogeneous target cohort, overlooking inherent heterogeneity within the target, which can exacerbate biases and unfairness in the presence of subgroups with imbalanced sample sizes. We develop a novel domain adaptation framework that addresses a more complicated target dataset that consists of heterogeneous and data-sparse subgroups and lacks gold-standard label observations. Our method simultaneously handles high-dimensionality, covariate shift, and outcome model heterogeneity by combining a model-assisted debiasing step used for covariate shift correction with an adaptive knowledge-guided sparsification procedure used to mitigate the issue of sample disparity. We also introduce a new model selection strategy to avoid negative knowledge transfer in the absence of labels in the target data. Our method is theoretically justified for being robust to nuisance model misspecification and adaptive to heterogeneity between the subgroups. Numerical experiments and two real-world applications, including genetic risk modeling of type 2 diabetes and prediction of mutation-induced protein stability changes, demonstrate the practical advantages of our method.

The z-transform of a sequence is a classical tool used in signal processing, control theory, computer science, and electrical engineering. It allows one to study sequences from their generating functions, with many operations that can be equivalently defined on the original sequence and its z-transform. In particular, the z-transform method focuses on asymptotic behaviors and allows the use of Taylor expansions. We present a sequence of results of increasing significance and difficulty for linear models and optimization algorithms, demonstrating the effectiveness and versatility of the z-transform method in deriving new asymptotic results. Starting from the simplest gradient descent iterations in an infinite-dimensional Hilbert space, we show how the spectral dimension characterizes the convergence behavior. We then extend the analysis to Nesterov acceleration, averaging techniques, and stochastic gradient descent.

Despite their empirical success across a wide range of generative tasks, the fundamental principles underlying the ability of diffusion models to learn data distributions are poorly understood. In this work, we develop a new mathematical framework that explains how diffusion models can effectively learn low-dimensional distributions from a finite number of training samples without suffering from the curse of dimensionality. Specifically, motivated by the intrinsic low-dimensional structure of image data, we theoretically analyze a setting in which the data distribution is modeled as a mixture of low-rank Gaussians. Under a suitable network parameterization, we show that optimizing the training objective of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples, where each subspace basis corresponds to the low-rank covariance of a Gaussian component. This equivalence allows us to show that the sample complexity for learning the underlying distribution scales linearly with the intrinsic dimension of the data, rather than exponentially with the ambient dimension. Our theoretical findings are further supported by empirical evidence that demonstrates phase transition phenomena in generalization on both synthetic and real-world image datasets. Moreover, we establish a correspondence between the learned subspace bases and semantic attributes of image data, providing a principled foundation for controllable image generation.

Contrastive learning leverages data augmentation to develop feature representation without relying on large labeled data sets. However, despite its empirical success, the theoretical foundations of contrastive learning remain incomplete, with many essential guarantees left unaddressed, particularly the realizability assumption concerning neural approximability of an optimal spectral contrastive loss solution. In this work, we overcome these limitations by analyzing pointwise and spectral consistency of the augmentation graph Laplacian. We establish that, under specific conditions for data generation and graph connectivity, as the augmented data set size increases, the augmentation graph Laplacian converges to a weighted Laplace-Beltrami operator on the natural data manifold. These consistency results ensure that the graph Laplacian spectrum effectively captures the manifold geometry. Consequently, they give way to a robust framework for establishing neural approximability, directly resolving the realizability assumption in a current paradigm.