In this work, we explore the mechanism of in-context learning (ICL) on out-of-distribution (OOD) tasks that were not encountered during training. To achieve this, we conduct synthetic experiments where the objective is to learn OOD mathematical functions through ICL using a GPT-2 model. We reveal that Transformers may struggle to learn OOD task functions through ICL. Specifically, ICL performance resembles implementing a function within the pretraining hypothesis space and optimizing it with gradient descent based on the in-context examples. Additionally, we investigate ICL's well-documented ability to learn unseen abstract labels in context. We demonstrate that such ability only manifests in the scenarios without distributional shifts and, therefore, may not serve as evidence of new-task-learning ability. Furthermore, we assess ICL's performance on OOD tasks when the model is pretrained on multiple tasks. Both empirical and theoretical analyses demonstrate the existence of the \textbf{low-test-error preference} of ICL, where it tends to implement the pretraining function that yields low test error in the testing context. We validate this through numerical experiments. This new theoretical result, combined with our empirical findings, elucidates the mechanism of ICL in addressing OOD tasks.

Relying on the representation power of neural networks, most recent works have often neglected several factors involved in haze degradation, such as transmission (the amount of light reaching an observer from a scene over distance) and atmospheric light. These factors are generally unknown, making dehazing problems ill-posed and creating inherent uncertainties. To account for such uncertainties and factors involved in haze degradation, we introduce a variational Bayesian framework for single image dehazing. We propose to take not only a clean image and but also transmission map as latent variables, the posterior distributions of which are parameterized by corresponding neural networks: dehazing and transmission networks, respectively. Based on a physical model for haze degradation, our variational Bayesian framework leads to a new objective function that encourages the cooperation between them, facilitating the joint training of and thereby boosting the performance of each other. In our framework, a dehazing network can estimate a clean image independently of a transmission map estimation during inference, introducing no overhead. Furthermore, our model-agnostic framework can be seamlessly incorporated with other existing dehazing networks, greatly enhancing the performance consistently across datasets and models.

Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: $X_t = {L^{\ast}}Y_{t}$, where $X_t, Y_t \in \mathbb{R}^p$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast}$ encodes the edge structure. Assuming $X_t$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L^{\ast}$ from temporally correlated samples of $Y_t$ via an $\ell_1$-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size $p$ significantly exceeds the number of samples $n$. We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on $(n, p, d)$, we show that the ML estimate recovers the sparsity pattern of $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast}$. We provide recovery guarantees for $L^\ast$ in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain).

Locally stationary processes (LSPs) provide a robust framework for modeling time-varying phenomena, allowing for smooth variations in statistical properties such as mean and variance over time. In this paper, we address the estimation of the conditional probability distribution of LSPs using Nadaraya-Watson (NW) type estimators. The NW estimator approximates the conditional distribution of a target variable given covariates through kernel smoothing techniques. We establish the convergence rate of the NW conditional probability estimator for LSPs in the univariate setting under the Wasserstein distance and extend this analysis to the multivariate case using the sliced Wasserstein distance. Theoretical results are supported by numerical experiments on both synthetic and real-world datasets, demonstrating the practical usefulness of the proposed estimators.

Tensor decompositions play a crucial role in numerous applications related to multi-way data analysis. By employing a Bayesian framework with sparsity-inducing priors, Bayesian Tensor Ring (BTR) factorization offers probabilistic estimates and an effective approach for automatically adapting the tensor ring rank during the learning process. However, previous BTR method employs an Automatic Relevance Determination (ARD) prior, which can lead to sub-optimal solutions. Besides, it solely focuses on continuous data, whereas many applications involve discrete data. More importantly, it relies on the Coordinate-Ascent Variational Inference (CAVI) algorithm, which is inadequate for handling large tensors with extensive observations. These limitations greatly limit its application scales and scopes, making it suitable only for small-scale problems, such as image/video completion. To address these issues, we propose a novel BTR model that incorporates a nonparametric Multiplicative Gamma Process (MGP) prior, known for its superior accuracy in identifying latent structures. To handle discrete data, we introduce the P\'olya-Gamma augmentation for closed-form updates. Furthermore, we develop an efficient Gibbs sampler for consistent posterior simulation, which reduces the computational complexity of previous VI algorithm by two orders, and an online EM algorithm that is scalable to extremely large tensors. To showcase the advantages of our model, we conduct extensive experiments on both simulation data and real-world applications.

Recent work in language modeling has raised the possibility of self-improvement, where a language models evaluates and refines its own generations to achieve higher performance without external feedback. It is impossible for this self-improvement to create information that is not already in the model, so why should we expect that this will lead to improved capabilities? We offer a new perspective on the capabilities of self-improvement through a lens we refer to as sharpening. Motivated by the observation that language models are often better at verifying response quality than they are at generating correct responses, we formalize self-improvement as using the model itself as a verifier during post-training in order to ``sharpen'' the model to one placing large mass on high-quality sequences, thereby amortizing the expensive inference-time computation of generating good sequences. We begin by introducing a new statistical framework for sharpening in which the learner aims to sharpen a pre-trained base policy via sample access, and establish fundamental limits. Then we analyze two natural families of self-improvement algorithms based on SFT and RLHF. We find that (i) the SFT-based approach is minimax optimal whenever the initial model has sufficient coverage, but (ii) the RLHF-based approach can improve over SFT-based self-improvement by leveraging online exploration, bypassing the need for coverage. Finally, we empirically validate the sharpening mechanism via inference-time and amortization experiments. We view these findings as a starting point toward a foundational understanding that can guide the design and evaluation of self-improvement algorithms.

Investigating the marginal causal effect of an intervention on an outcome from complex data remains challenging due to the inflexibility of employed models and the lack of complexity in causal benchmark datasets, which often fail to reproduce intricate real-world data patterns. In this paper we introduce Frugal Flows, a novel likelihood-based machine learning model that uses normalising flows to flexibly learn the data-generating process, while also directly inferring the marginal causal quantities from observational data. We propose that these models are exceptionally well suited for generating synthetic data to validate causal methods. They can create synthetic datasets that closely resemble the empirical dataset, while automatically and exactly satisfying a user-defined average treatment effect. To our knowledge, Frugal Flows are the first generative model to both learn flexible data representations and also exactly parameterise quantities such as the average treatment effect and the degree of unobserved confounding. We demonstrate the above with experiments on both simulated and real-world datasets.

Advancements in artificial intelligence, machine learning, and deep learning have catalyzed the transformation of big data analytics and management into pivotal domains for research and application. This work explores the theoretical foundations, methodological advancements, and practical implementations of these technologies, emphasizing their role in uncovering actionable insights from massive, high-dimensional datasets. The study presents a systematic overview of data preprocessing techniques, including data cleaning, normalization, integration, and dimensionality reduction, to prepare raw data for analysis. Core analytics methodologies such as classification, clustering, regression, and anomaly detection are examined, with a focus on algorithmic innovation and scalability. Furthermore, the text delves into state-of-the-art frameworks for data mining and predictive modeling, highlighting the role of neural networks, support vector machines, and ensemble methods in tackling complex analytical challenges. Special emphasis is placed on the convergence of big data with distributed computing paradigms, including cloud and edge computing, to address challenges in storage, computation, and real-time analytics. The integration of ethical considerations, including data privacy and compliance with global standards, ensures a holistic perspective on data management. Practical applications across healthcare, finance, marketing, and policy-making illustrate the real-world impact of these technologies. Through comprehensive case studies and Python-based implementations, this work equips researchers, practitioners, and data enthusiasts with the tools to navigate the complexities of modern data analytics. It bridges the gap between theory and practice, fostering the development of innovative solutions for managing and leveraging data in the era of artificial intelligence.

Introduction Reinforcement Learning (RL) is one of the most prominent and widely discussed methods in artificial intelligence, primarily focusing on how an agent learns to make decisions by interacting with an environment to maximize cumulative rewards [1]. RL has seen extensive applications in various domains, including autonomous driving [2], recommendation systems [3], unmanned aerial vehicles (UAVs) [4], financial trading [5], causal inference [6], and precision medicine [7,8]; see [9,10] for a review. The classic and simplified problem in RL is the stochastic bandit problems. Stochastic bandit problems exemplify the exploration-exploitation tradeoff dilemma, where an agent must choose between exploring new options to gather more information and exploiting known options to maximize rewards. The current review literature on stochastic bandit algorithms highlights applications in areas such as recommendation systems[11-13], experimental design[14], and precision medicine[8], causal inference[15]. Efficient bandit algorithms are designed from a statistical perspective. However, these aspects remain underexplored in existing reviews. This paper aims to address this gap by focusing on the probabilistic and statistical foundations of stochastic algorithms, with particular emphasis on concentration inequalities, minimax rate of regret upper bounds, small-sample statistical inferences, linear models, Bayesian optimization, statistical learning theory, design of experiments, the Neyman-Rubin causal model, functional data analysis, robust statistics, information theory, and so on.

Understanding how galaxies form and evolve is at the heart of modern astronomy. With the advent of large-scale surveys and simulations, remarkable progress has been made in the last few decades. Despite this, the physical processes behind the phenomena, and particularly their importance, remain far from known, as correlations have primarily been established rather than the underlying causality. We address this challenge by applying the causal inference framework. Specifically, we tackle the fundamental open question of whether galaxy formation and evolution depends more on nature (i.e., internal processes) or nurture (i.e., external processes), by estimating the causal effect of environment on star-formation rate in the IllustrisTNG simulations. To do so, we develop a comprehensive causal model and employ cutting-edge techniques from epidemiology to overcome the long-standing problem of disentangling nature and nurture. We find that the causal effect is negative and substantial, with environment suppressing the SFR by a maximal factor of $\sim100$. While the overall effect at $z=0$ is negative, in the early universe, environment is discovered to have a positive impact, boosting star formation by a factor of $\sim10$ at $z\sim1$ and by even greater amounts at higher redshifts. Furthermore, we show that: (i) nature also plays an important role, as ignoring it underestimates the causal effect in intermediate-density environments by a factor of $\sim2$, (ii) controlling for the stellar mass at a snapshot in time, as is common in the literature, is not only insufficient to disentangle nature and nurture but actually has an adverse effect, though (iii) stellar mass is an adequate proxy of the effects of nature. Finally, this work may prove a useful blueprint for extracting causal insights in other fields that deal with dynamical systems with closed feedback loops, such as the Earth's climate.