We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter $α\in (0,1)$. We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter $γ_t$ and the posterior contraction rate $ε_t$, and identify conditions on the Gaussian process prior under which $ε_t$ can be controlled. As special cases of our general bound, we recover regret of order $\tilde{\mathcal{O}}(T^{\frac{1}{2}})$ for the squared exponential kernel, $\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}} )$ for the Matérn-$ν$ kernel, and a bound of order $\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}})$ for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

The processing of high-dimensional streaming data commonly utilizes online streaming feature selection (OSFS) techniques. However, practical implementations often face challenges with data incompleteness due to equipment failures and technical constraints. Online Sparse Streaming Feature Selection (OS2FS) tackles this issue through latent factor analysis-based missing data imputation. Despite this advancement, existing OS2FS approaches exhibit substantial limitations in feature evaluation, resulting in performance deterioration. To address these shortcomings, this paper introduces a novel Online Differential Evolution for Sparse Feature Selection (ODESFS) in data streams, incorporating two key innovations: (1) missing value imputation using a latent factor analysis model, and (2) feature importance evaluation through differential evolution. Comprehensive experiments conducted on six real-world datasets demonstrate that ODESFS consistently outperforms state-of-the-art OSFS and OS2FS methods by selecting optimal feature subsets and achieving superior accuracy.

In real - world applications involving high - dimensional streaming dat a, online streaming feature selection (OSFS) is widely adopt ed. Yet, practical deployments frequently face data incompleteness due to sensor failures or technical constraints. While online sparse streaming feature selection (OS FS) mitigates this issue via latent factor analysis - based imputation, existing methods s truggle with uncertain feature - label correlations, leading to inflexible models and degraded performance. To address these gaps, this work proposes P OS FS -- an uncertainty - aware online sparse stream ing feature selection framework enhanced by particle swarm optimization (PSO). The approach introduces: 1) PSO - driven supervision to reduce uncertainty in feature - label relationships; 2) Three - way decision theory to manage feature fuzziness in supervised l earning. Rigorous testing on six real - world datasets confirms P OS FS outperforms conventional OSFS and OS FS techniques, delivering higher accuracy through more robust feature subset selection.

We develop an analysis of Thompson sampling for online learning under full feedback - also known as prediction with expert advice - where the learner's prior is defined over the space of an adversary's future actions, rather than the space of experts. We show regret decomposes into regret the learner expected a priori, plus a prior-robustness-type term we call excess regret. In the classical finite-expert setting, this recovers optimal rates. As an initial step towards practical online learning in settings with a potentially-uncountably-infinite number of experts, we show that Thompson sampling with a certain Gaussian process prior widely-used in the Bayesian optimization literature has a $\mathcal{O}(\beta\sqrt{T\log(1+\lambda)})$ rate against a $\beta$-bounded $\lambda$-Lipschitz adversary.

Ensuring equitable Artificial Intelligence (AI) in healthcare demands systems that make unbiased decisions across all demographic groups, bridging technical innovation with ethical principles. Foundation Models (FMs), trained on vast datasets through self-supervised learning, enable efficient adaptation across medical imaging tasks while reducing dependency on labeled data. These models demonstrate potential for enhancing fairness, though significant challenges remain in achieving consistent performance across demographic groups. Our review indicates that effective bias mitigation in FMs requires systematic interventions throughout all stages of development. While previous approaches focused primarily on model-level bias mitigation, our analysis reveals that fairness in FMs requires integrated interventions throughout the development pipeline, from data documentation to deployment protocols. This comprehensive framework advances current knowledge by demonstrating how systematic bias mitigation, combined with policy engagement, can effectively address both technical and institutional barriers to equitable AI in healthcare. The development of equitable FMs represents a critical step toward democratizing advanced healthcare technologies, particularly for underserved populations and regions with limited medical infrastructure and computational resources.

In this paper, we present an intuitive analysis of the optimization technique based on the quantization of an objective function. Quantization of an objective function is an effective optimization methodology that decreases the measure of a level set containing several saddle points and local minima and finds the optimal point at the limit level set. To investigate the dynamics of quantization-based optimization, we derive an overdamped Langevin dynamics model from an intuitive analysis to minimize the level set by iterative quantization. We claim that quantization-based optimization involves the quantities of thermodynamical and quantum mechanical optimization as the core methodologies of global optimization. Furthermore, on the basis of the proposed SDE, we provide thermodynamic and quantum mechanical analysis with Witten-Laplacian. The simulation results with the benchmark functions, which compare the performance of the nonlinear optimization, demonstrate the validity of the quantization-based optimization.

Adaptive optimizers have emerged as powerful tools in deep learning, dynamically adjusting the learning rate based on iterative gradients. These adaptive methods have significantly succeeded in various deep learning tasks, outperforming stochastic gradient descent (SGD). However, although AdaGrad is a cornerstone adaptive optimizer, its theoretical analysis is inadequate in addressing asymptotic convergence and non-asymptotic convergence rates on non-convex optimization. This study aims to provide a comprehensive analysis and complete picture of AdaGrad. We first introduce a novel stopping time technique from probabilistic theory to establish stability for the norm version of AdaGrad under milder conditions. We further derive two forms of asymptotic convergence: almost sure and mean-square. Furthermore, we demonstrate the near-optimal non-asymptotic convergence rate measured by the average-squared gradients in expectation, which is rarely explored and stronger than the existing high-probability results, under the mild assumptions. The techniques developed in this work are potentially independent of interest for future research on other adaptive stochastic algorithms.

Reinforcement Learning from Human Feedback (RLHF) learns from the preference signal provided by a probabilistic preference model, which takes a prompt and two responses as input, and produces a score indicating the preference of one response against another. So far, the most popular RLHF paradigm is reward-based, which starts with an initial step of reward modeling, and the constructed reward is then used to provide a reward signal for the subsequent reward optimization stage. However, the existence of a reward function is a strong assumption and the reward-based RLHF is limited in expressivity and cannot capture the real-world complicated human preference. In this work, we provide theoretical insights for a recently proposed learning paradigm, Nash learning from human feedback (NLHF), which considered a general preference model and formulated the alignment process as a game between two competitive LLMs. The learning objective is to find a policy that consistently generates responses preferred over any competing policy while staying close to the initial model. The objective is defined as the Nash equilibrium (NE) of the KL-regularized preference model. We aim to make the first attempt to study the theoretical learnability of the KL-regularized NLHF by considering both offline and online settings. For the offline learning from a pre-collected dataset, we propose algorithms that are efficient under suitable coverage conditions of the dataset. For batch online learning from iterative interactions with a preference oracle, our proposed algorithm enjoys a finite sample guarantee under the structural condition of the underlying preference model. Our results connect the new NLHF paradigm with traditional RL theory, and validate the potential of reward-model-free learning under general preference.

Let $\mathscr{F}_{n,d}$ be the class of all functions $f:\{-1,1\}^n\to[-1,1]$ on the $n$-dimensional discrete hypercube of degree at most $d$. In the first part of this paper, we prove that any (deterministic or randomized) algorithm which learns $\mathscr{F}_{n,d}$ with $L_2$-accuracy $\varepsilon$ requires at least $\Omega((1-\sqrt{\varepsilon})2^d\log n)$ queries for large enough $n$, thus establishing the sharpness as $n\to\infty$ of a recent upper bound of Eskenazis and Ivanisvili (2021). To do this, we show that the $L_2$-packing numbers $\mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon)$ of the concept class $\mathscr{F}_{n,d}$ satisfy the two-sided estimate $$c(1-\varepsilon)2^d\log n \leq \log \mathsf{M}(\mathscr{F}_{n,d},\|\cdot\|_{L_2},\varepsilon) \leq \frac{2^{Cd}\log n}{\varepsilon^4}$$ for large enough $n$, where $c, C>0$ are universal constants. In the second part of the paper, we present a logarithmic upper bound for the randomized query complexity of classes of bounded approximate polynomials whose Fourier spectra are concentrated on few subsets. As an application, we prove new estimates for the number of random queries required to learn approximate juntas of a given degree, functions with rapidly decaying Fourier tails and constant depth circuits of given size. Finally, we obtain bounds for the number of queries required to learn the polynomial class $\mathscr{F}_{n,d}$ without error in the query and random example models.

We consider stochastic approximations of sampling algorithms, such as Stochastic Gradient Langevin Dynamics (SGLD) and the Random Batch Method (RBM) for Interacting Particle Dynamcs (IPD). We observe that the noise introduced by the stochastic approximation is nearly Gaussian due to the Central Limit Theorem (CLT) while the driving Brownian motion is exactly Gaussian. We harness this structure to absorb the stochastic approximation error inside the diffusion process, and obtain improved convergence guarantees for these algorithms. For SGLD, we prove the first stable convergence rate in KL divergence without requiring uniform warm start, assuming the target density satisfies a Log-Sobolev Inequality. Our result implies superior first-order oracle complexity compared to prior works, under significantly milder assumptions. We also prove the first guarantees for SGLD under even weaker conditions such as H\"{o}lder smoothness and Poincare Inequality, thus bridging the gap between the state-of-the-art guarantees for LMC and SGLD. Our analysis motivates a new algorithm called covariance correction, which corrects for the additional noise introduced by the stochastic approximation by rescaling the strength of the diffusion. Finally, we apply our techniques to analyze RBM, and significantly improve upon the guarantees in prior works (such as removing exponential dependence on horizon), under minimal assumptions.