In many real-world settings, a centralized decision-maker must repeatedly allocate finite resources to a population over multiple time steps. Individuals who receive a resource derive some stochastic utility; to characterize the population-level effects of an allocation, the expected individual utilities are then aggregated using a social welfare function (SWF). We formalize this setting and present a general confidence sequence framework for SWF-based online learning and inference, valid for any monotonic, concave, and Lipschitz-continuous SWF. Our key insight is that monotonicity alone suffices to lift confidence sequences from individual utilities to anytime-valid bounds on optimal welfare. Building on this foundation, we propose SWF-UCB, a SWF-agnostic online learning algorithm that achieves near-optimal $\tilde{O}(n+\sqrt{nkT})$ regret (for $k$ resources distributed among $n$ individuals at each of $T$ time steps). We instantiate our framework on three normatively distinct SWF families: Weighted Power Mean, Kolm, and Gini, providing bespoke oracle algorithms for each. Experiments confirm $\sqrt{T}$ scaling and reveal rich interactions between $k$ and SWF parameters. This framework naturally supports inference applications such as sequential hypothesis testing, optimal stopping, and policy evaluation.

Overparameterized models have recently challenged conventional learning theory by exhibiting improved generalization beyond the interpolation limit, a phenomenon known as benign overfitting. This work introduces Adaptive Benign Overfitting (ABO), extending the recursive least-squares (RLS) framework to this regime through a numerically stable formulation based on orthogonal-triangular updates. A QR-based exponentially weighted RLS (QR-EWRLS) algorithm is introduced, combining random Fourier feature mappings with forgetting-factor regularization to enable online adaptation under non-stationary conditions. The orthogonal decomposition prevents the numerical divergence associated with covariance-form RLS while retaining adaptability to evolving data distributions. Experiments on nonlinear synthetic time series confirm that the proposed approach maintains bounded residuals and stable condition numbers while reproducing the double-descent behavior characteristic of overparameterized models. Applications to forecasting foreign exchange and electricity demand show that ABO is highly accurate (comparable to baseline kernel methods) while achieving speed improvements of between 20 and 40 percent. The results provide a unified view linking adaptive filtering, kernel approximation, and benign overfitting within a stable online learning framework.

We study a class of iterated empirical risk minimization (ERM) procedures in which two successive ERMs are performed on the same dataset, and the predictions of the first estimator enter as an argument in the loss function of the second. This setting, which arises naturally in active learning and reweighting schemes, introduces intricate statistical dependencies across samples and fundamentally distinguishes the problem from classical single-stage ERM analyses. For linear models trained with a broad class of convex losses on Gaussian mixture data, we derive a sharp asymptotic characterization of the test error in the high-dimensional regime where the sample size and ambient dimension scale proportionally. Our results provide explicit, fully asymptotic predictions for the performance of the second-stage estimator despite the reuse of data and the presence of prediction-dependent losses. We apply this theory to revisit a well-studied pool-based active learning problem, removing oracle and sample-splitting assumptions made in prior work. We uncover a fundamental tradeoff in how the labeling budget should be allocated across stages, and demonstrate a double-descent behavior of the test error driven purely by data selection, rather than model size or sample count.

The video menu at Dialogue Cafe teaches hearing people how to order a drink using sign language. The video menu at Dialogue Cafe teaches hearing people how to order a drink using sign language. 'Coffee is just the excuse': the deaf-run cafe where hearing people sign to order W esley Hartwell raised his fists to the barista and shook them next to his ears. He then lowered his fists, extended his thumbs and little fingers, and moved them up and down by his chest, as though milking a cow. Finally, he laid the fingers of one hand flat on his chin and flexed his wrist forward.

He catches nascent changes in the political weather. "During early, he kept telling me, 'Crime--there's something here,' " DeBoo told me. DeBoo studied the latest crime statistics and saw nothing unusual. He brushed off the worry. Then new numbers came out, showing a large pandemic spike in shoplifting and car theft, and concerns about crime exploded into the headlines. Last March, judging the winds, Newsom launched a podcast, "This Is Gavin Newsom."

What NAD+? Here's how it works in your body, why it matters, and if supplementation is worth the hype. It's more than likely that the NAD+ supplement craze has already crossed your path. The Biebers have infused it. Joe Rogan has podcasted about it. Gwyneth Paltrow swears by it and, of course, sells her own Youth-Boost NAD+ Peptide Rich Cream . NAD+ (short for nicotinamide adenine dinucleotide) is a coenzyme that your body makes naturally--it contributes to energy production and immune function, among other things. It reflects a broader shift in how people think about healthy aging and extending their healthspan overall .

Virtually every aspect of your health can be traced back to your microbiome. But some tests are better than others. Some of the recipes look tasty. I admit it: I'm a sucker for metrics. Fitness trackers that keep tabs on my steps and sleep? A DEXA scan to give me too much information about my body composition?

We study the learning dynamics of a multi-pass, mini-batch Stochastic Gradient Descent (SGD) procedure for empirical risk minimization in high-dimensional multi-index models with isotropic random data. In an asymptotic regime where the sample size $n$ and data dimension $d$ increase proportionally, for any sub-linear batch size $κ\asymp n^α$ where $α\in [0,1)$, and for a commensurate ``critical'' scaling of the learning rate, we provide an asymptotically exact characterization of the coordinate-wise dynamics of SGD. This characterization takes the form of a system of dynamical mean-field equations, driven by a scalar Poisson jump process that represents the asymptotic limit of SGD sampling noise. We develop an analogous characterization of the Stochastic Modified Equation (SME) which provides a Gaussian diffusion approximation to SGD. Our analyses imply that the limiting dynamics for SGD are the same for any batch size scaling $α\in [0,1)$, and that under a commensurate scaling of the learning rate, dynamics of SGD, SME, and gradient flow are mutually distinct, with those of SGD and SME coinciding in the special case of a linear model. We recover a known dynamical mean-field characterization of gradient flow in a limit of small learning rate, and of one-pass/online SGD in a limit of increasing sample size $n/d \to \infty$.

In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $β$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.

Modern large language models (LLMs) such as GPT, Claude, and Gemini have transformed the way we learn, work, and communicate. Y et, their ability to produce highly human-like text raises serious concerns about misinformation and academic integrity, making it an urgent need for reliable algorithms to detect LLMgenerated content. In this paper, we start by presenting a geometric approach to demystify rewrite-based detection algorithms, revealing their underlying rationale and demonstrating their generalization ability. Building on this insight, we introduce a novel rewrite-based detection algorithm that adaptively learns the distance between the original and rewritten text. Theoretically, we demonstrate that employing an adaptively learned distance function is more effective for detection than using a fixed distance. Empirically, we conduct extensive experiments with over 100 settings, and find that our approach demonstrates superior performance over baseline algorithms in the majority of scenarios. In particular, it achieves relative improvements from 57.8% to 80.6% over the strongest baseline across different target LLMs (e.g., GPT, Claude, and Gemini). The past few years have witnessed the emergence and rapid development of large language models (LLMs) such as GPT (Hurst et al., 2024), DeepSeek (Liu et al., 2024), Claude (Anthropic, 2024), Gemini (Comanici et al., 2025), Grok (xAI, 2025) and Qwen (Y ang et al., 2025). Their impact is everywhere, from education, academia and software development to healthcare and everyday life (Arora & Arora, 2023; Chan & Hu, 2023; Hou et al., 2024). On one side of the coin, LLMs can support users with conversational question answering, help students learn more effectively, draft emails, write computer code, prepare presentation slides and more. On the other side, their ability to closely mimic human-written text also raises serious concerns, including the generation of biased or harmful content, the spread of misinformation in the news ecosystem, and the challenges related to authorship attribution and intellectual property (Dave et al., 2023; Fang et al., 2024; Messeri & Crockett, 2024; Mahajan et al., 2025; Laurito et al., 2025). Addressing these concerns requires effective algorithms to distinguish between human-written and LLM-generated text, which has become an active and popular research direction in recent literature (see Crothers et al., 2023; Wu et al., 2025, for reviews).