AI-detection tools are getting better. Basically every recent, high-profile accusation of someone passing off AI-generated writing as their own has started in the same way: with a tool called Pangram. In March, when a horror novel from a major publishing house was pulled just days before its scheduled U.S. release date, it was in part because Pangram, an AI-detection program, had identified the text as AI-generated. Other people have fed text into Pangram to suggest that chatbots have been used to write articles in major newspapers including, multiple short stories awarded a prestigious literary prize, and most recently, significant chunks of Pope Leo XIV's encyclical warning about the dangers of AI. The tool is also used by universities to vet student work and scientific associations to scan research papers.

We study stochastic decision-theoretic online learning with full information and event-level pure differential privacy. A COLT open problem of Hu and Mehta asks to determine the optimal gap-dependent regret rate for stochastic decision-theoretic online learning under pure event-level differential privacy. For $K$ actions, losses in $[0,1]$, and a unique best action separated from the second-best action by gap $Δ_{\min}$, the known lower bound is of order $ \frac{\log K}{\min\{Δ_{\min},\varepsilon\}}, $ or equivalently, up to universal constants, of order \[ \frac{\log K}{Δ_{\min}}+\frac{\log K}{\varepsilon}. \] We give a horizon-free pure-DP algorithm and prove the explicit regret bound \[ \operatorname{Reg}_T \le 1000 \cdot \left(\frac{\log K}{Δ_{\min}}+\frac{\log K}{\varepsilon}\right) \] for every horizon $T$. The numerical constant is not optimized. The algorithm partitions time into blocks of exponentially increasing size, plays a single action throughout each block, and chooses the next action by an exponential mechanism applied to a data-independent random prefix of the previous block. The random prefix converts block regret into a sum, over all prefix lengths, of softmax selection errors. A single entropy-potential argument controls all privacy-dominated large-gap actions at cost $\log K/\varepsilon$.

The scaling limit where both the size of the training set $P$ and the width $N$ of a deep neural network grow at the same rate, the so-called proportional-width regime, has been intensely studied for shallow, single-hidden-layer networks. However, extending these non-perturbative results from shallow architectures to deep non-linear networks has proven very challenging. Here we present an effective approximate approach to predict the generalization performance of Bayesian multi-layer perceptrons (MLPs) of fixed depth $L$ on arbitrary high-dimensional data. We propose an equivalent Wishart Ansatz to capture the dominant stochastic fluctuations of the hierarchical empirical kernels of MLPs. This allows us to perform a large deviation analysis for the partition function of MLPs in the proportional limit, expressed in terms of a renormalized NNGP kernel. In this description, even strong representation learning in the proportional limit is encoded in at most $L$ scalar order parameters, determined self-consistently. Extending the approach to convolutional architectures (CNNs), we identify a hierarchical local kernel renormalization mechanism, which allows to quantify more complex data-dependent transformations of the large-width kernel in CNNs due to finite-width effects. We test our effective theory against sampling experiments from the Bayesian posterior of finite deep neural networks with depths $L \sim O(10)$ and $P\sim O(10^3)$ on classic benchmark datasets, finding overall very good agreement together with two distinct types of systematic deviations.

We study contextual bandits in the stochastic i.i.d.\ setting, where a learner observes contexts drawn from an unknown distribution, selects actions from a finite set $A$, and aims to identify an approximately optimal policy from a given class based on bandit feedback. Motivated by bandit multiclass classification with zero-one rewards, we focus on the \emph{$s$-sparse} setting in which, for every context, the reward vector has $L_1$-norm at most $s \ll |A|$. Our main result is the design of algorithms that, with high probability, output an $ε$-optimal policy compared to policy class $Π$ using $\tilde{O} ((s/ε^2 + |A|/ε)\log |Π|/δ)$ samples. We extend this bound to general Natarajan classes and complement it with a matching lower bound (up to logarithmic factors), thereby closing a substantial gap left by prior work (Erez et al., 2024, 2025), which incurred an additional $Θ(|A|^9)$ dependence. We obtain these results via two complementary approaches. First, we analyze contextual bandits through the lens of contextual decision making with structured observations, designing an exploration-by-optimization algorithm whose sample complexity is governed by the \emph{decision-estimation coefficient} (DEC; Foster et al., 2021, 2022). We show that, with $s$-sparse rewards, the induced model class admits a sharp DEC bound that scales with $s$ and directly yields the optimal rate. Since this approach is largely information-theoretic and involves solving complex min-max optimization problems, we also develop a second, more specialized algorithmic method based on a low-variance exploration technique. This approach leads to concrete, tractable algorithms and naturally extends to contextual combinatorial semi-bandits, leading to improved sample complexity guarantees for bandit multiclass list classification.

We study the Lipschitz bandit problem, where a learner sequentially maximizes an unknown Lipschitz function $f$ over a domain $\mathcal{X} \subset [0,1]^d$ using noisy pointwise evaluations. Existing regret bounds are either worst-case, scaling as $\tildeΘ \left ( T^{d+1/d+2}\right )$, or adaptive via the zooming dimension $d_z$, yielding $\tildeΘ \left ( T^{d_z+1/d_z+2}\right )$. However, such zooming-based guarantees are only partially instance-dependent, as they depend solely on the asymptotic growth of near-optimal level sets and fail to capture finer structural properties of $f$. We provide an analysis and an algorithm that characterizes the regret through integrals of the suboptimality gap of $f$ over its level sets. This yields regret bounds that adapt to the local growth of level sets, rather than only their asymptotic behavior. As a corollary, when the set of maximizers has dimension $d^\star>0$, we obtain improved adaptive rates of order $\tilde{\mathcal{O}} \left ( T^{d_z+1 / \max(d_z,d^\star)+2}\right )$ strictly improving over classical zooming bounds in this regime. Finally, we extend our analysis to the full-information setting (Lipschitz experts) and show how some of the regularity assumptions can be relaxed.

Cluster analysis is a widely applied machine learning technique to understand the existing patterns in the population of gamma-ray bursts (GRBs), in order to explore their physical sources. In the present scenario, the number of clusters corresponding to differentiable groups is still under conflict, in spite of numerous attempts with the state-of-the-art clustering procedures. This crucial unknown parameter needs to be evaluated, either directly or indirectly in terms of other tuning parameters, to produce the clusters in GRBs through implementation of an appropriate clustering algorithm. While most of the applied algorithms reached two physically explained groups of merger and collapsar predominated by the short and long bursts respectively, other statistical approaches violated this binary partition. However, physical establishment of any additional cluster(s) is not yet confirmed. Therefore, we propose a new algorithm, from a different stream of clustering referred to as `completely parameter-free', which carries out the classification of GRBs in a manner that has not been tried so far. It indicates two main groups, of short and long duration bursts from the BATSE sample, compatible with the merger-collapsar theory.

Finding approximations to an intractable probability distribution π of interest (usually known only up to a normalizing constant) is a key problem in scientific computing. Variational Inference stands out as a particularly attractive tool for this task, owing to its statistical and computational efficiency, and it has been the framework underlying many advances in computational statistics over the past half century (Parisi, 1980; Hinton and Van Camp, 1993; Jordan et al., 1999; Bishop and Nasrabadi, 2006). The central idea is to seek a tractable approximation to π within a chosen family of tractable distributions Q by minimizing a divergence to π over that'variational' family. Often, it is convenient or well-motivated to work with the family of product (or tensor, or factorized) distributions Q = P m, and define optimality through minimisation of the Kullback-Leibler (KL) divergence (also'relative entropy') min KL(ϱ||π): ϱ P m . A key practical aspect of working with this particular loss function is that in solving the associated optimisation problem, one is only required to compute expectations under the tractable variational distribution ϱ, rather than under the intractable target distribution π. In Bayesian statistics, π typically represents the joint posterior distribution of latent variables z Z and some parameters β B given observed data y Y. In these cases, we often choose m = 2 and seek the best variational approximation µ(dz) ν(dβ) to π to solve min KL(µ ν||π): µ P(Z), ν P(B) . The coordinate ascent variational inference algorithm (CAVI, Bishop and Nasrabadi, 2006; Blei et al., 2017) solves this problem by iteratively minimizing the Kullback-Leibler divergence with respect to one element at a time: given a starting point ν0, it iterates µk:= argmin

A central goal of modern causal inference is estimating heterogeneous treatment effects to answer questions like "how does an intervention affect each unit," rather than only on average. We study this problem with panel-data where we observe $n$ units across $m$ times under unknown, non-uniform treatment assignments. The data in this setting is naturally represented as a matrix of all unit--time treatment effects. Estimating heterogeneous treatment effects can then be expressed as obtaining a good estimation of each row's average in this matrix. This allows us to formulate the problem as matrix completion, which can be solved under natural low-rankness assumptions. However, existing matrix-completion guarantees are not powerful enough to get meaningful bounds for the per-row guarantee required for estimating the heterogeneous treatment effect; roughly speaking, they are only useful for estimating average treatment effect bounds, as also illustrated in a recent line of work. We give a simple, computationally efficient estimator that, without knowledge of the propensities and under standard low-rankness and regularity assumptions, achieves a row-wise $\ell_2$ error of $\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})$. Technically, our analysis establishes the first sharp row-wise $\ell_2$-perturbation bound for low-rank approximation, complementing existing spectral-, Frobenius-, and entrywise perturbation theory.

Frontier reasoning models are produced by posttraining base language models with reinforcement learning. Recent work has challenged this by showing that sampling from a sharpened version of the base model's distribution, a so-called power distribution, elicits comparable reasoning without additional training, curated datasets, or verifiers. However, making this method practical requires efficiently sampling from the power distribution. A sampler needs to "mix" to the power distribution, which necessitates moving between modes of the target distribution; intuitively, e.g., trying different reasoning strategies. The samplers proposed in prior works repeatedly select a "cut" position in the current reasoning trace uniformly at random and resample the suffix from that position onward. However, reasoning traces typically contain a few consequential decisions (e.g., the choice of proof strategy or algorithm), and we observe that a uniformly chosen cut tends to rewrite local details rather than revisit decision points. We introduce an algorithm (Entropy-Cut Metropolis-Hastings) that uses the base model's next-token entropy as a proxy to identify key decision points and resample from those positions. We empirically verify that entropy jumps are a useful proxy for decision points and, in a stylized model of reasoning, prove that our method's mixing time scales with the number of decisions in a trace rather than with the number of tokens, which can be much larger. Across MATH500, HumanEval, GPQA Diamond, and AIME26, our method consistently improves over baselines and RL-trained models.

Deep generative models offer powerful tools for multivariate data analysis, but their black-box architectures are often unidentified and difficult to interpret. We introduce the Deep Discrete Encoder (DDE) Copula, an identifiable and interpretable generative model for multivariate data with arbitrary marginal distributions. The model places a hierarchical directed network of binary latent variables inside a copula framework, enabling flexible dependence modeling for mixed discrete and continuous data. Estimation is based on rank likelihoods, which decouple marginal modeling from posterior inference on the DDE parameters and avoid specifying the marginal distributions. We establish conditions for identification of the DDE copula parameters, ensuring that layer-specific parameters provide meaningful summaries of multivariate dependence. We also prove quotient-space posterior consistency for continuous margins under the exact rank likelihood and treat the extended rank likelihood for tied or mixed margins as a generalized likelihood, with concentration under an additional contrast condition. For computation, we propose a stochastic expectation-maximization algorithm for \emph{maximum a posteriori} estimation, together with initialization strategies that improve convergence. To learn network dimension adaptively, we extend Bayesian rank-selection priors to infer layer-specific widths. Simulations show strong finite-sample performance, and a personality-survey analysis reveals interpretable hierarchical latent structure in complex multivariate data.