Papers

We study Langevin diffusion and Langevin Monte Carlo (LMC) when the target distribution changes over time. Under a log-Sobolev inequality (LSI), we derive non-asymptotic Rényi-divergence guarantees for tracking the current target. The framework covers continuous-time Langevin diffusion and its discretizations. We then apply the results to nonsmooth sampling based on successive Moreau envelopes. For this scheme, we give explicit choices of the smoothing parameters and step sizes, together with corresponding complexity bounds. To our knowledge, these are the first non-asymptotic Rényi-divergence tracking bounds for Langevin dynamics with discrete target updates.

The computation of Black-Scholes implied volatility is a fundamental task in quantitative finance, underpinning option valuation, model calibration and risk management. Although implied volatility is routinely used in practice, the inversion of the Black-Scholes pricing formula remains a challenging numerical problem, particularly in asymptotic regimes corresponding to extreme option prices, strikes or maturities, where the inverse map becomes highly sensitive to perturbations of the price. In this paper, we introduce a new family of asymptotically-informed neural-network architectures for implied-volatility computation. Exploiting the distinct behaviours of the Black-Scholes pricing function in different volatility regimes, we propose a family of architectures that learn a trainable partition of the price-log-moneyness domain through a system of gating functions and combines specialised local approximations of the implied-volatility function within each region. Extensive numerical experiments demonstrate that the proposed models consistently outperform standard feed-forward neural networks across a wide range of parameter domains, often by several orders of magnitude in relative accuracy while maintaining excellent generalisation properties. Furthermore, the neural-network outputs provide highly accurate initial guesses for a third-order Householder scheme, allowing near machine-precision implied-volatility computations after only two refinement iterations.

\begin{quote} {\bf\em Abstract\/}. ~A variety of models for insurance and other types of losses are special cases of the log-location-scale\/ family, with the lognormal and Pareto-$I$ distributions being the most prominent examples. The latter also serves as a primary example of infinite-mean models that often present challenges in risk management. In this paper, we utilize two asymptotic\/ theorems – the joint normality of sample quantiles (of i.i.d.\/ random variables) and the delta method – to construct nonlinear and linear regression frameworks for estimation and validation of log-location-scale distributions. Within these regression frameworks, four equally robust estimators – ordinary and generalized quantile (oQLS and gQLS) and log-quantile (log-oQLS and log-gQLS) least squares – are proposed. For log-location-scale loss models, the logarithmic transformation of quantiles approximates the nonlinear least squares solution exactly\/ and yields more accurate estimators. Also, the log-linear regression framework facilitates a convenient way to study the estimators' properties and to design a residuals-based goodness-of-fit test. Moreover, log-oQLS and log-gQLS have explicit formulas and can be easily computed for medium- ($n=10^3$), large- ($n=10^4$), and very large-size ($n > 10^6$) samples. Computational and statistical performances of the estimators, outlier-labeling rules, and the goodness-of-fit test are illustrated using simulated and real datasets. \vspace{2mm} {\bf\em Keywords\/}. ~Goodness-of-Fit; Outliers; Quantiles; Relative Efficiency; Robustness. \end{quote}

We study exact representations of $\mathrm{MAX}_N(x)=\max{x_1,\ldots,x_N}$ using two-hidden-layer ReLU neural networks. This problem has been studied in recent years in an attempt to characterize the exact number of hidden layers required to represent continuous piecewise linear functions. The best lower bound is 2, while the current upper bound is logarithmic in $N$. It remains completely open if the right answer is a constant number of hidden layers (possibly even 2!) or not. In fact, a recent breakthrough was the representation of $\mathrm{MAX}_5$ as a two-hidden-layer ReLU function obtained in [Bakaev et al., 2026], and the case of $\mathrm{MAX}_N$ was stated as open for $N\geq 6$ in that paper. Using a careful computer assisted search, we obtain two-hidden-layer ReLU representations of $\mathrm{MAX}_5, \mathrm{MAX}_6, \mathrm{MAX}_7$, and $\mathrm{MAX}_8$. We obtain these by considering rational linear combinations of terms of the form $\max\{\sum_{r=1}^{s}\max(x_{a_r},x_{b_r}),\sum_{r=1}^{s}\max(x_{c_r},x_{d_r})\}$, where $a_r,b_r,c_r,d_r\in\{1,\ldots,N\}$. Each inner maximum of two coordinates can be computed in a first hidden layer, and the outer maximum of the two side-sums can be computed in a second hidden layer. Consequently, every finite linear combination of these terms has a two-hidden-layer ReLU realization. An identity for $\mathrm{MAX}_N$ in this form therefore gives an exact two-hidden-layer ReLU representation of $\mathrm{MAX}_N$. Very recently, two-hidden-layer representations of $\mathrm{MAX}_N$ of the above form were obtained for all $N\leq 10$ in [Ruess et al., 2026]. Our representations are different and were developed independently. While our techniques share most of the high-level ideas presented in [Ruess et al., 2026], there are also some minor differences which may be of interest for future research on this problem.

Bayesian quantile regression based on the asymmetric Laplace (AL) distribution can be sensitive to extreme observations because of its exponentially decaying tails. We propose a robust error distribution constructed as a finite mixture of the AL distribution and a log-Pareto scale mixture of asymmetric Laplace distributions (LPAL). Unlike a direct log-Pareto extension of the normal location-scale representation of the AL distribution, the proposed AL-LPAL mixture preserves the prescribed quantile and exhibits log-regularly varying behavior in both tails. The LPAL component also has an unbounded density at the target quantile, yielding a distribution that combines sharp central concentration with super-heavy tails. We establish posterior robustness under arbitrarily extreme contamination and provide sufficient conditions for the existence of posterior moments of the regression coefficients and scale parameter. For posterior computation, we develop a Gibbs sampler using latent-variable augmentations and a computationally efficient mean-field variational Bayes approximation. Simulation studies show that the proposed method is competitive under moderate contamination and maintains stable point estimation with comparatively concentrated posterior intervals, particularly when severe contamination affects the quantile of interest. Applications to carbon dioxide and Boston housing data, using the same preprocessing as existing robust Bayesian quantile regression analyses, show favorable predictive performance across nearly all quantile levels and loss criteria considered.

In this paper, we study alternating regret in online convex optimization (OCO), motivated by the success of alternating learning dynamics in two-player games. Although previous works have shown that $o(\sqrt{T})$ alternating regret is achievable under various assumptions on the loss functions and feasible domains, the minimax regret rate has remained open even for the expert problem. In this paper, we resolve this question by showing matching lower and upper bounds for both the expert problem and general OCO. Somewhat surprisingly, for the $d$-expert problem, we show that the minimax alternating regret is $\Theta(\log d)$, independent of the horizon $T$. This significantly improves upon the best-known $\mathcal{O}(T^{1/3}\log^{2/3} d)$ established by Hait et al. [2025]. We further extend our results to general OCO over a $d$-dimensional compact convex set and prove that the worst-case minimax alternating regret is $\Theta\left(d\log \left(1+\frac{T}{d}\right)\right)$, also significantly improving upon the best-known $\mathcal{O}((d\log T)^{2/3}T^{1/3})$ upper bound and resolving the open problem posed by Cevher et al. [2023], Hait et al. [2025]. Technically, our upper bound for the expert problem is achieved by a corrected variant of Hedge, in which carefully designed correction terms cancel the unfavorable curvature arising in the alternating-regret analysis. We extend the same corrected-potential argument to continuous action sets to obtain the optimal alternating-regret rate for OCO. For the lower bounds, the expert construction repeatedly eliminates half of the candidate experts, while the OCO lower bound instance construction replaces this discrete elimination by a more involved multiscale construction on the unit disk.

Principal component analyses are often applied to spatial data towards inference on latent modes of spatial variation. These analyses are widespread across domains including spatial transcriptomics and environmental sciences, where the modes of spatial variation are represented by corresponding factors of gene expression or remotely sensed time series measurements. Many methods have been proposed for incorporating spatial information into a probabilistic PCA framework; however, there are three main drawbacks to currently available approaches. First, the loadings matrices are not orthogonal, and subsequent orthogonalization of those loadings corrupts the original prior spatial information. Furthermore, currently proposed methods assume stationarity in their spatial prior. Finally, current methods typically do not achieve linear-time computational complexity with respect to the number of spatial locations. To resolve these problems, we first parameterize the model directly with orthogonal loadings. For the prior distribution, we derive the sampling distribution of an SVD transformation with $k$ unique and $m-k$ repeated singular values. We then show under this model that the maximum a posteriori estimator for the orthogonal loadings is the eigendecomposition of $S + \frac{1}{n}\Sigma$, where $S$ is the empirical covariance matrix and $\Sigma$ is the prior spatial covariance. We develop a minorization-maximization-within-EM algorithm that is linear in computational complexity with respect to the number of spatial locations. We further extend our MM-EM algorithm to handle held-out locations and develop a validation strategy for optimizing the nonstationary prior covariance. Our methodology is used to infer the spatial distribution of direction-specific length scales in a human brain spatial transcriptomics case study, as well as a continental-scale phenology case study in sub-Saharan Africa.

Gromov-Wasserstein (GW) compares distributions through relations within each space. This pointwise comparison can be too sensitive in one-to-many settings, where several target outcomes refine one source state and their mean carries the geometry of interest. We introduce a weak GW framework that compares source relations with relations between the target conditional laws induced by a coupling. For inner-product relations, we retain the conditional means $m_\pi(x)=\mathbb{E}_\pi[Y\mid X=x]$. The resulting barycentric weak inner-product GW (wIGW) satisfies $\mathrm{wIGW}_{\mathrm{bar}}^2(\mu,\nu)=\inf_{\eta\preceq_{\mathrm{cx}}\nu}\mathrm{IGW}^2(\mu,\eta)$. Here $\eta\preceq_{\mathrm{cx}}\nu$ means that $\nu$ is a mean-preserving spread of $\eta$. Thus wIGW searches for an intermediate target geometry that can be refined into the prescribed target law without changing conditional means. Under finite second moments, minimizers exist and martingale gluing recovers an optimal coupling. With ridge regularization, moment duality gives an $A$-$B$ min-max problem whose inner step is weak optimal transport with a quadratic cost parameterized by $A$ and $B$; the outer problem optimizes these matrices. For finitely supported measures, we give an iterative algorithm. Under a quantitative ridge condition, the reduced problem is convex–concave, and the projected outer iteration satisfies an explicit contraction bound for inexact inner solves. Point cloud and graph feature refinement experiments illustrate how mean-preserving target refinements can have zero cost. A paired peripheral blood mononuclear cell (PBMC) multiome study evaluates atlas based cell type transfer through RNA/ATAC alignment in cell to cell and prototype to cell settings, with the prototype to cell setting representing the one-to-many case.

Multi-modal time series forecasting methods integrate auxiliary context into temporal predictions through increasingly sophisticated fusion mechanisms. A growing body of work reports substantial gains, yet it is often unclear whether they reflect genuine use of the context or incidental architectural effects. We ask a narrower, checkable question: when can auxiliary context help a forecaster at all? We identify two dataset-level conditions that must both hold: (1) the target is not dominated by a last-value shortcut (low autocorrelation rho_h), and (2) the context carries information about the target beyond history (non-zero conditional mutual information delta; when delta=0 no predictor can benefit—a distribution-free result). Through controlled experiments on MoME (a 14.3B-parameter mixture-of-experts model, 6 datasets, 10 seeds) and four additional fusion mechanisms implemented within a single-backbone testbed (5 datasets), we find that when both conditions hold, text-conditioned expert modulation contributes a sizeable MSE reduction; when either fails, the contribution collapses to the capacity floor of the modulation pathway and carries no context-attributable signal. We establish causality through two interventions: adding a shortcut to MoME suppresses routing contribution by 77-93% across 3 datasets; progressively corrupting context quality drives the context-specific benefit from +44% to negative. We validate the autocorrelation component of our diagnostic on 27 Monash Archive datasets. We provide a calibrated pre-training diagnostic that, on the datasets we test, yields no false positives in well-powered settings. We are explicit about the asymmetry of our evidence: the negative arm is broadly reliable, while the large positive magnitudes come from a single model family (MoME) and are corroborated only in direction by the testbed.

Machine learning is usually formalized through samples, while the persistent individual to which multiple observed or possible events refer often remains implicit. We propose the unit as an explicit primitive at the level of task semantics. A learning task first declares a population of persistent referents and a sameness criterion; the realized value $u$ denotes the selected referent. Supervised learning is the main formal specialization. Its semantic object is a family of unit-conditioned response laws. Homogeneity is the special case in which those laws coincide; a sample-only conditional is silent as to whether the world is homogeneous or the observed law is only the marginal of a heterogeneous family. What is learned from data is a pair $(T_\phi,R_\theta)$: a tokenizer that produces a contextual unit token and one shared response-law form that reads it. The structured class takes that form to be a simple relation in the token; a linear predictor is the running instance. The token is the learner-side representation through which the task-side unit affects prediction, while a learner specification that omits unit information is unit-insensitive; homogeneity remains a property of the world-side response family. When identity is unresolved, the world-side law mixes unit-conditioned targets, while the learner composes its shared form with a token. A trusted resolver may fix the unit and supply a lookup token; otherwise unit abduction forms a token of the same type from factual evidence. Unlinked single-row observations can fail to distinguish a heterogeneous unit world from a homogeneous pooled world; trusted same-unit pairs separate a restricted witness. The formal results concern this supervised specialization.

Optimizing expensive, high-dimensional black-box functions remains a central challenge in modern machine learning and scientific discovery. While local Bayesian optimization mitigates the curse of dimensionality, existing techniques often prioritize the probability of descent over the magnitude of progress. This leads to overly conservative steps that yield negligible improvement, wasting queries on directions that are nearly certain to descend but offer little decrease. We introduce Gradient Refinement and Progress-Aware Exploitation (GRAPE), a two-stage framework that first sharpens the local gradient posterior via a closed-form acquisition function, then selects update directions by maximizing the expected decrease conditional on descent. Theoretical analysis proves that this gradient refinement stage monotonically minimizes local uncertainty and that the progress-aware direction converges to true steepest descent as the posterior sharpens. Empirically, GRAPE demonstrates superior query efficiency across high-dimensional tasks: in black-box adversarial attacks, it achieves an average 5.4$\times$ speedup over baselines, and on large language model prompt optimization tasks, it outperforms the second best method by a reduction of 3.8 log-units in the final average regret.

Missing values, atypical observations, and heterogeneity across latent groups are common sources of complexity in regression data. The contaminated Gaussian cluster-weighted model (CG-CWM) provides a natural framework for handling atypical observations, including outliers and leverage points, in model-based clustering. We extend the CG-CWM to data with missing-at-random (MAR) values in both the response and covariate spaces. The proposed model provides clustering in regression analysis while distinguishing typical observations, outliers, and good and bad leverage points. By treating covariates as random, the model preserves assignment dependence, allowing them to contribute directly to cluster formation. Maximum likelihood estimation is performed through an expectation-conditional maximization (ECM) algorithm that accounts for four sources of incomplete information: missing responses and covariates, unknown component memberships, and latent contamination indicators. Conditional on these indicators, the joint distribution of responses and covariates is multivariate Gaussian, yielding closed-form conditional distributions for missing values and incorporating missingness uncertainty directly into parameter updates. Thus, missing values are handled within model fitting rather than by preliminary imputation. The framework provides clustering, clusterwise regression, model-based treatment of MAR values, and detection of atypical observations. Performance is assessed through numerical studies under varying levels of contamination and missingness patterns, and a real data application.

Time series forecasting models are widely used in high-stakes settings, yet their predictions remain difficult to interpret because existing post-hoc methods often ignore temporal dependence and fail to provide horizon-specific explanations. We propose a model-agnostic explainability framework that explains forecasting predictions by attributing each forecast horizon to temporally relevant historical lags. The framework models forecasting as a latent trajectory and introduces semantic flow to quantify how information evolves across time in the model's internal representations. By aggregating semantic flow, it constructs a lag-horizon attribution matrix that captures horizon-resolved temporal influence. To improve explainability, we further generate structure-preserving perturbations and fit sparse local surrogate models, producing human-readable and temporally coherent explanations. We evaluate the method using faithfulness and stability diagnostics across multiple benchmark datasets. Results show that the semantic-flow variant achieves competitive or superior faithfulness compared to standard post-hoc baselines, while being substantially more computationally efficient. Stability analysis further demonstrates that the explanations are robust and identifies regimes where interpretation should be applied with caution.

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every $\alpha\geq0$ and $\gamma>0$ and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive $\alpha$ and its benefit.

This note studies the geometry of full conformal prediction (FullCP) regions generated by an empirical energy-form pairwise score. Candidate-score convexity alone does not guarantee connected FullCP regions, even for empirical averages of losses convex in the candidate argument. For the energy-form score, each leave-one-out training comparison reduces exactly to a pairwise-dissimilarity sublevel condition. Under symmetry, a constant diagonal, a diagonal lower bound, and attainment of the associated Fréchet-type objective, all comparison regions share a minimizer; if they are convex, every nontrivial exact conformal region is star-shaped about that point. For power distances $\rho_\beta(x,y)=\|x-y\|^\beta$, this geometry holds for $\beta\ge1$, while the conventional energy score is strictly proper for $0<\beta<2$. For $d=1,\beta=1$, every nontrivial empirical-CRPS FullCP region is a nonempty closed interval (possibly $\mathbb R$ when $m=1$). For $1<\beta<2$ and $m\ge2$, explicit data-checkable derivative bounds give Lipschitz control of the radial exits and exact conformal radial function. Combined with directional root search and classical Lipschitz extensions, they yield certified inner and outer radial envelopes of width at most $\delta+2\widetilde Lh_{\mathcal U}$ and same-ray Hausdorff guarantees. An analytic two-dimensional example shows why preserving star-shaped but nonconvex geometry can matter. A staged two-dimensional study finds modest but systematic tightening of the generic certificate and frequent robust nonconvexity witnesses, with detected normalized radial departures typically small. The method is intended for low-dimensional multivariate outputs rather than high-dimensional scaling or runtime improvement.

Recent advances in artificial intelligence (AI) have sparked growing interest in its use for mathematical research. While some view this as a major opportunity for discovery, others have raised concerns about its impact on traditional research practices. Despite extensive debate, empirical evidence on how AI is actually being used in mathematics remains limited. To address this gap, we collected all 32,944 arXiv submissions posted between March 1 and August 20, 2026, whose primary or secondary categories included Mathematics. We identified 3,575 submissions that explicitly disclosed author use of AI, of which 1,712 involved at least one substantive mathematical contribution. Our analysis reveals several broad patterns. First, disclosed AI use increased sharply over the study period, with substantive use growing from 1.39% of Mathematics submissions in March to 14.09% through August 20. Second, substantive AI use is highly uneven across fields: Combinatorics has the largest number of such papers, while Metric Geometry has the highest substantive-use rate. Third, substantive AI use is geographically concentrated: under weighted author counts, the United States and China together account for about two-thirds of the recognized country weight. Fourth, AI is already being applied to open research problems: among 717 named open-problem records associated with substantive use, 71% are labeled as fully resolved based on the authors' descriptions, with proofs of the conjectured statement more common than counterexamples or disproofs. Finally, AI-system use is also highly concentrated, with OpenAI systems appearing most frequently, followed by Anthropic. Together, these findings suggest that AI-assisted mathematics is expanding rapidly but remains at an early and uneven stage of adoption.

Generative models are commonly ranked by Fréchet Inception Distance (FID) and Kernel Inception Distance (KID), yet FID's first-two-moment summary can miss distributional differences, and a reported scalar gap alone is not a calibrated test against sampling variation. FID's moment restriction has concrete consequences: on ImageNet, visually unrecognizable images optimized only to match the reference Inception mean and covariance obtain FID $24.7$ versus $58.6$ for held-out real images (lower is better). Moreover, FID and KID are scalar discrepancies that are unchanged when the two samples are exchanged and therefore do not encode the direction of a dispersion change: under-dispersion, as can occur in mode collapse, versus over-dispersion. We introduce ZID (Z-resolved Integrated Diagnostic), which combines six standardized location- and dispersion-sensitive arms from a rank graph (RISE) and Gaussian kernels (GPK at two bandwidths). Rather than asking one scalar to serve incompatible roles, ZID reports three linked outputs: an index for ranking departure magnitude, a permutation $p$-value for testing distributional equality, and a signed dispersion readout for diagnosis. In controlled experiments, ZID detects a broad range of departures, and its score tracks increasing severity along the corresponding sweeps, including cases in which FID is flat or reversed. On DiT-XL/2 and SiT-XL/2 guidance sweeps, ZID detects departure from real data, and its signed readout labels the high-guidance diversity collapse as under-dispersion.

Marginalized importance weighting evaluates a target policy by reweighting offline state-action samples with its discounted occupancy ratio, characterized by an adjoint Bellman equation. Existing minimax, primal-dual, and fitted fixed-point estimators can leave residual occupancy-balance violations because of function-class approximation, regularization, or incomplete optimization. These violations are difficult to diagnose and reduce because the objectives generally lack a direct supervised validation loss for hyperparameter tuning, model selection, and early stopping. We introduce isotonic Bellman calibration, a one-dimensional, model-agnostic post-processing method that reduces these violations while preserving the ranking information in any initial occupancy-ratio estimate. The method corrects the estimate's scale and shape by applying fitted occupancy-ratio evaluation (FORE) over a one-dimensional class of nondecreasing transformations. We characterize Bellman calibration as a conditional fixed-point property equivalent to occupancy-balance against every test function of the calibrated ratio. More generally, we derive a calibration-refinement bound showing that any fitted ratio with small calibration error performs nearly as well as the best post-processing based on its fitted values. For isotonic Bellman calibration, we establish finite-sample calibration guarantees and a KL oracle inequality relative to the best monotone transformation of the initial estimate. Consequently, isotonic Bellman calibration achieves small calibration error and KL risk within statistical error of the best monotone correction, with guarantees for downstream target-occupancy functionals, including policy-value estimation.

We study the fundamental problem of estimating the dependence structure of a centered stationary Gaussian process under local differential privacy (LDP). In this setting, the spectral density characterizes the dependence structure of the data and is the quantity to be estimated. Our main contribution is to close the open $\alpha^2$-versus-$\alpha^4$ gap between the previously known lower and upper bounds on the minimax rate. Specifically, we establish a minimax lower bound showing that, over Sobolev-type classes of spectral densities, the effective sample size in the high-privacy regime is $N\alpha^4$, rather than the usual $N\alpha^2$ arising for independent observations. This additional privacy cost is caused by the temporal dependence between the observations rather than by their marginal distributions. The proof relies on a contraction bound for privatized dependent Gaussian observations. Our second contribution is a matching upper bound, free of the polylogarithmic losses present in previous work. Rather than applying a generic privatization scheme to classical estimators, we construct a problem-specific procedure attaining the rate identified by our lower bound. Beyond closing the gaps in spectral density estimation, we apply the tools developed for this problem to several related questions. We (i) close the logarithmic gap for fixed-lag autocovariance estimation, (ii) show that the $\alpha^4$ cost arises locally around every spectral density bounded away from zero, and (iii) establish that classical asymptotic equivalence with an independent Gaussian experiment generally fails under LDP.

We uncover ELR collapse in language model pretraining: learning rate (LR) and parameter norm govern loss dynamics primarily through their ratio, the effective learning rate (ELR). When ELR is matched across runs, their loss trajectories collapse throughout training despite substantially different LRs and parameter norms. Across optimizers, architectures, datasets, and model scales, mean collapse errors are typically a few x 10^-3, below the seed-to-seed variation measured in a representative configuration. Systematic ablations identify normalization design and the timescale of LR-norm variation as key determinants of collapse precision. Controlled interventions further show that weight decay and Hyperball shape loss dynamics primarily through the ELR schedules they induce. Replacing LR with ELR enables a fitted functional scaling law (FSL) to transfer across norm-control methods. The resulting ELR-based FSL also explains delayed acceleration, a recurring effect of norm control. Together, these results establish ELR as a common coordinate linking LR scheduling, norm control, and loss dynamics.

In this work, we investigate rapid prediction of the dominant $n{=}0$ vertical instability growth rate in C-Mod and SPARC equilibria, where nonrigid free boundary response models are too slow for control cycle use. Using a Physics Attention Transformer trained on MEQ-FGE-L labels, we predict both the scalar growth rate and the associated two dimensional perturbed toroidal current density. We find mean absolute errors of 5.4~s$^{-1}$ on held out C-Mod equilibria and 12.7~s$^{-1}$ on synthetic SPARC cases, with spatial eigenfunction errors near 5%. We also compared PAT with operator based ML models : FNO2D and DeepONet, where we found PAT predicts a much lower normalised growth rate error and improved spatial reconstruction. These results indicate that PAT can reproduce MEQ-FGE-L outputs at control relevant latency and could support future studies of growth rate headroom monitoring and proximity aware shape control.

The Bayesian Ideal Observer (IO) establishes the theoretical upper bound on task performance for binary detection tasks. However, analytical computation of the IO test statistic is generally intractable. Numerical approaches based on Markov-chain Monte Carlo (MCMC) methods, including their recent deep generative model-based extensions, typically require extensive posterior sampling for each test image. Supervised learning has also been investigated to approximate the IO performance. However, such methods are typically trained for a specific detection task and signal and may require retraining when the task or signal changes. The score function, defined as the gradient of the log probability density, encodes the local geometry of the data distribution and is a fundamental quantity in modern score-based generative modeling. This work reformulates the IO test statistic in terms of the score function and introduces a score-based ideal observer (SIO). The proposed SIO uses a denoising convolutional neural network trained exclusively on signal-absent images to estimate the signal-absent score function. Once trained, the resulting score model can be used to approximate the IO test statistic for detection tasks involving arbitrary additive signals, without per-image posterior sampling or signal-specific retraining. Numerical studies consider a signal-known-exactly (SKE) detection task with a stochastic lumpy-background model. The results demonstrate that the proposed SIO can closely approximate the IO performance.

How can humans make sense of the rapid takeoff of artificial intelligence (AI)? We studied the sensemaking dynamics of AI through an open-ended, mixed-methods study with computational text analysis of millions of AI-related newspaper articles and social media posts grounded in 57 semi-structured interviews with AI professionals in 2021 and 2023–before and after the recent surge of public interest. We identify a range of sociological frames (interpretive schemas that structure collective cognition) and show how AI professionals use frames to address significant cognitive challenges, such as assigning responsibility for societal impacts. We develop a framework of three primary debates across which frames are adopted and contested: (i) the $\textit{method}$ of AI development, between frames of top-down expert systems and bottom-up emergent capabilities, (ii) the $\textit{mind}$ of an AI system, ranging from a passive tool to a humanlike "digital mind," and (iii) the $\textit{morality}$ of how AI is used, particularly the decision of whether to slow down or speed up AI development. As humanity enters the era of transformative AI, technologists and policymakers must account for the framing dynamics that will circumscribe our beliefs, values, and actions.

We settle the minimax-optimal alternating regret, a regret notion motivated by alternating learning dynamics in games, for both online linear optimization (OLO) and online convex optimization (OCO). For OLO over the probability simplex $\Delta_d$, we give an algorithm with $O(\log d)$ alternating regret that remains a constant for any time horizon $T$, and a matching lower bound. Our constant regret bound significantly improves previous results with $O(\log ^{2/3}d \cdot T^{1/3})$ regret [Cevher, Cutkosky, Kavis, Piliouras, Skoulakis, Viano, NeurIPS 2023, Hait, Li, Luo, Zhang, COLT 2025]. As a result, we obtain alternating learning dynamics with $O(\log d /T)$ convergence to Nash equilibria in two-player zero-sum games and $O(\log d /T)$ convergence to coarse correlated equilibria in two-player general-sum games. This is the first uncoupled learning dynamics with $O(1/T)$ convergence to CCE in two-player general-sum games, while all prior works suffer additional $\log T$ factors. For general OCO over a $d$-dimensional compact convex set, we give an algorithm with $O(d\log (1+T/d))$ alternating regret, improving the previous best of $\widetilde{O}(d^{2/3}T^{1/3})$. We also prove a matching lower bound of $\Omega(d\log (1+T/d))$, showing that the $\Omega(\log T)$ factor is unavoidable.

Retinal fundus photography is widely used for screening and monitoring ocular diseases, but many modern classification pipelines rely on deep latent representations and provide limited interpretability. This study develops an interpretable fundus image classification framework based on a ring-structured representation of the retinal vasculature centered on the optic disc. The method quantifies vessel geometry, color appearance, oxygenation-related vascular appearance, and vessel–background entropy within concentric retinal regions. These physiologically motivated descriptors are derived from vessel masks, image intensities, and optical-density measurements and aggregated across rings to capture spatial variation in vascular properties. Using only quantitative vascular descriptors, the proposed method achieved strong classification performance across three public fundus datasets. On HRF, it achieved 91.1% accuracy using automatically generated vessel masks, matching RETFound, a vision transformer pretrained on large-scale retinal fundus image data, under the same evaluation setting. Additional analyses suggest that pretrained image models are sensitive to acquisition-related spatial cues, including fundus scale and retinal position within the field of view, as well as broader non-vessel image characteristics. This framework may support interpretable disease classification, quantitative retinal phenotyping, and retinal biomarker discovery without requiring large task-specific training datasets.

Probabilistic models of Directed Acyclic Graphs (DAGs) with latent variables impose equality constraints on the observed data distribution beyond ordinary conditional independencies. These so-called Verma constraints arise in nested Markov models associated with Acyclic Directed Mixed Graphs, the latent projection of latent-variable DAGs. While nested Markov models have been extensively studied from the perspectives of graphical representation and causal identification, their implications for semiparametric efficiency theory remain less understood. We develop results toward establishing a semiparametric framework for statistical models defined by Verma constraints. Our key observation is that nested Markov constraints admit weighted conditional-moment representations under post-fixing distributions induced by graphical fixing operations. We show that fixing induces weighted orthogonality relations in L2(P), thereby converting Verma constraints into explicit tangent-space restrictions. Building on this representation, we characterize the tangent-space orthocomplement for models defined by a single nested Markov constraint through residualized weighted moment functions. This geometric formulation yields Hilbert-space characterizations of semiparametric efficient influence functions and efficiency bounds via orthogonal projection and equivalent minimum-variance formulations. We further discuss extensions to models involving multiple nested Markov constraints, for which we characterize a subspace of the orthocomplement as sums of the corresponding weighted orthogonality relations, while leaving the complete tangent-space characterization open. More broadly, our results connect nested graphical structure with semiparametric Hilbert-space geometry and provide a foundation for a general efficiency theory for nested Markov models. We illustrate the framework through several latent-variable DAGs.

The embedding dimension of categorical predictors is usually selected through heuristic tuning, although it directly affects model complexity, approximation quality, and finite-sample generalization. This paper formulates embedding dimension selection as a constrained allocation problem. The main contribution is to show that embedding capacity can be allocated across heterogeneous categorical predictors according to an explicit approximation-estimation tradeoff. We characterize approximation error through the singular-value tail of the latent category representation, while estimation error increases with total embedding complexity. Under a fixed global embedding budget and a tractable approximation model, this leads to a closed-form allocation rule in which the dimension assigned to each predictor is proportional to the square root of its approximation value relative to its parameter cost. Simulation experiments support the proposed approximation-estimation interpretation and show that the allocation rule improves budget efficiency relative to standard uniform and cardinality-based heuristics, particularly when the budget is binding and predictor heterogeneity is substantial. A real-data healthcare application further shows improvements in predictive accuracy and probabilistic calibration. Overall, the results establish embedding-dimension allocation as a principled finite-sample optimization problem rather than a purely heuristic modeling choice.

Credit risk models increasingly need to combine predictive accuracy with transparent explanations and auditable fairness constraints. Logistic regression remains attractive because its coefficients are easy to interpret, but it can miss nonlinear structure. Flexible models can improve prediction, but their explanations are often post-hoc and may not describe the decision rule itself. We introduce $\texttt{findr}$, short for flexible, interpretable deep regression, a semi-structured framework for binary credit risk modelling that decomposes the logit into an interpretable structured component and an orthogonal neural residual. The orthogonalisation separates coefficient-based effects from residual nonlinear variation, while an in-processing Wasserstein penalty mitigates group disparities by comparing score distributions during training. The framework also includes diagnostics that measure the structured component's contribution to logit variation, decision agreement, and local directional consistency. We evaluate $\texttt{findr}$ in a simulation study and on eight public credit datasets using score-level accuracy-fairness frontiers. The results show that $\texttt{findr}$ behaves close to logistic regression when the signal is approximately linear, while recovering much of the predictive gain of neural models when nonlinear structure is relevant. The diagnostics identify when coefficient-based explanations remain close to the full fitted model and when residual variation must also be examined. These findings support semi-structured modelling as a practical way to make performance, fairness, and interpretability trade-offs explicit in credit risk decisions.

The STAR (Student-Teacher Achievement Ratio) experiment (1985, Tennessee, USA) is a landmark hierarchical dataset designed to assess the impact of class size on student outcomes, with observations nested within classes. To encode class-level interventions in such hierarchical settings, we develop a complete, scalable, open-source pipeline for Hierarchical Structural Causal Models (HSCM) that bridges symbolic identification and practical estimation. Our approach integrates graph transformations, pyAgrum's do-calculus for automatic identification of causal effects, adaptation of symbolic expression into closed-form HSCM formulas, and numerical estimation from fitted local probability models. A key innovation is our adapted Abstract Syntax Tree (AST), which decomposes pyAgrum's identified formulas into independent density, expectation, and marginalization tasks, enabling parallel and scalable computation. We validate the pipeline on canonical HSCM motifs and benchmark scenarios with known ground truth, then apply it to STAR kindergarten mathematics outcomes. The results show that flat baselines (ignoring hierarchy) recover associations but fail to encode class-level interventions, and that symbolic identification alone is not enough for practical Hierarchical Structural Causal inference; scalable estimation and numerical stability checks are central parts of the scientific object.

We introduce a novel latent space network model for analyzing multivariate time series of neural spike-train data. The methodology is motivated by an experimental study in mice, where neuronal responses were collected under a sequence of visual discrimination tasks. We adopt a latent variable framework to model the firing rates of aggregated brain areas, while simultaneously inferring the interactions between regions via a hidden network structure. This interaction network is embedded in a geometric latent space, enabling interpretable visualizations and novel model-based summaries. The proposed framework provides an intuitive interpretation of the latent variables, which bear a conceptual connection to node eigen-centrality measures. To capture temporal dependence, we incorporate a nested hidden Markov structure that can flexibly represent non-linear shifts that are induced by the changing of experimental conditions. We further establish theoretical properties of the model by deriving sufficient conditions that prevent degeneracy, thereby guiding our model assumptions. Overall, the proposed methodology provides a unified framework to characterize brain activity, its temporal dynamics, and spillover effects through a hidden latent space network model.