Papers

This paper compares a series of contemporary portfolio construction approaches by employing ten U.S. stocks (TSLA, WMT, BAC, GS, LLY, MRK, GOOG, META, AAPL and XOM) in a time frame from September 2023 to December 2025. The paper explores both basic mean-variance optimization, constrained optimization, Fama French five factor regression modeling, Monte Carlo simulation, and the Black-Litterman model to determine how constraints to a solution, risk factors to a strategy, simulated approximations, and specific market views may all impact the outcome of portfolio allocation, performance and stability. Overall, the results show that standard optimization may result in highly concentrated portfolios, while constrained optimization leads to changes in portfolio allocations by altering the efficient frontier, five factor regression models suggest that a basic investment style of defensive large value and profitability exposure, Monte Carlo approximation is a viable technique to arrive at mean-variance optimal portfolios provided the simulations are high enough especially under a box constraint, the Black Litterman portfolio approach produces more economically intuitive allocations and greater stability compared to standard mean-variance optimization as the approach balances equilibrium returns with investor views.

Under standard graphical assumptions, the Markov boundary of a target variable is the smallest set of features that renders every other feature redundant. Once the boundary is observed, the target is conditionally independent of the rest of the table. This is a tempting object for tabular prediction, since it names exactly the columns a model should need. Yet modern regressors are still trained on the full feature set. We ask whether the Markov boundary is genuinely useful for prediction on SCM3K, a 3,450-task synthetic SCM benchmark with feature counts from 40 to 1000 and six SCM families, evaluated with six regressors. The answer is more nuanced than the theory suggests. Restricting a regressor to the oracle boundary often improves prediction substantially, and the improvement grows as the feature space becomes larger and sparser. But the natural pipeline of recovering the boundary with causal discovery and training on the recovered mask does not deliver. Existing estimators exhaust the compute budget before reaching the regime where the boundary helps most, and even where they run they rarely beat the full feature set. We trace this to three causes. Discovery optimizes structural recovery rather than prediction. False negatives and false positives carry sharply asymmetric predictive cost. The exact boundary is only one of many feature sets that beat all features. We then develop what these facts imply for prediction-aligned feature selection and for tabular models that learn to use causal structure.

Longitudinal studies frequently incorporate covariates that evolve over time, creating complex dependence structures between outcomes and predictors. When covariates are time dependent, standard power analysis tools–largely developed for generalized estimating equations (GEE)–can yield misleading results because they do not account for the moment based structure required for valid marginal inference. Generalized Method of Moments (GMM) provides a flexible and efficient framework for estimating marginal effects in the presence of time dependent covariates, yet no practical tools exist for conducting power analysis under GMM. This paper introduces a modern, implementable framework for power estimation in longitudinal studies with time dependent covariates using GMM. Two complementary approaches are developed: a Wald based method that leverages the asymptotic normality of GMM estimators, and a distance metric method based on quadratic forms of sample and population moment conditions. Both approaches require only limited distributional assumptions and rely on valid moment conditions rather than full likelihood specification. We outline the theoretical foundations, provide step by step implementation guidance, and illustrate the methods using data from the Osteoarthritis Initiative. A simulation framework is presented for evaluating empirical performance. These methods fill a critical gap in the longitudinal modeling literature by offering applied researchers a practical, distribution light approach to power estimation when time dependent covariates are present and GMM is the preferred estimation technique.

Pairwise human-preference platforms such as Chatbot Arena have become central to large language model (LLM) evaluation, yet reliable task-specific ranking remains challenging. Global leaderboards mask task heterogeneity, while ranking each fine-grained task independently is unstable under sparse, imbalanced comparisons. We propose a low-rank framework for task-specific LLM ranking from sparse pairwise comparisons, modeling the task-by-model ability matrix $\Theta^\star \in \mathbb{R}^{d_t \times d_m}$ as low rank so that information is shared across related tasks while task-specific differences are preserved. We first develop a max-norm ($\ell_\infty$) accurate estimator for the latent scores, combining a convex initializer with alternating-minimization refinement, and prove task-wise top-$K$ recovery guarantees under sparse sampling. Our main contribution is an uncertainty quantification framework for task-specific ranking. We construct cross-fitted one-step debiased estimators for fixed score contrasts – such as the task-specific ability gap between two models – yielding asymptotically valid confidence intervals that attain the semiparametric efficiency bound. We then extend the inference to the high-dimensional ranking regime, where per-task ranks and top-$K$ membership are determined by many dependent score-gap hypotheses. Using Gaussian and multiplier-bootstrap calibration, we obtain simultaneous confidence sets for per-task ranks and valid top-$K$ membership tests across many tasks and models. Experiments on synthetic data and Chatbot Arena show that low-rank sharing improves sample efficiency over independent task-wise Bradley-Terry estimation and produces tighter, better-calibrated ranking certificates, with the largest gains in the sparse regime typical of real LLM benchmarks.

This paper develops a framework for differentially private $e$-values under Gaussian differential privacy ($\mu$-GDP). We characterize the canonical noise mechanism, establishing that optimal multiplicative perturbation follows a Gaussian distribution. Using this distribution, we derive a globally sharp rejection threshold that strictly improves upon the standard Markov bound. Asymptotic analysis shows that in low-sensitivity regimes, the calibrated private test achieves a net power gain over the non-private baseline. For multiple testing, we introduce a recursive peeling algorithm that adaptively concentrates the privacy budget on the most promising hypotheses. This construction guarantees rigorous $\mu$-GDP and yields valid private $e$-values compatible with standard multiple testing procedures. Simulations and a genome-wide association study confirm that the method controls the false discovery rate while improving upon naive all-noisy privatization and recovering power close to non-private benchmarks.

The scaling exponent $\alpha$ in neural scaling laws $L(N) \propto N^{-\alpha}$ is commonly treated as a fixed constant set by architecture and data. We present evidence that $\alpha$ depends systematically on the optimizer. In controlled random-feature regression experiments – the canonical theoretical framework for neural scaling – we measure $\alpha$ across five optimizer variants and six spectral conditions. Preconditioned optimizers consistently yield steeper scaling (larger $\alpha$), with the $\alpha$-shift increasing across most of the tested spectral range, peaking near $s = 1.5$, and remaining large at $s = 2.0$. At $s \approx 1.0$ (characteristic of natural language), the full natural gradient achieves $\alpha \approx 0.31$ versus $\alpha \approx 0.12$ for gradient descent – a $2.6\times$ larger fitted exponent that, within the random-feature model, compounds with each model-size doubling. Whether and how this exponent shift transfers to large-scale LLM training – where recent evidence suggests the advantage may attenuate with scale – remains an important open question. Our results imply that scaling-law forecasts should account for optimizer choice, and we provide a spectral diagnostic predicting when advanced optimizers will pay off.

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For population near-minimizers we prove convergence to the terminal-constrained problem as the penalty diverges, and show that the same limit is recovered almost surely when the learned controls are evaluated on independent finite samples under explicit coupling conditions on the penalty, random-feature count and sample size. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for a class of well-behaved priors, where the empirical estimator converges to the Bayes-optimal predictor under asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.

We consider continuous time to treatment initiation. This can commonly occur in preventive medicine, such as disease screening and vaccination; it can also occur with non-fatal health conditions such as HIV infection without the onset of AIDS. While traditional causal inference focused on `when to treat' and its effects, we consider the incremental causal effect when the intensity of time to treatment initiation is intervened upon. We derive the efficient influence function for this estimand and develop an estimation framework that accommodates flexible machine learning methods while achieving fast convergence rates. Valid confidence bands are obtained leveraging empirical process theory. We illustrate our approach via simulation, and apply it to cervical cancer screening data to study the incremental effect of time to subsequent HPV testing on cervical intraepithelial neoplasia detection.

Residual-based goodness-of-fit tests for parametric time-series models are often complicated by parameter-estimation effects, which can alter the limiting behavior of diagnostic statistics. We propose a sample-splitting generalized spectral test (in the spirit of Escanciano(2006)) for assessing conditional mean specification in linear and nonlinear time-series models. The procedure estimates the model parameter on a fitting subsample and constructs a generalized spectral Cramer-von Mises statistic from residuals computed on a checking/testing subsample. The statistic aggregates pairwise conditional mean restrictions over all lags and is therefore bandwidth-free and free of truncation-lag selection. Under mild regularity conditions and a score-alignment condition, the residual-based process has the same limiting null distribution as the infeasible oracle process based on the true errors. Although the resulting limiting law is still non-pivotal, it can be consistently approximated by a simple multiplier bootstrap that does not require generating bootstrap time series or re-estimating parameters. Such an oracle-equivalence property is in sharp contrast to the original full-sample test, for which parameter estimation contributes an additional first-order term to the limiting process, and requires re-estimating parameters in each bootstrapped sample. We further establish consistency of the proposed test against fixed alternatives and nontrivial power against local alternatives. Extensive simulations and real data analyses show that the proposed test controls size well, has comparable power, and delivers substantial computational savings in models where repeated estimation is costly.

In demographic literature, forecast uncertainty is often quantified with a statistical model. This model-based approach may potentially suffer from drawbacks, namely model misspecification, selection effect, and lack of finite-sample validity. We introduce a model-agnostic and distribution-free procedure, conformal prediction, for constructing prediction intervals for a functional time series. In the family of conformal prediction, split conformal prediction divides the data into training, validation, and test sets. Within the validation set, we can select optimal tuning parameters by calibrating the empirical coverage probabilities to match their nominal values. With the selected optimal tuning parameters, we then construct the prediction intervals using the same forecasting model for the holdout data in the testing set. Without sample splitting, sequential conformal prediction sequentially updates the predicted quantiles via an autoregressive process. Using Australian age- and sex-specific log mortality rates, we evaluate and compare the interval forecast accuracy, as measured by empirical coverage probability, coverage probability difference and mean interval score, between the two variants of conformal prediction.

Exact Kriging and conditional simulation (CS) for uncertainty quantification are computationally infeasible for modern spatial analyses with large numbers of observations and dense prediction grids. We present a rapid approximation to the Kriging prediction step for stationary Gaussian processes for a regular prediction grid by approximating each off-grid covariance vector by a sparse linear combination of on-grid covariances within a local $L$-order neighborhood of $M = (2L)^2$ neighboring grid points. This reformulation reduces complexity from $O(N n^3)$ to $O(N \log N + nM + M^3)$ while preserving accuracy. A factorial study shows that approximation error decreases systematically with increased Matérn smoothness, neighbor order $L$, and grid resolution, aligning with bounds from kernel approximation theory. In a North American summer-rainfall application ($n=1368$), our method produces predictions visually indistinguishable from exact Kriging with point-wise errors on the order of $10^{-5}$ inches and achieves more than $150$ times speedups at a $350\times350$ grid, also outperforming Vecchia and LatticeKrig predictions. Embedded in a fast CS scheme, the approach reproduces Kriging standard errors and scales favorably with both $n$ and $N$. We recommend a practical workflow that uses a fast method for parameter estimation followed by our rapid predictor for fine-grid mapping and uncertainty quantification.

Fraud detection models in payment networks train on chargeback labels that are systematically biased. Every label must survive three sequential gates: authorization (declined transactions generate no labels), issuer reporting (unreported fraud is invisible), and delay (pending chargebacks are missing at training time). Labels that do arrive may be corrupted by first-party misuse or issuer misclassification. A companion paper [arXiv:2605.27557] proved that these four impairments impose a minimax lower bound on detection performance. This paper asks: can that bound be achieved? We formalize the observation pipeline as a sequential missing-data problem with three propensity stages and a corruption layer, and construct the Sequential Triply Robust (STR) estimator. The STR corrects for all four impairments simultaneously and achieves the semiparametric efficiency bound – no estimator can have lower asymptotic variance. It is sequentially triply robust: at each gate, consistency requires only that either the propensity model or the outcome regression is correctly specified, not both. We provide corruption correction via noise-rate-adjusted pseudo-labels, empirical Bayes shrinkage to stabilize inverse-propensity weights for small issuers, a plug-in variance estimator yielding valid confidence intervals, and a Bernstein concentration inequality for finite-sample guarantees. On the operational side, we derive the optimal training delay – the maturity window that minimizes the sum of label-quality loss and model staleness – and prove that the STR permits training on data that is days old rather than months old, decoupling model freshness from the chargeback maturity cycle. The STR provably dominates naive chargeback-based training in mean squared error for any sample size.

Regression is a fundamental tool in scientific research. Ordinary least squares (OLS), one of the most widely used regression methods, enjoys several desirable properties, including the best linear unbiased estimator (BLUE) property. It is well known that, under the assumptions of the standard model, the OLS is conditionally unbiased given the covariates, i.e., $\mathbb{E}(\widehat Y-Y\mid X=x)=0$. However, an often-overlooked property of OLS is that the prediction error is generally not unbiased conditional on the outcome, i.e., $\mathbb{E}(\widehat Y-Y\mid Y=y)\neq 0$. As a consequence of minimizing mean squared error, OLS predictions are systematically shrunk toward the outcome mean, which explains the classical phenomenon of regression to the mean (RTM): large outcome values tend to be underpredicted, whereas small outcome values tend to be overpredicted. This conditional prediction bias creates a nonignorable problem for predicted outcome-based inference, where scientific inference is performed using the predicted outcome $\widehat Y$ and another variable $W$. In applications such as brain-age analysis and causal inference, we show that inference based on regression-predicted outcomes can be systematically biased. To address this issue, we propose outcome-calibrated regression (OCR), a new regression framework with a closed-form solution that directly enforces outcome calibration. The proposed OCR estimator eliminates conditional prediction bias with respect to the outcome and enables valid inference using regression-predicted outcomes.

Many applications require statistically valid inference across many related tasks, while using only a handful of high-quality labels per hypothesis. In AI evaluation, these tasks may correspond to model behaviors across prompts, subgroups, or hypotheses; in social science surveys, they may correspond to related questions, populations, or measurement conditions. Prediction-powered inference (PPI) uses abundant but inexpensive proxy measurements to improve inference from limited, ground-truth labels, but commonly used methods treat tasks independently and therefore fail to exploit shared structure across related tasks. This limitation is especially important in settings where only a small number of labels are available per task. To address this issue, we introduce a multi-task prediction-powered inference framework that uses labeled data from related tasks to improve power while preserving task-specific inference. Our methods exploit the shared structure in the proxy-ground-truth relationship through cross-task recalibration, while retaining within-task rectification and power tuning to construct accurate point estimates and confidence intervals. We prove that efficiency gains beyond power-tuned PPI are only possible when the proxy-ground-truth relationship contains nonlinear structure; affine cross-task recalibrations are asymptotically equivalent to using the original proxy. We complement our theoretical findings with experiments on synthetic and semi-synthetic datasets, as well as a case study auditing language models on election-related information during the 2024 U.S. presidential election. Using a large human-annotation study, we show that cross-task recalibration can substantially reduce confidence interval widths when labels are scarce.

Besides the classical motivation of fusing evidence from multiple sources, modern inferential procedures based on randomization, resampling, and data splitting often introduce analyst-generated multiplicity, where aggregating outputs across random realizations can improve robustness and stability. This emphasizes the importance of developing principled strategies for fusing measures of evidence across different inferential settings, while preserving the key properties of the adopted inferential framework. The present paper addresses this problem in the context of inferential models (IMs), a possibilistic approach for provably valid statistical inference. Although the fusion of possibility measures has been extensively studied in the possibility-theory literature, existing methods do not, in general, preserve IM validity. We propose a general validity-preserving framework for possibilistic fusion, motivated by the ranking–validification construction underlying IMs. We study the implementation of this framework under independence, arbitrary dependence, and exchangeability of the available IMs, thereby providing a unified approach for IM fusion across a broad range of practically relevant scenarios. The proposed framework also reveals important efficiency considerations, showing that intuitive and commonly used fusion operators may become inefficient in the IM context, so that alternative choices can sometimes be advantageous, including ones that might not appear natural from a purely intuitive standpoint.

Conformal prediction is a framework for providing prediction intervals with distribution-free validity, guaranteeing predictive coverage for data drawn from any distribution. Its two main variants are full conformal prediction and split conformal prediction (also called transductive and inductive). Full conformal prediction is widely considered to be statistically more efficient (since split conformal prediction requires data splitting, and therefore can lead to wider prediction intervals due to the resulting loss in sample size), but its implementation is computationally prohibitive, as it requires the underlying model to be refit for every candidate value in the response space. Existing computational shortcuts, such as using a discrete grid of values to approximate the full conformal prediction construction, frequently lack theoretical guarantees on marginal coverage and can fail in practice. To address this limitation, we introduce a novel class of approximations to the full conformal prediction method, based on the idea of tournaments, which enables the construction of prediction sets with a rigorous marginal coverage guarantee of $1-2\alpha$. Under stability conditions, the theoretical coverage guarantee tightens to approximately $1-\alpha$. This new framework generalizes the existing method of leave-one-out cross-conformal prediction, while allowing for flexible use of various existing approximation strategies.

Marine corrosion significantly reduces a ship's availability, increases costs of operation and could impact safety. Protective coatings mitigate these risks, but their effectiveness deteriorates over time. Early detection of coating breakdown is crucial to prevent costly repairs and safety concerns. While corrosion itself is well-understood, coating degradation remains under-investigated due to insufficient long-term data. This work addresses this knowledge gap by enhancing coating defect prediction and optimizing inspection planning for ships. The Power Law Non-Homogeneous Poisson Process (PL-NHPP) is utilized for modeling coating defect arrivals. Unlike prior studies, we employ a hierarchical Bayesian approach for parameter fitting, effectively addressing limitations associated with scarce real-world data. Furthermore, we optimize inspection planning by incorporating out-of-service costs and potential costs increases due to delayed repairs. The efficacy of these methods is evaluated through a comprehensive case study involving a recently commissioned fleet with limited historical data. This research contributes to the advancement of condition-based maintenance (CBM) strategies for ships by enabling more accurate prediction of coating breakdowns and optimizing inspection schedules early in the life of the fleet. This approach ultimately improves operational efficiency and reduces life-cycle costs.

Storage tanks for hazardous liquids are common in industry and agriculture. During a pollution incident, liquid may drain from a storage tank through a small hole, crack, or pipe. After containing the leak, estimating the discharged volume of liquid is essential for public safety, regulatory assessment, and remediation. When the original inventory of liquid is unknown, this constitutes an inverse problem. In this work, we present a framework for inferring the initial liquid level in a partially drained tank from the observed final liquid level after a pollution incident and an estimate of the drainage duration. Because the drainage dynamics, model parameters, and observations are uncertain, we employ Bayesian statistical inversion to combine prior physical knowledge with experimental liquid level time series data to predict the initial liquid level with quantified uncertainty. We use a physics-based model based on Torricelli's law to describe the tank-draining dynamics and augment it with an empirical discrepancy function to account for missing or imperfectly modeled physics. In our experiments with a tank draining of water, we found that our inferred initial liquid level was accurate, although uncertainty increased with drainage duration. Beyond its application to pollution forensics, this work may also serve as a hands-on classroom project illustrating dynamic modeling, model discrepancy, and Bayesian inference.

Bridging the gap between visual realism and physical understanding is a core challenge for video-based world models. We study the structural identifiability of continuous-time physical laws from raw pixels, focusing on whether an encoder-only pipeline can uniquely recover the parameters of second-order linear ODEs. We prove that a level-set slope-coverage condition ensures the learned latent space is locally affine to the true physical state, enabling exact parameter recovery. Our theory provides the first characterization of minimal data requirements across damping regimes, establishing that underdamped systems are identifiable from a single video clip, whereas other regimes require three diverse trajectories. We further introduce a variance-floor regularizer to stabilize the decoder-free objective and prevent latent collapse. Validated on synthetic and real-world data, our approach demonstrates that interpretable physical constants can be reliably estimated from video without the need for compute-intensive pixel reconstruction, ensuring both physical correctness and transparency. Code is available at https://github.com/wenjiewang3/PhysicsFromVideo.

We develop a natural Bayesian multiplicity-correcting prior distribution within the probabilistic forward stepwise representation of model space priors for regression problems. The proposed prior, obtained from making an analogy to the Holm procedure, exhibits behavior closely aligned with that of the Matryoshka doll prior. We compare both priors to several other priors, including some recently put forward as objective choices for model space prior probabilities. Our comparisons indicate that adequate multiplicity correction requires a degree of sparsity that many recommended priors do not provide, and we argue that multiplicity correction itself offers a principled and transparent criterion for specifying model space priors in regression.

Response times collected in computerised assessments provide information about the underlying response process and may exhibit within-person variation over the course of a test. We propose a latent variable model for log response times that incorporates individual-specific change-points. The model extends the log-normal response time model by allowing an item-specific shift in the mean structure after an unobserved change-point. The change-point is treated as a discrete latent variable, and its distribution is modeled as a function of latent speed. Estimation is carried out using marginal maximum likelihood. The framework yields posterior distributions for change-point locations, allowing uncertainty to be quantified at the individual level, and supports statistical inference for the change-point effect parameters. A simulation study examines parameter recovery and change-point estimation under varying boundary conditions, prevalence of changers, sample sizes, and test lengths. The results show accurate recovery of item and structural parameters. The proposed model provides a unified approach to modeling response times with within-person changes in behaviour.

Spatial individual-level models (ILMs) provide a flexible framework for modeling infectious disease transmission across populations with known locations. Bayesian inference for these models relies on Markov chain Monte Carlo (MCMC), which requires repeated likelihood evaluation and, when parts of the epidemic trajectory are unobserved, data-augmented sampling over high-dimensional latent variables. This computational cost limits the applicability of MCMC to large populations and to settings requiring inference across multiple outbreaks. We propose using neural posterior estimation (NPE) for amortized Bayesian inference in spatial ILMs. NPE trains a conditional normalizing flow on simulated data to approximate the posterior directly, bypassing likelihood evaluation at inference time. We compare two embedding architectures: a convolutional neural network (CNN) operating on the population-level incidence curve and a graph neural network (GNN) operating on individual-level infection and location data. In a simulation study under full observation, stochastic removals, and partial observation, both variants achieve empirical coverage close to the nominal level, with the GNN embedding yielding lower error and narrower credible intervals for the spatial transmission parameters. Data-augmented MCMC is the most accurate when its chains converge, and GNN-NPE is closer to MCMC than CNN-NPE at a small fraction of the per-epidemic cost of MCMC. We apply the framework to the observed 2001 UK foot-and-mouth disease outbreak on 1,177 farms, where MCMC with the exact likelihood provides a reference posterior. Both NPE variants recover the reproduction number estimated by MCMC. GNN-NPE gives a narrower posterior for the spatial decay of transmission than CNN-NPE but is shifted relative to the MCMC reference.

Randomly initialized neural networks induce a prior over functions, but the predictor used in practice is produced only after training. We ask how much of this initial bias survives the training pipeline. To make the question measurable, we introduce initialization memory: the dependence of the validation-selected predictor on the scale of the random initialization. We perform controlled CIFAR-10 experiments on ResNets where initialization memory already sharply separates training regimes. Low-learning-rate SGD can interpolate while still remembering its initialization: on ResNet-9 with batch size $b=128$, test accuracy varies by $26.5$ percentage points across initialization scales despite $\ge99.5\%$ training accuracy. This is not undertraining: extending the same low-learning-rate regime to $5{,}000$ epochs leaves the spread essentially unchanged. In contrast, Adam-family methods largely erase the dependence. SGD can also be made to forget when larger learning rates are paired with explicit $L_2$ norm control. We interpret these findings in terms of the time scale of forgetting: gradient-flow-like dynamics can preserve initialization memory, whereas stochastic finite-step effects, explicit norm decay, and adaptive preconditioning erase it on scales governed by the size of explicit or implicit regularization. The practical inductive bias of a trained network is therefore not the architectural prior alone, but the architectural prior after being filtered by the forgetting dynamics of the training pipeline; and the same regularizers that improve generalization are precisely those that erase memory of initialization.

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 $\Delta_{\min}$, the known lower bound is of order $ \frac{\log K}{\min\{\Delta_{\min},\varepsilon\}}, $ or equivalently, up to universal constants, of order \[ \frac{\log K}{\Delta_{\min}}+\frac{\log K}{\varepsilon}. \] We give a horizon-free pure-DP algorithm and prove the explicit regret bound \[ \operatorname{Reg}_T \le 1000 \cdot \left(\frac{\log K}{\Delta_{\min}}+\frac{\log K}{\varepsilon}\right) \] 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 $\log K/\varepsilon$.

Question-answering services built on retrieval-augmented generation (RAG), in which a language model answers from retrieved documents, are inspected continuously and upgraded repeatedly, so their reliability guarantee must survive both. We study federated conformal RAG: nodes holding private corpora score candidate answers with a shared language model and send compressed scores to a hub that returns an answer set; a miss omits the true answer. We formulate monitoring as a sequential test: alarm when misses exceed a certified bound on the miss rate (the allowance). But a conformal certificate covers one frozen configuration at one look fixed in advance, so it gives no anytime-valid alarm; a threshold calibrated for one model certifies nothing about its replacement; and re-certifying each upgrade at full error level compounds failures. We propose Anytime-FC-RAG with three auditable rules: pick each configuration before its fresh calibration, charge every certificate attempt to one trajectory-wide budget $\delta_{\rm cal}$, and fix threshold, allowance and bet before each query. One betting wealth $E_t$, growing in expectation only when misses exceed the allowance, never resets across deployments. Our certified adaptive-refresh theorem shows that, with evidence budget $\delta_e$, alarming when $E_t$ reaches $1/\delta_e$ has false-alarm probability at most $\delta_e + \delta_{\rm cal}$ under a random number of history-selected, evidence-driven refreshes of model, retriever, corpus, score or threshold; $\delta_{\rm cal}$ pays for wrong certificates. With an exact order-statistic certificate the bound is distribution-free, architecture-agnostic and about misses, not per-input coverage or distribution shift as such. On Qwen2.5/MMLU-Pro the alarm separated audited-harmful from benign shifts completely at a 5% budget; in a model swap, only fresh calibration kept a downgrade's miss rate in check.

Generalized additive index models (GAIMs) offer a flexible semiparametric framework for capturing complex data relationships, balancing the interpretability of parametric models with the flexibility of nonparametric approaches. However, classical stage-wise estimation procedures for GAIMs suffer from computational inefficiencies due to their sequential nature and reliance on nonparametric smoothing. To overcome these drawbacks, we propose efficient, simultaneous estimation algorithms for GAIMs. By leveraging basis expansion, we cast the semiparametric estimation task as a finite-dimensional optimization problem solvable by first-order methods such as gradient descent (GD). Furthermore, we introduce a variational inequality (VI) estimation algorithm, extending the VI framework from generalized linear models to GAIMs. We provide a unified convergence result to a stationary point for both algorithms. Numerical experiments highlight the computational and statistical advantages of our methods over classical stage-wise procedures, and reveal the potential benefits of the VI-based approach over GD for non-canonical link functions.

When surveillance data of infectious disease incidence (e.g. weekly case counts) are disaggregated by demographic indicators, disparities in long-run health outcomes between these groups become apparent. Accurate identification of high-risk subpopulations would enable policy-makers to target interventions early in an epidemic; but, temporal models of disease incidence typically lack robust treatment of multivariate (i.e. subpopulation-level) outcomes. We propose a novel Bayesian latent-variable extension of the endemic-epidemic (“EE”) modeling framework commonly used for this purpose. Specifically, we augment the EE model class with explicit representation of unobserved individual-level transmissibility; explicit separation of disease incidence and prevalence; and parametric estimation of between-demographic-groups mixing structure. The resulting model may be tailored for either rare-disease (highly-endemic) contexts or outbreak-driven (highly-epidemic) contexts, and is capable of inferring social contact mixing patterns from incidence data alone, including mixing patterns among multiply-stratified data. To demonstrate, we conduct a simulation study comparing our model to an existing doubly-stratified EE model in the intended rare-disease application regime. We then compare our inference to the competitor's for real incidence data of norovirus gastroenteritis in Berlin, 2011-2015, disaggregated by six age groups and twelve geographic regions. Finally, we report inference of our model on COVID-19 incidence recorded in Michigan during the first year of the pandemic, disaggregated by six age groups and sixty-six geographic regions.

We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is separated from zero, it implies that the density of the maximum is uniformly controlled at all quantiles above ${2}/{3}$, which is sufficient for many hypothesis testing applications; it yields validity of Gaussian and bootstrap approximations for maxima of high-dimensional sums at test levels $\alpha < {1}/{3}$ without further restricting the covariance. Under these same conditions, the argument is extended to show that the maximum absolute value of a Gaussian vector has a uniformly bounded density on the real line. The method also produces new bounds on the variance of the maximum. We discuss implications for high-dimensional correlation testing, time-uniform sequential inference, and non-parametric confidence bands under latent, low-dimensional structure.

Model-based reinforcement learning (MBRL) agents typically learn world models by minimizing predictive loss. However, powerful RL optimizers inevitably exploit minor model inaccuracies, leading to simulator exploitation and a reality gap where policies succeed in simulation but fail in the real world. We propose that the objective for learning simulators should be strategic robustness rather than predictive accuracy, and formulate this as a zero-sum minimax game between a model player and an adversarial policy player. We provide a comprehensive theoretical analysis: (1) an online learning guarantee showing the game is learnable with sublinear regret bounds; (2) a tractable critic-based simplification bounding the global policy-value gap by the local critic's loss; and (3) an Error-MDP duality, proving that finding the worst-case policy is formally dual to a standard RL problem where the reward is the one-step critic error. This duality yields a provably convergent active data selection algorithm. Experiments on continuous control tasks demonstrate that our approach reduces prediction error in strategically important regions by $1.5$-$2.2\times$ and enables policies trained purely in simulation to match near-optimal real-world performance.