Papers

Bayesian optimization uses posterior uncertainty to decide where to explore, but this uncertainty is not spatially neutral. On bounded domains, Gaussian-process posterior variance is typically larger near faces, edges, and corners, giving variance-based acquisition functions a systematic boundary bias. This spatial preference is usually inherited from the surrogate rather than chosen as part of the exploration strategy. We separate these roles by keeping the predictive model fixed and modifying the spatial preference of the exploration term. We control the preference with spatial weight fields and amortize their construction with neural surrogates. Controlled experiments show that the Gaussian process' default boundary bias changes which parts of the domain are explored and which optima are reached. We compare three alternative exploration principles and find, across standard benchmark functions and stochastic robot-pushing tasks, that they can often lower regret, particularly early in the optimization. The spatial structure of posterior uncertainty is therefore not a neutral default: it shapes the exploration strategy and can materially affect optimization performance.

This paper develops strong Gaussian approximations (KMT type) for constant learning rate stochastic gradient descent (SGD). We work with heavy tailed noise and assume that the number of iterates grows and the learning rate shrinks simultaneously. Our work constructs simple Gaussian processes on the same space and bounds, in probability, the maximum difference between the SGD errors and their respective limits. While there exist central limit theorems for constant rate SGD centered at the optimum, these only provide individual iterate inference. By contrast, our quenched KMT approximations allow for finite sample, simultaneous, whole-trajectory-level inference and imply stochastic process convergence (Functional Central Limit Theorems (FCLTs)). For accumulated errors centered at the true optimum, we first obtain a KMT-type approximation bound when the learning rate scales as $\gamma_n=\frac{c}{\sqrt n}+o(n^{-1/2})$ for a constant $c \in \mathbb{R}^+$ and $n$ iterations. The result reveals a trade off between controlling the bias of SGD and the initialization effect, with no learning rate scaling generally making both contributions vanish in the accumulated error limit. To eliminate these contributions, we consider a Burn in scheme and Richardson-Romberg extrapolation and prove respective KMT and FCLT theorems. We also prove a finite-horizon KMT-type approximation and FCLT for the rescaled iterate trajectory to a stationary Ornstein–Uhlenbeck (OU) process. We consider applications in risk control and simultaneous inference and perform simulations to illustrate our results.

Determinantal point processes (DPPs) provide tractable distributions for nite random sets, but the usual symmetric-kernel specication imposes nonpositive pairwise dependence. Statistical research on nonsymmetric DPPs, which can accommodate positive dependence, remains limited. We study identication and estimation for this class. First, we establish the equivalence of two sucient conditions for a nonsymmetric kernel to dene a valid DPP. Second, we develop the identication theory for nonsymmetric DPPs. Third, we develop cardinality-truncated maximum-likelihood estimation and estimation by averaged stochastic gradient descent. Under suitable identication and regularity conditions, we establish consistency and asymptotic normality for both estimators. Simulations examine identication, the eects of truncation, and the nite-sample performance of the estimators. We illustrate the methods with two cyber applications at markedly dierent sampling frequencies: weekly sector-by-breach data and second-by-second enterprise authentication data with more than ve million set observations.

The Copula Active Subspaces (CAS) method of Part I estimates a non-Gaussian observation-noise density $\pi_{\boldsymbol{n}}$ from residual samples. Its KL divergence $\mathcal{D}_{\mathrm{KL}}(\pi_{\boldsymbol{n}}\,\|\,\widehat\pi_{\boldsymbol{n}})$ admits an exact decomposition into marginal, projection and reduced-density terms and bounds the posterior KL divergence averaged over the true predictive distribution. We derive a priori bounds for these terms, together with estimates or bounds from residual samples. Weighted nearest-neighbour entropy estimates are used for the marginal and reduced-density terms, with asymptotic standard errors in exact rank coordinates. For the projection error, the score is fitted on one subsample, and its mean squared magnitude outside the retained subspace is estimated on the other. Corrections for score and rank-transform errors then give a theoretical upper bound with probability at least $1-\delta$. Tests on three known noise laws in dimension $d=20$ use the estimated rank transform. In $11$ of $12$ combinations of example and sample size, the sum of the two estimated terms and the trace bound exceeds the reference divergence after averaging over the estimated models. In these tests, score-estimation error is the largest contribution to looseness. The overestimate from the Gaussian log-Sobolev inequality alone is at most a factor of $1.98$.

As watermarking of LLM-generated text enters deployment, users increasingly need to check text whose source is unknown. Existing detectors read the watermark through a specific tokenizer or embedding pipeline, and even if all providers adopted the same watermarking method, differences in these representations would leave their watermarks incompatible, and the text alone does not reveal which representation produced it. We propose ByteMark, a unified watermarking scheme that defines its signal on the UTF-8 bytes of published text. Models with different tokenizers embed this signal under one public specification and key, and a single detector verifies the text from the text and key alone. When the tokenizer or embedding model of a baseline detector is replaced, its detection degrades sharply, whereas ByteMark, which uses neither, retains its detection rate; ByteMark also remains detectable when participating models continue or rewrite the text.

Modern learning problems increasingly leverage data from heterogeneous sources to improve robustness and generalizability under distribution shift, often with high-dimensional parameters admitting low-rank structure. Existing group distributionally robust optimization (group-DRO) approaches optimize against adverse source mixtures, but extending this principle to low-rank matrix estimation with a broad class of losses leads to a challenging nonconvex-concave problem for which methods and theory remain limited. We propose Prior-Informed generalizable Low-rank representation Learning via Adversarial Reweighting (PILLAR), which smooths the group-DRO objective by regularizing adversarial source weights toward a user-specified reference distribution while learning a shared low-rank representation. By tempering extreme adversarial reweighting, PILLAR improves estimation stability and reduces the potential conservativeness of group-DRO. Using Burer-Monteiro factorization, we optimize the nonconvex-concave objective by bifactored gradient descent with mirror ascent. We establish linear convergence to a statistical neighborhood of the population saddle point and derive an error bound that separates low-rank estimation from adversarial reweighting. We further show that PILLAR improves on vanilla group DRO when the latter's excess conservativeness dominates source heterogeneity and prior misspecification. Simulations and an electronic health record study of suicide-attempt risk demonstrate improved worst-source and transported predictive accuracy.

Neural learners for classification and regression are often criticized as black boxes. We propose a probabilistic decomposition of neural learners that accounts for heterogeneous predictability across the feature space. A training procedure combining the EM algorithm, stochastic gradient descent, and maximum likelihood estimation is developed to jointly learn the predictable component and an input-dependent reliability score. We show that the optimal decomposed learner recovers the response distribution within the predictable component, while the reliability score characterizes the predictability of individual observations. We further extend the framework to incorporate localized feature transformations. Experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed framework.

Sharing time-to-event data can leak sensitive information about individual participants in clinical studies. To protect the privacy of data, differential privacy (DP) framework offers rigorous protection for data sharing. The local variant of differential privacy is particularly appealing because each record is perturbed before it leaves the data owner or contributor. However, the injected perturbation distorts the true distribution of the source data, yielding low utility and invalid statistical inference with the released data. In this work, we propose a nonparametric estimator for the true survivor function via the deconvolution kernel function with locally differential private (LDP) right-censored survival data in which both the event times and the event indicators are privatized. Our estimator simultaneously de-biases the perturbation applied to the times and the errors induced by flipping the event indicators due to LDP. Theoretically, we establish the uniform consistency and point-wise asymptotic normality of the de-biased distribution estimator, and derive its uniform convergence rate together with the optimal bandwidth attaining that rate. Simulation studies and the analysis of a semi-synthetic dataset generated from the METABRIC breast cancer cohort demonstrate the performance of the proposed estimator across privacy budgets, censoring levels, and sample sizes. By the post-processing property of DP, the proposed de-biased estimator inherits the privacy guarantee while restoring valid inference on the survivor function.

Augmentation-based self-supervised learning can recover the content shared across views without separating its underlying factors into individual coordinates. We ask whether augmentations can be designed to make those factors identifiable. Our approach allows content factors to fluctuate across augmented views while preserving their mean for each source instance. Each instance receives a fixed, recorded set of augmentation strengths that controls how much its factors fluctuate. When factors respond differently to changes in these strengths, their patterns of variation provide information that distinguishes them. Under suitable decoder and support assumptions, we prove that these differences identify individual factors in a conditional Gaussian model, up to permutation, scaling, and translation. We then develop a learner that uses the recorded strengths to estimate these patterns jointly across instances, enabling learning from few views per instance. Synthetic experiments demonstrate near-perfect factor recovery through nonlinear mixing. Controlled dSprites experiments show improved coordinate alignment using augmentations that target known factors, even without telling the learner which augmentation corresponds to which representation coordinate. These results suggest a route to component identification in self-supervised learning: augmentation design can supply identifying information beyond shared content alone.

Variance-based methods recover a Gaussian linear structural equation model (SEM) with equal or suitably constrained error variances, either by recursively identifying the source nodes with smallest conditional variance (top-down method) or sink nodes with highest conditional variance (bottom-up method). We show that the use of conditional variance is incidental: replacing it with the conditional risk, i.e., the minimal expected loss of linear prediction under the negative log-likelihood of the noise distribution, extends both methods to non-Gaussian noise. Existing methods are the least-squares special case. This yields two main results. First, exploiting the noise distribution improves structure recovery: the matched-loss procedures recover the SEM more accurately than their least-squares counterparts and the LiNGAM methods, which are designed for non-Gaussianity. Second, the two methods, interchangeable in the Gaussian setting, separate under non-Gaussian noise: the top-down method is correct for every noise density with finite differential entropy, whereas the bottom-up method requires a regularity condition and can fail without it, for instance under mean-zero exponential or Cauchy noise. Simulations confirm both results.

Generative modelling involves two complementary tasks: learning a probabilistic approximation of an unknown data distribution from observations, and designing a tractable mechanism for sampling from this approximation. We present a stochastic-analysis perspective on score-based generative models, connecting ideas from classical sampling theory, score estimation, and diffusion time reversal. We first explain how score functions link Langevin sampling with score matching, and how Gaussian smoothing leads to denoising score matching and multiscale sampling methods. We then emphasize the distinction between dynamics preserving a fixed distribution and dynamics transporting a prescribed family of marginals. This motivates a path-space treatment of diffusion time reversal. Finally, we connect the ideal reverse diffusion with practical generative algorithms.

We study the convergence of Markov chains in the geometry induced by Hilbertian embeddings associated with positive definite kernels, also called Maximum Mean Discrepancy (MMD). Under a contraction assumption in the kernel-induced geometry, and a standard spectral gap assumption, we establish non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure. The obtained bounds separate the roles of the two conditions: the spectral gap determines the effective sample size in the leading term, while the kernel contraction enters only through a linear correction. As an application, we study one-step prediction of dynamical systems from a single trajectory, deriving fast learning rates for a nonlinear least-squares estimator. The rates highlight the interplay between mixing properties in the kernel-induced geometry and the classical source and capacity conditions used in the analysis of least-squares.

We study the last iterate of standard asynchronous Q-learning from a single trajectory of a finite Markov decision process. The trajectory follows a fixed behavior policy whose state–action chain is irreducible and may be periodic. For discount factor $\gamma$, write $H=(1-\gamma)^{-1}$, and let $\mu_{\min}$ be the minimum stationary state–action probability and $\mathfrak h_\star$ the largest expected hitting time of the reverse state–action chain. We prove that zero-initialized Q-learning with a constant step size achieves sup-norm error at most $\varepsilon$ with high probability using $\widetilde{O}(H^4/(\mu_{\min}\varepsilon^2)+\mathfrak h_\star)$ transitions, for $0<\varepsilon\le1$. The statistical term retains the quartic effective-horizon dependence of standard Q-learning, while initialization cost is additive in $\mathfrak h_\star$. For aperiodic behavior chains, a comparison with mixing time yields a transient term $\widetilde{O}(t_{\mathrm{mix}}/\mu_{\min})$. An empirical state–action count rule selects a constant step size for a subsequent training run and attains the same rate without prior knowledge of model parameters, target accuracy, or confidence level. The output is the last training iterate, with both stages included in the sample count. The proof uses two deterministic Bellman envelopes to separate statistical error from fixed-policy initialization error and adapts reverse-time hitting-cost compensation from TD learning to state–action propagation. An exploration lower bound holds for arbitrary predictable step sizes. Together with the existing statistical lower bound and its extension to pilot-selected constant steps, it yields matching mean-square rates up to logarithmic factors on the stated large-sample parameter classes.

Many scientific models can be simulated but have intractable likelihoods, making frequentist uncertainty quantification difficult. We construct exact anytime-valid confidence sequences for the parameters of such models. At each observation, fresh simulations from each candidate null conformalize a positive predictable score into a conditional e-value; their product is a test martingale, which we invert over the parameter. Validity holds for any such score, whatever its accuracy. We show that the oracle likelihood-ratio score maximizes expected log growth, which motivates approximating it with an amortized neural ratio estimator. On synthetic benchmarks, an audit of model routing in an LLM API, and microlensing data, the resulting confidence sequences stay valid under continuous monitoring, where fixed-sample methods do not, and neural scores remain competitive with exact likelihood ratios and application-specific scores.

We establish finite-sample guarantees for conditional diffusion models learned from data and implemented with finitely many denoising steps. Our approach aligns the score-training objective with the sampling grid, allowing statistical estimation and numerical discretization errors to be analyzed together. Under a smooth conditional density model, we derive high-probability bounds on the generated distribution's error and quantify the required number of score evaluations. For every smoothness index $\beta>0$, a suitable grid yields averaged conditional Kullback–Leibler (KL) divergence $\widetilde O_p(n^{-2\beta/(d_x+d_y+2\beta)})$ relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error $\widetilde O_p(n^{-\beta/(d_x+d_y+2\beta)})$. Here $n$ is the sample size, $d_x,d_y$ are the data and conditioning dimensions, and errors are averaged over the conditioning variable. The total-variation bound matches the rate established in existing conditional distribution-estimation theory, up to logarithmic factors.

Gaussian process solvers for linear partial differential equations (PDEs) return posteriors rather than point estimates, but they condition only on equality information: the PDE residual, boundary data, and measurements. Many solutions of interest also obey inequality constraints, including nonnegativity, monotonicity, and absence of arbitrage, which equality conditioning cannot enforce. We prove that conditioning a Gaussian process prior jointly on linear equality observations and a finite-dimensional inequality event is equivalent to conditioning sequentially in terms of Borel measures on the solution space. The physics-informed posterior is therefore a valid prior for the inequality update at virtual points. The combined law is non-Gaussian yet computable through truncated-Gaussian moments, and neither the PDE conditioning nor the inequality update requires modification. We evaluated the construction on a Poisson benchmark, option-price surfaces, and air-quality fields. Under incomplete boundary data, a monotonicity constraint lowered the error of a two-dimensional Poisson reconstruction by $63\%$. On a real S&P 500 option chain thinned to $20$ retained quotes, no-arbitrage constraints reduced held-out extrapolation error by $22\%$ and cut the calendar-arbitrage rate from $10\%$ to $1\%$. On four real air-quality snapshots, nonnegativity reduced a Gaussian moment-based sub-zero-tail proxy by $74\%$ at the sparsest monitor count. Changes in mean accuracy and constraint diagnostics differed across data regimes. The probability of the noisy virtual constraint event under the equality posterior measures agreement with the specified bounds before the update. The construction turns admissibility from a post-processing step into conditioning information for probabilistic PDE solvers.

What can an adversary recover from a model trained with DP-SGD? Its $(\varepsilon,\delta)$ guarantee does not answer this question directly. We derive a distortion bound that can be computed before training and applies even to an adversary who observes the parameter trajectory during training, knows every other training record and the population distribution from which the training records are drawn. The analyst may choose the representation in which error is measured, allowing the bound to target the aspects of reconstruction that are privacy-sensitive in a given application. Our information-theoretic framework separates two contributions: the information about a record revealed by training, measured through conditional mutual information, and the uncertainty already present in the data distribution, measured through source entropy. Rate-distortion theory combines them into a lower bound on the expected error of every reconstruction attack. For DP-SGD, the leakage bound depends only on the sampling rates and noise multipliers, while the source entropy is lower-bounded from a sample with finite-sample confidence. An orthogonal block construction keeps the resulting bound informative for high-dimensional images. We instantiate the framework in pixel space and in LPIPS feature space, obtaining, to our knowledge, the first lower bound on reconstruction distortion measured directly by LPIPS. On a range of image datasets the bounds establish non-trivial unavoidable expected reconstruction error within heuristically recommended privacy-budget ranges and beyond. Experiments compare the bounds with strong reconstruction attacks and identify the principal sources of the remaining slack.

Prior-data fitted networks (PFNs) are trained to approximate the Bayesian posterior predictive distribution (PPD) of a synthetic prior, yet results about the approximation quality are surprisingly thin. Taking an empirical risk minimization perspective, we decompose the deployment error of PFNs into prior, context, and amortization error, and measure the latter against a reference PPD under a realistic prior. Results show that PFNs generally recover the targeted PPD very well. Nevertheless, we find that parts of the residual amortization error persist, especially in rare settings under the prior pretraining distribution, even when scaling the model size, which manifests in disagreement between independently pretrained PFNs and leaves uncertainty about the predictor at inference time. As a remedy, we study multiple types of Bayesian model averages and find that these can consistently reduce this uncertainty and improve predictions up to full-size TabICLv2 models.

Neural networks (NNs), while highly flexible and versatile, are often poorly calibrated. Post-hoc calibration methods adjust the outputs of an already trained model using a separate calibration dataset and additional calibration parameters. This process adds overhead that requires separate optimization of the calibrator and subsequent model selection. In this paper, we show that standard (post-hoc) calibrators can be trained end-to-end with the backbone via a bilevel Stackelberg formulation. This formulation eliminates the need for a separate calibration set or a dedicated calibration loss by jointly optimizing accuracy and calibration using cross-entropy, introducing only one additional hyperparameter: the calibrator learning rate. Although our framework supports calibrators ranging from temperature and matrix scaling to NN layers, our results show that using the final classification layer itself as the calibrator yields accurate, well-calibrated models with no additional parameters. Our approach outperforms SOTA calibrators with minimal added complexity, while also avoiding the multiplicative overhead of separate calibration components in settings such as multi-task learning.

Linearizing softmax attention offers a route to efficient Transformer inference, but preserving pretrained attention remains challenging. We identify an important source of approximation error overlooked by standard linearization methods: a small number of outlier dimensions contribute disproportionately to query-key dot products and account for a large fraction of the approximation error. Motivated by this heterogeneity, we propose Outlier-Aware Attention Linearization (OAL), a dimension-aware method for constructing quadratic attention approximations. Starting from a second-order Taylor approximation, we introduce a Hadamard-product formulation that makes individual query-key contributions explicit. This formulation relaxes globally shared coefficients to support dimension-specific linear terms and cross-dimensional quadratic interactions while preserving separable evaluation. To retain this flexibility with fewer parameters, OAL groups dimensions with similar contribution scales to share coefficients, while allowing distinct treatment of the few outliers. The resulting positive quadratic kernel is calibrated against pretrained attention outputs and admits linear-time evaluation with respect to sequence length for fixed head dimensions. On Qwen2.5-0.5B, OAL reduces frozen-attention output relative MSE by approximately 8.9% compared with DiJiang. Replacing middle-layer attention modules and jointly fine-tuning the kernel coefficients with LoRA further lowers WikiText-103 perplexity from 31.71 to 27.50 compared with a learned global quadratic, while improving accuracy on PIQA, HellaSwag, and ARC-E. These results demonstrate the value of outlier-aware dimension grouping for both attention approximation and pretrained-model conversion. The code is available at https://github.com/wuqianyue09/OAL

Latent field models are essential for spatial statistics, but the posterior distribution of the latent field, even with a Gaussian process (GP) prior, is analytically intractable whenever the likelihood is non-Gaussian. Existing approximations are either too slow or too restrictive for large non-Gaussian datasets. We present a scalable variational inference framework that mirrors the sparse nearest-neighbor autoregressive structure of the prior in the variational family, uses normalizing-flow conditionals for expressiveness, and amortizes all variational parameters via a single shared neural conditioner. Doubly stochastic optimization with reduced ancestor sets achieves polylogarithmic complexity per iteration.

Reinforcement learning with verifiable rewards (RLVR) has substantially improved LLM reasoning, yet growing evidence suggests it mainly performs distribution sharpening: concentrating probability mass on reasoning paths the base model already produces. Recent work therefore seeks RLVR's gains without needing reward signals, by sampling from the power distribution $\pi_\alpha \propto \pi_{\text{base}}^\alpha$ via Markov chain Monte Carlo or sequential Monte Carlo. We show that these samplers fail to converge to $\pi_\alpha$ under realistic compute budgets, and that their gains are largely recreated via simple Best-of-$N$ with appropriate grading . This suggests they succeed by selecting high-probability sequences, not by approximating $\pi_\alpha$, raising a natural question: is the power distribution itself a worthwhile target? We test this hypothesis directly by fine-tuning the base model to match $\pi_\alpha$, sidestepping sampler convergence issues entirely. Despite approximating the target far more faithfully than existing samplers, this strategy yields gains that are noisy and specific to certain models and sampling tasks. Together, our results suggest that the value of power sampling lies not in faithfully targeting $\pi_\alpha$, but in the simpler goal of extracting sequences the base model already deems likely.

The external validity of a study is the extent to which its findings generalize to unseen experimental conditions, such as datasets. We formalize the results of a study together with its research question, and define external validity as the probability that two independent and same-sized replications of the study produce results whose maximum mean discrepancy (MMD) is below a given tolerance. This approach, centered on the choice of a space of results and of an appropriate reproducing kernel for the MMD, is flexible enough to be deployed on multiple different types of results and research questions. Additionally, we establish an asymptotic power-law relation between the size of a study and the quantiles of the MMD, which we use to estimate the desired size of an externally valid study from a small-scale preliminary study. We deploy our framework to rankings with ties and discuss kernels suited to different research questions. Experiments on existing case studies and on a simulated newly designed study illustrate the applicability of the framework. We release our code as part of the genexpy Python module.

We discuss three simple connections between generative diffusion and optimal transport. First, the joint law of the data and of their Ornstein-Uhlenbeck noising is, at every time, the solution of an entropic optimal-transport problem, at a temperature set by the noise level. Backward diffusion can consequently be read as an annealing which gradually decreases the entropic regularization. Second, we study the speciation transition–how the modes of the data distribution emerge during generation–and compare how it happens in diffusion and flow matching with respect to optimal transport. Using Gaussian mixtures as high-dimensional example, we show that the dynamical instability through which diffusion and flow matching separate the modes becomes, for optimal transport, an exponentially thin boundary layer at the beginning of generation, which is rounded by entropic regularization. Third, considering a generic forward dynamics with a general confining potential, we show that the backward process driven by the score also satisfies a drift-based variational principle whose minimal cost is the entropy difference between initial and final distributions.

Correlation matrices combine nonlinear positive-definiteness constraints with a simultaneous row-column permutation symmetry. Their generation therefore involves two distinct problems. First, every output must remain on the correlation elliptope, irrespective of how its variables are ordered. Second, the generator must learn the domain-specific distribution occupied by empirical data inside that valid space. Existing financial image generators mix these problems: they learn ordered matrix entries and repair invalid draws afterward. We introduce CorrFlow, which separates them. A generalized Fisher transform maps full-rank correlation matrices bijectively to unconstrained coordinates, guaranteeing valid outputs without projection. A graph flow-matching network learns the empirical distribution in those coordinates and is permutation equivariant: reordering the variables only reorders the output. We compare a conditional CorrGAN reimplementation, image diffusion, and CorrFlow on $80\times80$ Taiwan equity matrices using five seeds per neural method. CorrFlow produces 100% valid draws without repair and obtains feature sliced-Wasserstein distance $.712\pm.141$, versus $4.339\pm.252$ for CorrGAN and $4.610\pm.191$ for image diffusion after their invalid outputs are projected back to the correlation space. It also improves this distance over historical lookup, while lookup retains higher local coverage. These results support the combined construction under the evaluated training budget; they do not establish superiority over fully tuned image models or isolate the contribution of each component.

We analyze distributional drift in nonstationary weighted risk minimization by decomposing the excess risk into estimation, drift, and approximation components. The drift error is commonly bounded through discrepancy-type quantities that control changes of the risk over classes of predictors. We instead relate the drift error directly to the movement of the best predictor within the model class and analyze its behavior in parametrized regression models. Our analysis shows that discrepancy-type quantities can be overly conservative: unlike our oracle drift, they may scale at a larger order than the resulting excess-risk contribution and can respond to changes in the risk surface that do not affect the optimal predictor. For linear regression, we characterize this gap sharply and combine the resulting drift bounds with estimation guarantees. We further extend the analysis to nonparametric regression.

Uncertainty quantification for electric vehicle (EV) charging demand is essential to turn the increasing pressure of EV adoption on power systems into a source of flexibility. We study a large-scale dataset of public charging stations in Scotland, where demand exhibits a hierarchical structure, organized across multiple aggregation levels, from individual charging points to regional and national scales. The charging infrastructure is continuously evolving, with new stations deployed and existing ones occasionally removed, which calls for forecasting and uncertainty quantification methods tailored to this evolving structure. To address these challenges, we combine hierarchical forecasting with recent advances in conformal prediction with missing data. The conformal procedures considered produce valid prediction intervals at every node of the hierarchy and the algorithm CP-MNR-Nested successfully accounts for the time-varying hierarchical structure of the data.

When a language model is distilled from a teacher that shares its base parameters, the student can inherit behavioural traits of the teacher—including misalignment—even from data that is semantically unrelated to the trait (Cloud et al., 2026). All existing defences require knowing the trait. We remove this requirement and prove that a subliminally transferred trait can be recovered from the parameter updates alone, without supervision, by exploiting a single asymmetry: the trait is expressed consistently across diverse datasets generated by the teacher, whereas task-specific and shared capabilities vary. The resulting estimator returns a weight-space direction that can be amplified to discover and interpret the trait or projected out to suppress it. Applied to the subliminal-learning benchmarks of (Cloud et al., 2026), our method recovers the transmitted animal-preference traits without any trait labels and suppresses their expression with almost no impact to task performance.

The BROJA partial information decomposition provides an operationally motivated way to separate the information that two sources carry about a target into unique, redundant, and synergistic components. For continuous and high-dimensional variables, however, computing the BROJA decomposition requires solving a difficult optimization problem. Previous work showed that, when each source-target pair is jointly Gaussian, the problem can be reduced to finite-dimensional numerical optimization. Here, we derive a closed-form solution to this optimization problem, enabling direct computation of the BROJA decomposition in the general multivariate Gaussian setting. The resulting closed-form expression enables speedups exceeding two orders of magnitude for high-dimensional Gaussian BROJA PID. Code is available at https://github.com/brainsandmachines/Eigen_PID.

Compression schemes provide a general framework for deriving data-dependent risk bounds for learning algorithms. For learning rules based on preferent compression, existing results upper-bound the risk in terms of a sample-dependent complexity $\boldsymbol{k}$, defined as the cardinality of the compression augmented by the training examples on which the reconstructed hypothesis is inappropriate. These results, however, yield a margin of order $O(\sqrt{\ln N/N})$ above $\boldsymbol{k}/N$. In this paper, we establish that the logarithmic factor is not intrinsic. We prove that, for every fixed confidence level $1-\delta$, the risk is bounded with probability at least $1-\delta$ by $$ \frac{\boldsymbol{k}}{N} + O\left(\frac{1}{\sqrt{N}}\right), $$ uniformly over all possible values of $\boldsymbol{k}$. Thus, risk estimates for learning algorithms based on preferent compression attain the same law-of-large-numbers rate as the empirical estimate of the probability of a fixed event. We also show that the leading term $\boldsymbol{k}/N$ is necessary and investigate its relation to the empirical risk. In particular, we identify regimes in which the empirical risk retains the $O(1/\sqrt{N})$ convergence rate, as well as regimes in which it converges more slowly or ceases to be consistent, while the complexity-based estimator preserves the $O(1/\sqrt{N})$ rate.