Papers

This paper studies the complexity of convex optimization using lazy second-order oracles (Doikov, Chayti, and Jaggi, ICML 2023), where an algorithm queries gradients every iteration and Hessians once per $m$ iterations. Under this setting, we show a lower bound of $\Omega(m+ m^{1/7} \epsilon^{-2/7})$ on the number of total iterations to find an $\epsilon$-solution using a novel block zero-chain construction. Then we propose a novel method that achieves a new upper bound of $\tilde{\mathcal{O}}(m+ m^{1/7} \epsilon^{-2/7})$, which significantly improves the prior one (Chen, Liu, Luo, and Zhang, COLT 2026) of $\tilde{\mathcal{O}}(m+ m^{13/21} \epsilon^{-2/7})$ and is tight up to logarithmic factors.

Gaussian Processes (GPs) are a powerful tool for modelling and quantifying uncertainty in functional relationships. However, they require practitioners to make a number of design decisions, such as the choice of the kernel and the observation model. Suboptimal choices can produce misspecified models that do not capture the underlying data generating process. We introduce Predictively Oriented Gaussian Processes (PrO-GPs), which treat predictive uncertainty as the primary inferential target and provide a robust alternative to standard GPs. Although direct computation of a PrO posterior for nonparametric models is intractable, we derive a reduced formulation and practical sampling scheme for efficient computation. Through synthetic and real data experiments, we show that PrO-GPs produce better calibrated predictive distributions under model misspecification compared to standard GP approaches.

Hawkes processes are widely used to model temporal point patterns exhibiting self-exciting behavior. In this work, we consider a Hawkes model with a baseline intensity depending on a set of dynamic covariates and an exponential triggering function. A Bayesian approach is applied to variable selection in this context. Spike-and-slab priors are proposed to enable a parsimonious choice of relevant variables. We also provide practical simulation procedures for the fitted covariate-driven Hawkes model, which support posterior predictive checks and probabilistic forecasting. This approach is illustrated by the example of spore release of Venturia inaequalis, the fungus responsible for apple scab disease. We study the dependence of primary spore releases on environmental covariates and highlight the self-exciting behavior of subsequent releases. The Bayesian framework also provides a natural basis to forecast future spore releases. By delivering interpretable posterior inclusion probabilities alongside uncertainty-aware predictive risk bands, this new methodology provides a rigorous foundation for managing temporal event risks driven by both external environmental forcing and internal contagion.

Gross Domestic Product (GDP) is widely used to guide economic and social decision-making, but it provides only a partial view of wellbeing. For this reason, the European Union (EU) has launched initiatives to monitor sustainable and inclusive wellbeing beyond GDP, developing indicator frameworks that cover dimensions such as health, education, environment, and social inclusion. These indicators are naturally grouped into thematic areas, and policymakers are interested in understanding how these areas contribute to global wellbeing and which indicators explain differences across countries. However, such data are high-dimensional, contain missing values, and may include anomalies affecting entire observations, specific thematic areas, or individual indicators. We introduce blockwise outliers and propose bloccPCA, a robust multiblock PCA method that simultaneously handles casewise, blockwise, and cellwise outliers, as well as missing values. The method provides robust global components to summarize the overall structure of wellbeing, while preserving thematic-area contributions through robust blockcomponents. It also yields diagnostic tools to identify whether anomalies arise at the case, block, or cell level. Monte Carlo simulations and an application to the EU wellbeing dataset show that bloccPCA provides valuable insights into sustainable and inclusive wellbeing.

Clustering variables in causal graphs reduces the size of the graph and simplifies causal inference. However, arbitrary clustering can alter crucial causal relations among variables and lead to erroneous conclusions. While the identifiability of a causal effect in the clustered graph implies the identifiability in the original graph under mild conditions, nonidentifiability in clustered graph does not imply nonidentifiability in the original graph without further assumptions. When both identifiability and nonidentifiability are preserved, the clustering operation is called identification invariant. We present a broad class of clustering operations that are identification invariant based on conditions related to the c-components of the original graph. Finally, we demonstrate use of the results in practical settings.

Neural network-based approaches have emerged as efficient alternatives to traditional optimization-based procedures for the calibration of stochastic volatility models. However, existing work has focused primarily on predictive accuracy, with comparatively little attention devoted to understanding the structure of the learned inverse calibration mappings. In this work, we analyze neural calibration mappings for the Heston and rough Heston models across multilayer perceptron, highway, and softmax-parametrized highway architectures, using complementary Shapley-based methods from explainable AI. Specifically, we consider SHAP and $\nu$SHAP explanations, which capture distinct, complementary notions of feature relevance, corresponding to sensitivity and sufficiency of feature subsets, respectively. Short maturities and smile wings consistently dominate parameter inference, and the dominant attribution structure remains qualitatively stable across architectures despite differences in predictive accuracy and parameter count. Parameter-specific differences between SHAP and $\nu$SHAP further reveal how distinct regions of the implied volatility surface contribute to parameter recovery and expose substantial redundancy in the calibration input. Building on this redundancy, we show that $\nu$SHAP explanations can guide a significant reduction in input dimensionality for the rough Heston model while matching calibration accuracy relative to the full implied volatility surface. These findings demonstrate that complementary Shapley-based methods provide structural insight into learned inverse calibration mappings beyond predictive error metrics, and offer a practical route to feature selection in neural calibration problems.

We introduce a set of consequence-based measures that quantify the misclassification arising from imperfect reliability in multi-item psychometric scales. Particular attention is given to the probability of individuals in extreme latent-trait percentiles being correctly identified from their observed scores. These cost functions provide a principled and interpretable way to characterise how inadequate reliability distorts classification and reduces the informational value of test scores. We also examine methodological issues in estimating reliability under common-factor models, with emphasis on McDonald's omega and the construction of accurate confidence intervals. To address the computational burden of model fitting, especially in small samples, we derive a modified version of Cronbach's alpha that closely approximates omega when the common-factor model holds. This estimator has comparable sampling variability to omega while requiring no parameter estimation, offering a practical and computationally efficient alternative. y

Large language models (LLMs) can support rubric-based screening of CVs, but applying a high-capability model to every candidate and criterion is costly. We present CreateScore, a domain-theory-informed Bayesian network for criterion-level LLM routing. A hand-specified directed acyclic graph with Dirichlet-multinomial conditional probability tables converts CV evidence into posterior uncertainty; low-uncertainty decisions are resolved by a local 8B model and uncertain ones are escalated to a 120B reference model. The graph is causally motivated, but the system performs standard Bayesian conditioning, not causal inference. The escalation threshold is calibrated on a training fold (target: 70% resolved locally) and then frozen. On 200 synthetic Data Science CVs (139 training and 61 test candidates, five criteria), 77.7% of criterion decisions were resolved locally (237 of 305). Relative to a reference condition in which the 120B model adjudicated every criterion, routed escalation reduced token use by 65.2% and raised exact score agreement from 32.8% (8B alone) to 42.6% (95% CI 31.0-55.1%); at n = 61 the gain was not statistically distinguishable. The uncertainty signal did not, however, identify the decisions on which the 8B model erred: disagreement with the reference was 16.2% among escalated and 19.4% among locally resolved decisions (AUROC 0.47, 95% CI 0.39-0.56), no better than random selection. We also document how an earlier evaluation was invalidated when truncated reasoning-model outputs were silently replaced by local labels, and we recommend safeguards for cascade evaluation. CreateScore is supported as an auditable cost-reduction mechanism, not yet as a targeted error detector, and is not an autonomous hiring system.

Test-driven development gives AI coding agents executable requirements for implementing software. Because these agents can adapt their implementations to the examples they observe, passing a predetermined collection of tests can leave substantial parts of the intended behavior unimplemented. We propose Generative Test-Driven Development (GTDD), a formulation of test-driven development in which a separate testing agent generates new inputs after each candidate implementation is fixed, using a human-specified behavioral contract and the feedback from earlier rounds. A trusted evaluator checks these inputs, returns reduced counterexamples to the coding agent, and saves them for regression testing, so development continually confronts failures beyond the initial examples. We characterize the evidence that this process provides through a finite-population analysis of false acceptance under adaptive candidate selection. The resulting bounds quantify how test visibility and repeated feedback affect acceptance, and show that fresh random audits after candidate commitment control false acceptance across development rounds. In a paired experiment on a stateful key-value store, both policies that regenerated tests during development ended with lower mean failure rates than the policy whose tests were generated once by the same language model, and giving the tester the candidate's source produced no detectable additional improvement. Further conditions requesting equal numbers of tests did not isolate any single feature of the policies as the source of this difference. GTDD combines this adaptive development feedback with established regression tests and an independent acceptance rule.

Cross-fitting is routine in much of applied research. While conventional confidence intervals that ignore cross-fold dependence are asymptotically valid in several settings, they undercover in many applications that share a common form of nonregularity: from the classic cross-validation problem of testing whether a fitted model outperforms another, to testing for heterogeneous treatment effects with machine learning, to estimating the value of a potentially non-unique optimal treatment regime. Exploiting a new locality condition, I show that a large class of cross-fitting estimators still satisfies a central limit theorem despite the nonregularity, but with an asymptotic variance that must be adjusted for the cross-fold correlation. Then, I propose a method for estimating this correlation and construct new confidence intervals that attain asymptotically nominal coverage. Finally, I show that the proposed confidence intervals attain approximately nominal coverage in a simulation study with random forests and neural networks.

This paper investigates non-parametric regression estimation when the explanatory variable takes values in a separable Hilbert space and observations are sampled over an increasing regular spatial lattice. Under a rigorous field-to-field independence setup between covariates and errors, we explore the structural and asymptotic concentration properties of the functional $k$-nearest-neighbour ($k$-NN) estimator. By establishing a sharp pathwise deterministic sandwiching framework for the data-driven random bandwidth, we successfully decouple the local infinite-dimensional small-ball profile from the polynomial covariance decay of the neighborhood indicators. Pointwise convergence rates are derived across short-range, critical, and long-range spatial regimes, revealing a combined penalty term that reflects both covariate spatial interaction and response error memory. Furthermore, uniform consistency over compact subsets is established for unbounded sub-Gaussian error processes through metric entropy and indicator boundary shell chaining. Structured polynomial simulations confirm our theoretical rates and exemplify the precise mechanics of the spatial long-range bottleneck.

With the advance of measurement technologies and increasing computing power, large spatial data with heterogeneous structures are often collected over high-dimensional domains. Existing Gaussian process (GP) models and computational strategies are often inadequate for analyzing such datasets in multi-dimensional domains. To address these challenges, we develop a Bayesian residual tree GP methodology called ResTGP for large spatial data with potentially heterogeneous structures in multi-dimensional domains. The key idea is to decompose a Gaussian process at a cascade of resolutions along a dyadic tree through iteratively computing predictive and residual processes so that the residual process on each tree node, both interior and leaf, becomes sufficient for the finer-level dependency within that node. This allows characterization of the underlying covariance structure in a flexible, multi-scale manner while achieving divide-and-conquer on the data domain, which leads to computational efficiency. To allow efficient tree inference, we introduce a computational strategy for Bayesian inference based on recursive message passing, which scales linearly with the sample size given the tree. This paper also proves posterior consistency of the model for estimating continuous functions in a nonparametric regression framework. Extensive numerical examples and the storm surge application confirm the advantages of the proposed method.

The Muon optimizer applies spectral orthogonalization to matrix-valued updates and has shown strong performance in large-scale neural network training, yet the mechanisms of this transformation in feature learning remain poorly understood. In this work, we investigate this question through a tractable factual-recall model, where a fact maps each subject-relation pair to an answer, and a linear transformer learns the subject- and relation-dependent information required to recover this mapping. The transformer is optimized with gradient flow (GF), spectral GF, or Sign GF, which are continuous-time limits of gradient descent, Muon, and Adam, respectively. Prior studies (Nichani et al., 2025) have shown that when the number of subjects exceeds the number of relations, GF learns relation-dependent information before subject-dependent information, producing a feature-separation phase during training. We characterize this separation with the learning times when the subject- and relation-dependent components of the prediction reach a target accuracy. With $S$ subjects and $R$ relations, GF has a learning-time ratio of $\widetilde{\Theta}(\sqrt{S/R})$, whereas Spectral GF reduces this ratio to $\widetilde{\Theta}(1)$. In addition, for fixed $S$ and $R$, the subject- and relation-dependent errors decay as $1/(T\log T)$ in training time $T$ under GF, but as $\exp(-\mathrm{poly}(T))$ under spectral GF. Finally, we show that GF and spectral GF are equivariant under orthogonal transformations of the token embeddings, whereas Sign GF is not: Different orthonormal embeddings can potentially produce no feature separation, a large feature-separation phase, or even a reversed learning order. These results provide a mechanistic view of how spectral orthogonalization can fundamentally reshape feature-learning dynamics.

High-dimensional Gaussian DAG learning faces a statistical-computational gap: methods with sharp sample complexity rely on computationally expensive subset search and a supplied indegree bound, whereas polynomial-time alternatives have less favorable sample complexity. We introduce HOST, an efficient DAG learning algorithm that replaces subset search with nodewise hold-out scoring and convex regression, without requiring a supplied indegree bound. Our key insight is that recovering a correct ordering does not require uniformly small estimation errors in ordering scores but only one-sided control of those errors. In the ordering step, HOST exploits the fact that score estimation using hold-out samples inflates ordering scores in expectation, which is the favorable direction for candidates that should not yet be selected. Given the ordering, HOST recovers parents by recursively removing indirect effects from total effects between two nodes. Under suitable conditions, HOST exactly recovers a $p$-node DAG of maximum indegree $d$ with sample complexity of order $d\log p$ in polynomial time. Experiments show that HOST achieves competitive graph recovery while exhibiting favorable runtime scaling.

For a given point forecaster, proportional-integral-derivative (PID) calibration configurations can attain similar overall coverage yet produce different interval widths. We introduce PID-Expert, an online aggregation method for improving the efficiency of conformal prediction intervals for time series. PID-Expert combines thresholds from a fixed library of PID calibrators, each evolving under its own coverage feedback. Aggregation weights depend on normalized interval width and miscoverage, while a shared multiplier adapts the miscoverage penalty using feedback from the reported interval. We establish local regret bounds for weighted expert losses under the realized multiplier sequence and, separately, a pathwise upper bound on time-averaged aggregate miscoverage, with control in expectation and almost surely under a stability condition. Across four simulation settings and two real-data applications, PID-Expert produces narrower mean intervals than a prespecified Conformal PID benchmark while keeping overall empirical coverage close to the nominal level. Compared with expert selection and equal averaging, aggregation generally provides a more balanced coverage–width trade-off across forecasting settings. Rolling analyses further reveal local coverage–efficiency trade-offs, particularly following abrupt distributional shifts. Overall, PID-Expert reduces reliance on a single PID configuration while improving interval efficiency.

We study fixed-confidence best-arm identification under strict 1-bit feedback constraints. At each round, the learner selects an arm and a query set, and receives only a single bit indicating whether the sampled reward belongs to that set. We consider a distribution-free finite-variance setting with arm-wise localization, where direct empirical mean estimation is no longer available and clipping becomes unavoidable. We first formulate a time-uniform 1-bit mean-estimation primitive based on randomized threshold queries and a clipped tail-integral identity. We then embed this primitive into candidate-challenger best-arm identification algorithms. A fixed-clipping algorithm gives a simple anytime $(\epsilon,\delta)$-PAC guarantee, while a phased adaptive-clipping algorithm matches the clipping level to the current resolution and yields a gap-adaptive sample complexity. We also prove a $K$-arm worst-case information-theoretic lower bound showing that the logarithmic penalty caused by finite-variance 1-bit feedback is intrinsic. This bound matches the leading dependence of the phased algorithm up to lower-order $\log\log$ factors.

Diffusion models are highly developed in continuous spaces for image and video domains. Recently, major advances have been made for discrete diffusion models for categorical data, specifically in the language domain. In contrast, diffusion models for discrete integer-valued data are less developed, despite the prevalence of this modality, ranging from images and music to gene counts. We introduce Jumping Up and Down (JUD)—a new family of denoiser-based diffusion models for discrete ordinal data. This is the first family of diffusion models for ordinal data which centers around training denoisers, which at the same time allows for bi-directional (up and down) perturbations of the data. The simplicity of the training objective, combined with the flexibility of bi-directional perturbations, leads us to obtain competitive results across different data modalities.

Objective perturbation adds a random linear term to a regularized empirical risk and releases the exact perturbed minimizer. We study the finite computation obtained by releasing the $N$-th iterate of deterministic gradient descent on $w\mapsto F(w;S)+\langle z,w\rangle$, where $z\sim\mathcal N(0,\sigma^2I_d)$ is drawn once before optimization. For strongly convex and smooth objectives with Lipschitz Hessian, we prove an explicit condition under which the map $z\mapsto w_N$ is a $C^1$-diffeomorphism on the bounded domains used in the privacy argument, with a quantitative lower bound on the smallest singular value of its Jacobian. This permits a direct change-of-variables analysis of the finite iterate. For generalized linear models, the resulting privacy-profile bound has no explicit ambient-dimension factor once the iteration condition holds, and its finite-iteration correction decreases geometrically. By letting the free truncation parameter grow slowly with $N$, we recover the corresponding exact-minimizer certificate in the limit. We also bound the expected excess empirical risk by $d\sigma^2/(2\mu)$ plus a geometrically decreasing optimization term, and transfer the result to population risk without an additional multiplicative condition-number factor in the leading statistical terms.

Boolean factor models, including prominent subfamilies such as Boolean matrix decompositions and cognitive diagnosis models, find broad applications ranging from social sciences to engineering. Despite their flexibility, a key challenge lies in establishing the identifiability of their graphical structures, which specify how latent variables influence observed variables. Existing identifiability conditions typically rely on the strong assumption of pure nodes, which may be unrealistic in many applications. We develop a novel approach leveraging the Hasse diagram to represent the distribution of observed variables and transform identifiability into a graph isomorphism challenge. Based on this, we establish sufficient and necessary graphical identifiability conditions that do not require pure nodes. We further derive equivalent algebraic conditions and develop an efficient Boolean satisfiability (SAT)-based verification algorithm. We extend the analysis to probabilistic Boolean factor models, establishing conditions for jointly identifying the graphical structure and additional model parameters without requiring pure nodes. Our results substantially broaden the class of identifiable and interpretable Boolean factor models by removing the pure-node requirement, yielding new theoretical insights, while also providing practitioners with a concrete and easily implementable tool to assess model identifiability.

Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution $P_{\mathrm{data}}$ from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance, for all $p\geq 1$. Specifically, given $n$ i.i.d. samples from $P_{\mathrm{data}}$, we show that, for every $d>d_p^\ast(P_{\mathrm{data}})$ and appropriately chosen network architectures and hyperparameters, the learned distribution $\widehat{P}^{\mathrm{FM}}$ satisfies $ \mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/\xi)\bigr)^{1/(2p)}$ with probability at least $1-\xi$, where $d_p^\ast(P_{\mathrm{data}})$ denotes the Wasserstein-$p$ dimension of the target measure. Our results demonstrate that flow matching naturally adapts to the intrinsic geometry of data and mitigates the curse of dimensionality, as the convergence exponent depends on the intrinsic rather than ambient dimension. These guarantees remain meaningful in high-dimensional regimes and provide a theoretical explanation for the empirical success of flow matching on structured data distributions under substantially more relaxed assumptions than those in existing analyses.

Covariance estimation is a key component of many applications in system identification and data-driven control. Although heavy-tailed distributions may lack a covariance matrix to estimate, the shape matrix provides a well-defined, scale-free generalization for the broad family of elliptical distributions. In this setting, practitioners often employ Tyler's M-estimator (TME), which is defined implicitly and is computed using a fixed-point iteration. The spatial sign covariance matrix estimator (SSCM) offers a much simpler alternative: it corresponds to one iteration of TME. Yet, the SSCM has been largely regarded as an inferior estimator due to its statistical inconsistency under fixed-dimensional asymptotics. By contrast, using second-order tools from random matrix theory, we establish that, under standard assumptions, SSCM asymptotically dominates TME in Frobenius risk when dimension and sample size grow proportionally. This advantage arises from implicit regularization, which reduces variance at the cost of a bias that vanishes as the dimension grows. Numerical experiments support the theoretical predictions even at moderate dimensions.

Post-training reasoning LLMs often requires making a tradeoff between costly supervision and compute inefficiency: supervised finetuning typically requires reasoning traces, which is data inefficient since in many applications one only has prompt-answer tuples and lacks the groundtruth chain-of-thought (CoT) sequence. On the other hand, reinforcement learning (with verifiable rewards, for example) can operate without CoT data but it is much more compute-inefficient since it requires multiple autoregressive completions per optimization step, none of which are guaranteed to give a non-zero policy gradient. In this work we explore a middleground, a self-distillation technique that uses answer-only data, treating CoT as latent variables. This relies on training a partner “inverse reasoning LLM" that generates the original's reasoning traces conditioned on the answer. The inverse model is trained once and reused across incoming batches of answer-only data, with all generation happening offline and up front, making the approach particularly suited to continual post-training. Using Qwen3-0.6B on the Countdown and GSM8K tasks, we show that after GRPO training has saturated, a single offline distillation round from the inverse model roughly doubles pass@1 on hard prompts and, on Countdown, outperforms rejection-sampling self-distillation on every hard-tail metric at every mix ratio (with smaller gains on GSM8K); rejection sampling by construction has no signal on problems the model never solves on its own. Continued GRPO on the fresh data retains an edge at hard pass@1 but trails distillation at pass@8 and pass@32 while regenerating rollouts at every step. As an added bonus we see that the inverse model has a second application as an accurate verifier of correctness. When used for best-of-$n$ selection at test time, it raises selection accuracy on Countdown from 0.753 (majority voting) to 0.969.

To commit to buying external data or participate in collaborative learning, one must decide whether the additional data will improve prediction enough to justify the cost. This comes with several challenges: (i) the decision often relies only on aggregated statistics available publicly, rather than individual-level data; (ii) covariate and model shifts can induce negative transfer, so the additional data deteriorates rather than improves performance; (iii) if the data is sensitive, its privatization requires the injection of noise, which can also offset the benefit of a larger sample size. In this paper, we model the problem of dataset selection through high-dimensional regression with multiple heterogeneous sources and a weighted ridge estimator. Our approach uses only summary statistics and it gives privacy guarantees either on labels only or jointly on features and labels, in terms of $\rho$-zero-concentrated differential privacy. The main technical contribution is a deterministic equivalent of the test error, which captures the interactions between sample size, covariance structure, model shift, regularization and privacy noise. Our theory allows to optimize hyperparameters (weights and ridge regularizers) and, more broadly, to decide when private external datasets are useful without accessing the data itself but only relying on population-level quantities. This provides a theoretically tractable foundation for private transfer learning, which we support via experiments on both synthetic and real-world datasets.

This paper studies the finite-sample bias of plug-in estimators for joint entropy, conditional entropy, and mutual information for finitely supported discrete random variables. Using the univariate reduction framework developed in our companion works, we derive explicit first-order bias formulas. For a pair (X,Y) with support sizes r and s, the biases are respectively -(rs-1)/(2n), -r(s-1)/(2n), and (r-1)(s-1)/(2n). We then propose bias-corrected estimators for the three measures and prove that the uncorrected and corrected versions are asymptotically equivalent, differing by a deterministic term of order O(1/n); consequently, they share the same asymptotic distribution. A simulation study validates the theoretical results, including a dedicated independence study and the relationship with Wilks' theorem. An application to a motor insurance portfolio, where X is the driver's age class and Y the claim severity class, illustrates the improvement brought by the bias correction for risk classification.

In multiple-baseline experiments, staggered adoption designs, and stepped-wedge trials, every unit switches once from baseline to treatment at a randomized onset time and remains treated afterward. The treatment onsets in these designs are absorbing. Since such experiments often involve few heterogeneous units, randomization tests of the sharp null of no effect on any unit-period cell are common. Rejecting this null, however, implies that some effect exists somewhere, not that it is large, positive for most unit-periods, or persistent. To improve on this limitation, we study what can be established via randomization tests for bounded and quantile nulls on cell effects in absorbing onset designs. Under the assumption that a cell's potential outcome depends on its current treatment status and not on when treatment began, we establish finite-sample validity for both tests. We then develop the power theory for the bounded-null tests with a fixed number of units $N$ and a growing series length $T$ admitting $K_T$ onsets. The $p$-value against a fixed alternative decays as $K_T^{-N}$ up to a factor polylogarithmic in $T$, with lower bounds of the same order, so the onset window, not the series length, determines the power. Theory-implied guidelines for experiment planning are also provided.

Inference-time control steers a pretrained generative model towards a target distribution without retraining. We study tilted targets $\pi_0\propto G_0\,p_0$, where $p_0$ is the sampler output distribution and $G_0$ is an evaluable reweighting function. Existing approaches rely on sequential annealing with sequential Monte Carlo (SMC) or parallel annealing with replica exchange (RE). Sequential control is exact but needs large particle populations, whereas no exact parallel control method exists: existing RE corrections approximate an intractable time reversal and are biased. We propose Exact Non-equilibrium COntrol with Replica Exchange (ENCORE), the first exact parallel control method. Each replica stores its generation trajectory, so the upward move is a truncation and the intractable time reversal is never simulated. We prove target invariance and show that the resulting dynamics are those of non-equilibrium replica exchange with the exact time reversal as forward proposal. Under regularity conditions, our diffusion analysis shows that both sequential and parallel control become unstable under refinement of the time discretisation without guidance, whereas guided proposals remain stable and yield diagnostics for tuning the schedule and the computational budget. Across synthetic targets, Boltzmann sampling of biomolecules, and image generation, ENCORE achieves competitive accuracy and diversity, remains robust to sampler perturbations, and applies to distilled samplers where existing RE corrections are unavailable.

Why does supervised fine-tuning (SFT) lead to more forgetting than reinforcement fine-tuning (RFT), even when all teacher demonstrations are semantically correct? We study this question on classification tasks where tokens within each semantic class express the same semantic answer in different styles. The tasks share an underlying semantic rule but differ in their prompt distributions and teachers' stylistic preferences. Using a tractable linear-softmax policy, we derive an exact decomposition of the updates into semantic and style components. We show that, at a common policy and prompt, SFT and RFT have parallel semantic updates but differ in their style dynamics. Starting from a policy with no within-class style preference, RFT with exact policy gradients preserves this symmetry, whereas SFT with a nonuniform teacher develops off-axis style drift along a nonzero task mean under population updates. We use this drift to establish a separation under explicit conditions: for population updates from a common perfectly fitted checkpoint, SFT forgetting admits a strictly positive lower bound over a finite training interval, while RFT retains zero semantic error. Simulations over task sequences support these theoretical predictions.

Differential privacy (DP) admits several notions, but the choice among them may affect both privacy analysis and utility. In this paper, we consider four mainstream curve-based privacy notions within a unified information-theoretic framework. For a fixed ordered pair of output distributions, we establish information equivalence among the two directional privacy profiles of $(\varepsilon,\delta)$-DP, the pair of hypothesis-testing trade-off functions, and the extended privacy-loss distribution. The exact Rényi differential privacy (RDP) curve joins this equivalence class whenever it is finite at some order greater than one. Under this mild condition, choosing among these notions changes only their semantic interpretation and computational requirements. In contrast, taking the maximum of the directional privacy profiles or compressing the RDP curve into a single zero-concentrated differential privacy (zCDP) parameter can lose information. We quantify the information loss between the exact RDP curve and its zCDP bound for standard noise mechanisms. This gap is zero for Gaussian noise but generally positive for Gaussian-mixture, Laplace, discrete Gaussian, and Poisson-subsampled Gaussian mechanisms. Moreover, this gap grows linearly with the number of independently composed mechanisms. Our information-theoretic perspective has practical consequences. At the same certified privacy level, retaining the full RDP curve rather than using zCDP reduces the required noise variance by up to $45\%$ for Gaussian-mixture noise in workloads comparable in size to the American Community Survey. For DP-SGD on Fashion-MNIST under Poisson subsampling, an RDP-based privacy accountant improves test accuracy by up to $8.73$ percentage points compared to a zCDP-based accountant when both are calibrated to the same $(\varepsilon,\delta)$ guarantee.

Sampling from the posterior is the central task of computational Bayesian inverse problems. The standard workhorse in Bayesian inference - Markov chain Monte Carlo (MCMC) - is sequential, yields correlated samples, and must be rerun for each observation. Conditional flow matching offers an amortized alternative: a transport map, trained once on joint samples of parameter and data, that yields independent approximate posterior samples for any observation at negligible online cost, without new likelihood evaluations. We give a careful, MCMC-literate assessment of flow matching for PDE-based inverse problems with function-valued parameters. Exploiting the flow's tractable density, we derive computable accuracy estimates of the underlying approximate posterior in total-variation distance and Kullback-Leibler divergence and, moreover, propose a hybrid sampler that is asymptotically exact by Metropolization. We validate the accuracy estimates and demonstrate the amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka-Volterra model.

Recent advances in deep learning for time series prediction have amplified the need for reliable uncertainty quantification. Conformal prediction has gained attention as a distribution-free framework for constructing prediction intervals with coverage guarantees. However, its coverage guarantees rely on data exchangeability, an assumption generally violated in time series. Active research has focused on developing conformal prediction methods for time series that overcome this limitation. While deep sequence models, such as recurrent neural networks and Transformers, have often been used in conformal prediction for time series, limited work has systematically studied how deep sequence models can be utilized in conformal prediction for time series. In this work, we systematically investigate the use of deep sequence models in conformal prediction for time series through three approaches: conditional quantile regression, conditional quantile function estimation, and localized conformal prediction. We provide a theoretical analysis establishing asymptotic conditional coverage guarantees for all three approaches under suitable assumptions. Through comprehensive experiments on real-world datasets, we demonstrate the effectiveness of leveraging deep sequence models into conformal prediction for time series.