Papers

Background Decision curve analysis (DCA) evaluates prediction model usefulness using net benefit (NB). Population-level NB is often interpreted as a proxy for population-level expected utility, but this assumes comparability of utility across subgroups. We aimed to characterize when subgroup utility heterogeneity may invalidate population-level DCA conclusions. Methods We compared prediction-driven treatment with default strategies in populations containing subgroups with different utility values. We derived an inconsistency region: combinations of subgroup-specific {\Delta}NB values for which population-level NB and utility favor different strategies. We also developed a practical robustness framework. Results Opposite signs of subgroup-specific {\Delta}NB provide a warning signal for possible inconsistency. The inconsistency region is larger when subgroup sizes are more similar and subgroup-specific \(a-c\) values are more different, where \(a-c\) is the incremental utility of a true positive relative to a false negative. When \(a-c\) differs across subgroups, population-level NB combines quantities on different implicit utility scales and may conflict with population utility. A real-world case study illustrates the problem. Conclusions Using population-level NB as a proxy for population utility implicitly assumes homogeneous \(a-c\) across subgroups. Subgroup DCA and our framework can identify and assess when utility heterogeneity may invalidate population-level conclusions.

In observational causal inference, covariate balancing is widely used to reduce source-target covariate shift, but under weak overlap in high dimensions, stronger balance can induce concentrated weights and increase variance. Balance measures how well the target covariate distribution is represented, but does not by itself determine how reliably the counterfactual mean can be estimated. We develop risk-calibrated balancing for the average treatment effect on the treated, which applies ridge augmentation to any normalised base weights and selects its penalty using conditional prediction risk of the counterfactual mean. Under a random-effects predictive model, we derive an exact finite-sample decomposition of this risk into residual covariate imbalance and weight-induced variance. For design-independent base weights under proportional asymptotics, we characterise how limiting risk depends on source and target covariance geometry, population mean shift, and weight concentration. For covariate-adaptive base weights, we develop a uniformly consistent target-aware risk estimator whose minimiser attains vanishing scaled oracle excess risk. Simulations show that the high-dimensional risk predictions remain informative for adaptive balancing and that target-aware tuning generally reduces excess target risk. Empirical analyses of job-training and single-cell perturbation data show that risk-calibrated balancing generally improves on the corresponding base estimators, with larger gains under weaker overlap.

Classical statistical theory is insufficient to explain the successes of general-purpose AI models, because it depends on handcrafted inductive biases that it cannot justify. No Free Lunch (NFL) theorems force any learner that beats chance on some environments to underperform on others. We might hope that past experience informs which environments to expect, but NFL applies equally to meta-learning. Thus, any method that makes meaningful predictions necessarily begins with an inductive bias external to the data. Choosing to bias toward short programs yields Solomonoff induction (SI), whose performance is competitive against all computable learners - albeit up to "constants" that become large when comparing against specialized methods that exploit background information. We therefore relativize SI to an information vantage point, biasing toward short programs with access to all preexisting information. This reframes the inductive bias: instead of seeking some absolute notion of simplicity, we favor accessibility with respect to our vantage point. An algorithm can only outpredict the relativized SI to the extent that its code contains additional information about the data, and no algorithm can generate such information. While SI is incomputable and hence not a practical algorithm, it provides a formal optimum for inference in the limit of infinite compute, and there is evidence to suggest that frontier AI systems roughly approximate it. Thus, the only known answer to meta-NFL is rooted in algorithmic information theory, which we should expect to play a fundamental role in explaining the generalization behavior of modern (and future) AI systems.

On-policy distillation (OPD) is becoming an important component of large language model (LLM) post-training for transferring the reasoning capability of a strong teacher LLM to a weaker student LLM. OPD trains the student by minimizing the reverse KL divergence between the teacher and the student via rollouts generated by the student's policy. However, estimating the gradient of the reverse KL divergence in OPD remains a challenge. Using only the sampled token from the student-generated rollout is computationally cheap but provides limited distributional supervision, which will degrade accuracy. In addition, using the full vocabulary provides complete distributional supervision but is computationally expensive. Therefore, recent works propose Top-$k$ OPD (TK-OPD) that use selected top-$k$ tokens, which provides richer distributional supervision than sampled-token estimation at substantially lower computational cost than full-vocabulary estimation. Unfortunately, using only the selected top-$k$ tokens induces bias, leading to accuracy degradation, as the probability mass outside the selected top-$k$ tokens is discarded. To address the bias of TK-OPD, we propose Tail-Corrected Top-$k$ On-Policy Distillation (TT-OPD). It preserves the advantages of TK-OPD, including rich distributional supervision and low computational cost, while providing an unbiased estimator of the gradient of the reverse KL divergence. The key insight of TT-OPD is to use not only the selected top-$k$ tokens, but also the sampled token from the student-generated rollout, thereby recovering the discarded probability mass in expectation, avoiding the bias. Experimental results demonstrate that TT-OPD significantly outperforms other tested OPD variants.

Non-stationary online learning has attracted much attention in recent years, as static regret is insufficient to guide algorithm design in changing environments. To address this limitation, interval regret and dynamic regret have been introduced as two representative performance metrics that strengthen static regret in complementary directions. Interval regret requires an online algorithm to achieve competitive static regret over every local time interval, whereas dynamic regret evaluates performance against an arbitrary sequence of time-varying comparators. Despite their importance, the relation between these metrics has long remained unclear. Prior work has often regarded interval regret as the stronger notion, based on the intuition that local guarantees should naturally induce global guarantees. Consequently, it is widely conjectured that an algorithm with optimal interval regret should automatically attain optimal dynamic regret. In this paper, we first establish a negative result that refutes this intuition of a metric-level implication. Specifically, for both convex and curved functions (including exp-concave and strongly convex functions), we show that there exist instances in which an algorithm with optimal interval regret nevertheless fails to achieve optimal dynamic regret. We then show how to leverage local adaptivity to obtain optimal dynamic regret. In particular, optimal dynamic regret can be attained by invoking an interval regret minimization process over an enlarged Euclidean ball containing the original convex feasible domain and using a suitable domain-converted surrogate loss. This reduction applies to both convex and curved functions. As a byproduct, we obtain the first proper and efficient algorithm with optimal dynamic regret for exp-concave functions, improving prior results while significantly simplifying the analysis.

Sampling latency compounds in diffusion workflows, where users generate and discard many candidates before keeping one. Surprisingly, the poor outputs of standard few-step samplers do not reflect a lack of reconstruction capacity: by optimizing only the initial noise, frozen 3-4-step samplers can closely reproduce their corresponding full-step outputs. Building on this finding, we learn corrections to the initial noise and denoising updates using endpoint supervision, improving correspondence with full-step outputs generated from the same noise and prompt. The resulting previews allow users to screen candidates cheaply and reserve full-step generation for promising ones. Input correction also transfers across sampling budgets without retraining. Experiments show substantial improvements in reference fidelity, including 53-78% lower reconstruction MSE than retrained LD3 on unconditional benchmarks, alongside improved ranking preservation and candidate selection on SD1.5, SDXL, and FLUX.1-dev.

Language model post-training is often bottlenecked by the need for human-collected preference data, which is expensive and difficult to scale. Reinforcement learning from AI feedback (RLAIF) style approaches that leverage pseudo labels offer an abundant alternative but introduce systematic biases that degrade downstream alignment. Recent general-purpose semi-supervised methods correct for teacher bias using a small set of human-labeled examples, but suffer from high variance especially when human annotations are scarce. To this end, we propose ABC-Align, leveraging abundant pseudo label signal to minimize variance and applying a lightweight, adaptive correction grounded in the human-labeled subset. The correction strength is tuned automatically during training using plug-in estimates of the relevant bias–variance quantities. On LLM alignment with RLHF, DPO, and GRPO where human feedback is scarce, we empirically demonstrate that ABC-Align achieves superior performance over prior semi-supervised baselines in a series of experiments on an increasing scale. Our code is available at https://github.com/SewoongLab/abc-align .

Split Federated Learning (SFL) enables resource-constrained clients to participate in collaborative training, but vanilla SFL exchanges smashed data and gradients at every batch, which incurs significant communication overhead. Recent methods reduce this overhead with an auxiliary network at the client-side cut layer. However, we identify that this approach makes the client optimize a local objective that differs from the end-to-end objective, which fundamentally limits the collaborative training between the client and the server. We propose Compensated Feedback based SFL (CoeF-SFL), a communication-efficient framework that retains the end-to-end objective without any auxiliary network. In CoeF-SFL, the client and the server exchange the smashed data and the gradients once per round and reuse them during local training. Since this reuse makes the gradients stale on the client side, we compensate them with a curvature-based correction in the activation space and develop two variants. CoeF-D approximates the Hessian with a diagonal gradient outer product, while CoeF-J exploits the tractable Jacobian-based Hessian of a surrogate loss that upper-bounds the true loss. We provide the theoretical background of each method, characterizing its compensation. Across vision and language tasks, model capacities, cut layers, and data distributions, CoeF-SFL significantly outperforms auxiliary-network-based methods under the same communication frequency, and the improvement is most substantial on vision tasks. Code is available at https://anonymous.4open.science/r/CoeF-SFL-2686/README.md

Building on the James–Stein approach to leading eigenvector estimation (Goldberg and Kercheval, Proc. Natl. Acad. Sci. USA 120, e2207046120, 2023), we develop a data-adaptive augmented James–Stein shrinkage framework for estimating leading eigenvectors and eigenspaces under a generalized spiked population model, in the high-dimensional regime where the dimension $p$ and sample size $n$ grow proportionally. For each spiked eigenvector, we construct an augmented target subspace that combines auxiliary information, either from domain knowledge or prior information, with the remaining sample spiked eigenvectors. This augmentation allows information shared across the sample spiked components to be exploited while retaining a fully data-driven shrinkage rule. We show that the resulting eigenvector estimator strictly improves upon standard PCA whenever the target subspace contains nonvanishing information about the population eigenvector, while asymptotically reverting to PCA when the target is uninformative. The individual estimators further yield a nested sequence of estimators for all leading spiked eigenspaces, with analogous dominance properties. The proposed estimator also strictly improves upon the existing HDLSS-motivated shrinkage estimator in the proportional high-dimensional regime. Simulation studies demonstrate substantial finite-sample gains and robustness to misspecification of the number of spikes.

Linear mixed-effect panel data models are widely used in longitudinal biomedical, environmental, and social studies, but are often sensitive to data contamination and numerous covariates. To address these challenges, we propose a robust variable selection approach based on a least squares approximation (LSA) of the density power divergence (DPD) objective function combined with the Adaptive LASSO penalty. The LSA converts the nonlinear DPD objective into a computationally efficient quadratic approximation while preserving the robustness of DPD estimation. Under suitable regularity conditions, the proposed DPD Adaptive LASSO-LSA estimator is shown to possess oracle properties, including selection consistency and asymptotic normality. Simulation studies demonstrate improved robustness, better sparsity recovery, and significantly enhanced computational efficiency than penalized likelihood methods, while maintaining performance comparable to existing DPD-based approaches. An application to the SWAN bone mineral density dataset illustrates the practical relevance of the proposed methodology for robust estimation and reliable identification of important covariates.

Functional data are intrinsically infinite-dimensional, and often exhibit phase variation, where corresponding events occur at different times across observations. Existing linear dimension reduction methods struggle with nonlinear amplitude variation, while functional autoencoders without an explicit warp entangle temporal misalignment with shape. We propose the Amplitude–Phase Functional Autoencoders (AP-FAE), an unsupervised framework for functional data that spans both univariate and multivariate cases, with emphasis on the multivariate setting, and factorizes the latent space into separate amplitude and phase embeddings derived from all channels. A smooth functional decoder reconstructs channel-specific amplitude functions in canonical time, and a shared monotone, endpoint-preserving warp captures phase variation. We prove a bound linking amplitude recovery to registration, reconstruction, and noise errors, and validate it numerically. Across synthetic data and six real-world benchmarks, AP-FAE outperforms state-of-the-art baselines on most clustering and alignment metrics and on all reconstruction metrics. Clustering with amplitude embeddings alone consistently surpasses joint amplitude–phase clustering, confirming the benefit of explicit disentanglement. Code is available at https://anonymous.4open.science/r/APFAE-418C/}{https://anonymous.4open.science/r/APFAE-418C/.

Formulaic alpha discovery is a core challenge in quantitative trading, as identifying alphas that work well together remains difficult. Recent reinforcement learning (RL) methods formulate this task as a Markov decision process (MDP), but two important issues remain unresolved. First, as the alpha pool evolves, the reward function changes accordingly, making the MDP inherently non-stationary. Second, most existing methods optimize a single objective, typically predictive power, while ignoring other important properties of a high-quality alpha pool. Motivated by these challenges, we propose AlphaPareto, an RL method for formulaic alpha discovery. To address non-stationarity, AlphaPareto augments the state to include both the alpha under construction and the current alpha pool, and applies a large language model (LLM) to encode the pool. This design allows the agent to adapt to the evolving search environment. To overcome the limitation of single-objective reward design, AlphaPareto replaces the scalar reward with a multi-objective vector-valued reward that simultaneously captures predictive power, temporal stability, perturbation robustness, and diversity, and optimizes these objectives through a Pareto-regularized learning procedure. Empirical applications to real-world datasets show that our AlphaPareto method outperforms its competitors.

Multivariate time series anomaly detection (MTAD) is crucial for ensuring the safe and reliable operation of complex systems. Many existing methods learn normal patterns by training reconstruction or forecasting models on predominantly normal data. However, a large portion of these approaches rely on deterministic models and their associated point-wise output errors for anomaly scoring. Since real-world multivariate time series are inherently stochastic due to measurement noise and intrinsic system randomness, purely error-based scores can be unreliable, as large errors may arise from benign fluctuations rather than true anomalies. Probabilistic approaches address this limitation by quantifying uncertainty in model outputs. In particular, probabilistic state-space models (PSSMs) provide a principled framework by modeling stochastic system dynamics through latent state transitions and measurement noise via emission models. Despite this advantage, existing PSSM-based MTAD methods often struggle to capture long-range temporal dependencies and inter-variable dependencies, as they typically rely on noise-sensitive recurrent architectures and lack explicit cross-variable structure modeling. To address these limitations, we propose Graph-Transformer-Enhanced Probabilistic State-Space Model (GT-PSSM), a novel PSSM-based MTAD method that tightly integrates PSSM-based probabilistic modeling of stochastic dynamics with Graph Transformer-based learning of temporal and inter-variable dependencies. By jointly modeling stochasticity, long-range temporal dependence, and variable interactions within a unified probabilistic framework, GT-PSSM enables more robust anomaly detection.

A non-significant coefficient in a dynamic panel model need not imply the absence of a relationship. It may instead reflect heterogeneous effects averaged toward zero, reciprocal dynamics overlooked by a recursive specification, or relationships masked by the omission of correlated covariates. Standard linear estimators cannot distinguish among these possibilities. We develop an inferential workflow for political panel data that resolves this ambiguity by combining flexible autoregressive estimation, forecast-necessity testing, functional characterization, and same-data linear benchmarking. The workflow first identifies relationships required for out-of-sample prediction, then characterizes their functional form across political contexts, and finally, distinguishes differences arising from estimator flexibility from those due to model specification. Applied to the causal sequence model of democratization on the V-Dem panel of 113 countries, the workflow reproduces the model's central finding - the protective belt of civil society, the rule of law, and institutionalized parties - while recovering reciprocal relationships from democracy to its institutional supports that a linear model cannot detect. Most importantly, three weak published direct effects, of which two are null, and one is marginally significant, receive three different diagnoses: one dissolves under the full specification, one reflects heterogeneous effects averaged toward zero, and one was masked by the reduced variable set. The workflow corrects the published record in both directions, removing one relationship and recovering two. More broadly, the workflow provides a framework for evaluating dynamic political theories under a model class capable of representing nonlinear and reciprocal mechanisms while preserving relationship-level interpretation and explicit inferential standards.

What determines the unavoidable sample cost of learning cyclic causal structure? For cyclic linear non-Gaussian models, we study exact condensation recovery from observational data: identifying the strongly connected component (SCC) partition and all edges between components. We establish the first information-theoretic lower bounds on sample complexity for this target. For $p$ variables, maximum SCC size $s_{\max}$, and maximum external-parent count $d_B$, any estimator requires order $s_{\max}\log(ep/s_{\max})+d_B\log(ep/d_B)$ samples in the worst case over a regular model class. These bounds distinguish the costs of SCC membership and external-parent selection. Under principal invertibility and without correlation faithfulness, we establish a population block-exogeneity principle that identifies unknown root SCCs through residual independence and inclusion minimality. A sparse-adjustment characterization shows that small adjustment sets suffice to identify SCCs and their direct external parents, without regressing on all previously recovered variables. These characterizations yield BlockExo, which attains a structurally matching sample bound without knowing $s_{\max}$ or $d_B$ under suitable conditions. Simulations support the structural dependence of our sample bound and demonstrate BlockExo's sample-efficient recovery in comparisons with other methods for cyclic causal discovery.

Non-negative matrix factorization (NMF) is a singular statistical model whose Bayesian asymptotics are governed by the real log canonical threshold (RLCT). We study the local geometry of the factorization map and derive an upper bound for the RLCT of NMF. Let $H$ be the model inner dimension and $H_0$ the non-negative rank of the true $M\times N$ matrix. Assuming that the true matrix admits a strictly positive factorization of inner dimension $H_0$ in the interior of the parameter domain, we prove, for smooth positive priors, that $\lambda\leq \{(H-H_0)\min(M,N)+H_0(M+N-H_0)\}/2$. This bound strictly improves the previous bound when $H_0\geq3$. The proof uses a local analytic normal form that separates independent linear coordinates from a residual matrix product. When $H=H_0$ also equals the ordinary rank of the true matrix, we obtain the exact value $\lambda=H_0(M+N-H_0)/2$. Under the standard assumptions of singular learning theory, these results bound the leading coefficients of the expected Bayesian generalization error and the Bayesian free energy.

Retrieval-augmented generation (RAG) can ground large language models in external evidence, but retrieved context does not guarantee that generated claims are factually supported. This problem is especially relevant in multi-hop RAG, where retrieval and reasoning proceed through multiple dependent stages. We study whether claim-level conformal factuality control, previously developed for RAG, remains effective in this setting. We apply split-conformal claim filtering to multi-hop RAG and evaluate it on HotpotQA, Natural Questions, and TriviaQA using Llama 3.1 8B and GPT-4o-mini, together with a single-hop reference experiment. Across all six multi-hop model-dataset configurations, increasingly stringent conformal targets consistently increase the fraction of responses whose retained claims are fully supported. At the 95% target, this rate ranges from 95.80% to 97.20%, compared with 55.60%-76.03% without filtering. However, the improvement is strongly selective: only 4.41%-31.09% of generated claims are retained and 9.70%-51.40% of responses remain non-empty at the 95% target. These results show that conformal factuality extends to multi-hop RAG, while demonstrating that nominal reliability must be interpreted jointly with claim retention and abstention.

We test whether a multivariate vector X conforms to a specified distribution F, a problem in copula modelling and density forecasting. The Rosenblatt transform reduces it to a test of uniformity, but depends on an arbitrary coordinate ordering that strongly affects power under dependence. We study order randomization: applying the transform under many random orderings and merging the evidence with dependence-robust rules. Reordering conserves the total Mahalanobis signal energy and merely redistributes it, so one ordering is a lucky or unlucky draw. In simulations we observe significant gains in calibrated power over both the expected single random ordering and order-invariant references. Two ingredients are essential: a two-sided base statistic, and a re-estimating parametric bootstrap that restores level under an estimated null and unlocks the gain. The calibrated pooled tests are robust to the departure's shape; no order-invariant reference we compare is: the symmetric-root test collapses on diffuse departures, while the shape-flat chi-squared test trails on concentrated ones. We apply it to a Gaussian foreign-exchange risk model over a decade of daily data on nine currencies, where it detects episodes such as Brexit and COVID. Though we focus on Gaussian nulls, the procedure extends to any null whose conditional distributions can be computed and simulated from.

Equivalent probability requests can lead to different decisions even when both reports are valid. We introduce probability contracts, a benchmark that connects exact finite-world posteriors, validated event alignment, interface coherence, and failure-aware decision evaluation. Across four model-interface configurations on 1,000 worlds, Kev has lower aggregate canonical posterior error than Jev but larger complement and coarsening residuals; accuracy ordering varies by stratum. Jev's Event and Choice interfaces change the binary action on 32.8% of valid pairs at defer cost 0.10. Post-hoc analyses show that disagreement certifies only 11-52% of mean binary pair error and does not consistently outperform confidence for selection. An action-region characterization and a standard scoring-rule identity explain averaging's expected Brier guarantee relative to random interface selection, but not a decision-loss guarantee at each cost. The loss contrast takes both signs on a 99-cost grid for every configuration; small penalties where both policies beat deferral have pointwise intervals containing zero. Secondary checks specified before collection include a separate 400-root cohort, where Event/Choice effects remain configuration-dependent. A joint surface-order and answer-ID intervention shifts posttrained probabilities. Both bounded reasoning arms yield no valid probability reports, leaving their probability accuracy undefined. Probability contracts make these distinctions measurable by evaluating event semantics, posterior error, coverage, and decision cost together.

This paper proposes Wasserstein Causal Forests (WCF) for settings in which each unit's outcome is itself a probability distribution. This study also defines finite-grid transformed average and conditional average treatment effects, including a reference-distance contrast that asks whether treatment moves unit-level distributions toward a prespecified benchmark. Simulations cover null effects, location and shape changes, limited overlap, equal-mean but different laws, heterogeneous effects, multimodality, and structural zeros. WCF is most accurate on the conditional-law metric in most reported designs and sharply improves reference-effect estimation in the principal location-and-shape settings, but it is less accurate than the forest baselines for multimodal settings. WCF is applied to the famous Project STAR \citep{word1990state}, revealing that small classes alter more than the mean: they raise within-grade mathematics achievement by $0.161$ standard deviations on average (SE $0.028$); but the gain is not a uniform location shift, it is larger in the upper part of the classroom score distribution ($+0.179$ at the ninetieth percentile versus $+0.091$ at the tenth) and, in descriptive stratum estimates, largest in the schools serving the most economically disadvantaged students (highest free-lunch quartile, $+0.262$, versus $+0.092$ to $+0.164$ elsewhere), while overall dispersion is essentially unchanged.

We present a three-tier progression of Hidden Markov Models: maximum-likelihood (Baum-Welch), variational Bayesian (VBEM), and a neural variational extension (Neural VBEM), that model each customer's transaction history as a trajectory through a small number of latent behavioural regimes, one of which is empirically identified as fraud-associated. The Neural VBEM HMM replaces the fixed Gaussian-multinomial emission family with a learned encoder, compressing a 741-dimensional transaction representation into a 64-dimensional latent space in which the VBEM HMM's posterior operates; a UMAP projection of this space reveals that the discovered regimes are not discrete clusters but ordered segments of a single continuous behavioural manifold, with confirmed fraud concentrated at its extreme. We show that the model's natural output, that is, the posterior probability of regime membership, is routinely mistaken for a fraud probability, and quantify the resulting miscalibration (the regime-membership interpretation error, MRIE); a corrected posterior-predictive score, closes most of this gap. We further distinguish batch (smoothed) inference, which uses look-ahead unavailable at deployment time, from filtered (forward-only) inference, and report both. On IEEE-CIS transaction data, the neural tier achieves a 14.4$\times$ fraud enrichment in its identified regime; while its AUPRC trails a discriminative XGBoost baseline, we show this gap is structural and not incidental, and argue the model is best positioned as a calibrated triage and interpretability layer rather than a drop-in ranking replacement.

Existing interpretable graph additive models still face limitations in either computational scalability or modeling flexibility. In terms of structural modeling, previous approaches either face quadratic scaling costs or sacrifice explicit source-to-target contribution decomposition. In terms of feature components, they rely either on per-feature neural networks or on single shared bases with limited feature specialization. We address both problems by introducing HARMONIA: Interpretable Graph Learning through Mixtures of Neural Bases, an interpretable-by-design framework. For feature modeling, HARMONIA introduces a Mixture of Neural Bases (MoNB), which routes features to specialized basis experts, enabling parameter sharing without sacrificing feature-specific specialization. For structural modeling, HARMONIA uses Relative Random Walk Probabilities (RRWP) to capture multi-hop and multi-path relationships, and proposes Sparse RRWP Aggregation (SRA) to compute these interactions through sparse graph propagation without quadratic pairwise complexity. HARMONIA retains a simple additive form in which predictions decompose into feature responses modulated by structural influence. Empirically, HARMONIA achieves stronger explanation recovery than existing interpretable graph baselines while maintaining competitive predictive performance and scaling to graphs with millions of nodes. These results show that interpretable graph learning can remain both faithful and scalable without sacrificing predictive effectiveness.

Testing whether two populations of networks share the same edge probabilities is a basic problem in network inference. How hard it is depends on the norm used to measure the difference. For the inhomogeneous Erdős–Rényi (IER) model, the optimal sample complexity is known for every integer $L_r$ norm and for $1\le r<2$. For non-integral $r>2$, however, the known upper and lower bounds do not match, and the lower bound was conjectured to be tight. We study this gap for two-sample testing on aligned vertices. We propose a test that runs two published statistics, of orders $2$ and $\lceil r\rceil$, on the same data and rejects if either one rejects. Its thresholds come from Hölder interpolation, so that both statistics have the same sample cost. We prove that this test attains the conjectured rate. Combined with earlier results, this shows that for every fixed $r\ge1$ the minimax sample complexity is of order $n^{\max\{4/r-1,\,2/r\}}/\epsilon^2$, even when the separation changes with $n$. In simulations with $n$ between 32 and 256, the number of graphs needed for 80% power at level $0.05$ grows with $n$ at a rate consistent with the theory. For $r=2.5$, for example, the fitted exponent is $0.78$, against the theoretical value $0.8$. Interestingly, the two statistics split the work as the interpolation argument suggests: the higher-order statistic is more powerful when only a few edges change, and the $L_2$ statistic when many edges change.

Plug-and-Play (PnP) algorithms are a class of iterative methods for inverse imaging. Within an optimization algorithm, they combine a flexible fidelity term, encoding the forward operator, and a pretrained image denoiser, in order to deal with more severe corruptions such as blurring or downsampling when reconstructing an image. This work studies provably convergent PnP methods for mixed-noise forward processes by using the infimal convolution as a fidelity term, providing a statistical interpretation as a joint maximum a-posteriori estimator over both noises, and preserving the Bayesian MAP interpretation of PnP methods. Independently, we extend the PnP formulation to multiple prior terms using the Davis–Yin three-operator splitting. This extension can be combined with either standard fidelities or the proposed mixed-noise infimal-convolution fidelities. We verify that these generalized PnP methods are convergent under standard Kurdyka–Lojasiewicz conditions. Numerical experiments on Laplace-Gaussian and Poisson–Gaussian noise demonstrate stable convergence of single-prior and multiple-prior PnP methods, and divergence under fidelity mismatch. Furthermore, PnP with infimal convolution fidelities are able to scale to noise up to 38% standard deviation, with multiple priors reaching different stationary points that qualitatively preserve more textural properties.

This paper conducts an empirical mediation analysis to examine how U.S. stock-market sectors responded to the 2025 tariff-policy episode. Using S&P 500 stock data along with hundreds of firm-level financial variables, we study whether sector membership was associated with differences in stock returns and whether these differences were statistically related to firm financial characteristics. For each of five economically distinct tariff-policy windows, we use high-dimensional linear causal mediation models to estimate the direct sector effects. We use a linear regression model to estimate the total sector effects, and then estimate the indrect sector effects by using the difference between the total sector effects and the direct sector effects. To accommodate a large number of candidate mediators, we use a partially penalized least-squares procedure that regularizes financial-variable coefficients while unpenalizes coefficients of all sector indicators. Wald and F-type tests are used to examine whether indirect and direct effects are significant or not, respectively. The results show that tariff news did not produce one uniform market response. Sector effects vary across five economically distinct tariff-policy windows. The clearest findings are in the direct sector effects. Technology was the sector most negatively affected during the initial decline and escalation collapse, but it also rebounded most strongly during the April 9 relief rally. The results indicate that in addition to Technology, Energy and other trade-exposed sectors were particularly affected during the initial decline and escalation phases. The selected mediators provide evidence that growth, investment, liquidity, profitability, and cash-flow characteristics helped characterize firms associated with sector-level return differences.

Conformal prediction provides marginal coverage guarantees, yet practitioners may wonder if the observed coverage is truly abnormal or consistent with sampling variation. Inference for realized coverage has received comparatively little attention, especially for time series. We study split conformal prediction with adjacent calibration and test sets of temporally dependent data. Using the functional dependence measure, we derive non-asymptotic bounds on marginal coverage error without mixing assumptions, which can be difficult to verify and may fail even for simple short-memory models. We establish a Bahadur representation to derive, to our knowledge, the first central limit theorem for realized coverage of split conformal prediction under temporal dependence. A consistent block-based estimator of the standard error yields an asymptotically justified test. We further study long-memory time series, which remain understudied in conformal prediction. For Gaussian linear processes, we show how very strong temporal dependence can lead to a non-Gaussian limiting law of realized coverage and establish block-sampling inference with an estimated normalization. The resulting theory explains how temporal dependence changes coverage uncertainty.

We study conformalized quantile regression and conformalized median regression that fit a model on one block of a time series and calibrate the conformal interval on the adjacent block. The existing theory of conformal prediction for time series rests largely on mixing conditions, which are hard to verify from a time-series model and fail for many standard processes, including simple ones with short memory. We replace this theoretical toolbox with the functional dependence measure, which in principle accommodates long-memory observations. The accuracy of the conformal interval length for time series has been understudied. To the best of our knowledge, this paper is the first work that establishes non-asymptotic coverage guarantees and accuracy of interval length simultaneously for split conformal regression on time series. Furthermore, for Gaussian linear processes with long memory, where both the estimation of the center and its calibration converge slowly, we establish a sharper rate for the length error. We show that the calibrated length converges faster than the estimated center itself, and provide a matching lower bound for the usual centers when the calibration block is sufficiently large relative to the training block. To our knowledge, this is the first theoretical analysis of conformal interval length dedicated to long memory.

The proportion mediated (PM), defined as the indirect effect divided by the total effect, is intuitive when direct and indirect effects operate in the same direction. Under inconsistent mediation, however, opposing pathways partially cancel each other, causing PM to fall outside the unit interval or become unstable as the total effect approaches zero. We introduce two complementary effect-scale measures: the Mediated Magnitude Share (MMS), which quantifies the absolute magnitude of an indirect pathway relative to the combined absolute magnitudes of the direct and indirect pathways, and the degree of suppression, which quantifies the fraction of this combined magnitude that is cancelled by opposing pathways. We establish the bounds and limiting behavior of both measures in the two-pathway case, show that MMS reduces to PM under consistent mediation, and extend both measures to settings with multiple indirect pathways. For binary outcomes, we extend the framework to odds-ratio and risk-ratio scales by using logarithms to convert multiplicative effects into additive ones, enabling direct application of MMS and the degree of suppression. Together, these measures provide bounded and interpretable characterizations of mediation systems in which pathway directions are not uniform.

Modeling temporal dependence and uncertainty is central to forecasting with world models. The Visual Bayesian Regression World Model combines visual features, physical histories and known covariates through interpretable regression, within a modular architecture supporting trend, seasonal and cycle dynamics. Visual compression reduces representation dimension, while Bayesian variable selection reduces active regression dimension. Posterior prediction combines forecasts across predictor subsets using their posterior probabilities as weights and accounts for parameter uncertainty and future disturbances. The model forecasts joint visual–physical states recursively and physical targets directly. Across four forecasting tasks spanning object motion, vegetation greenness and solar power, ViBR-WM achieves lower mean overall physical-target error than Temporal Straightening, ConvLSTM, PredRNN and SimVP on every task. Repeated fitting and resampling support these overall gains.

Weak-to-strong generalization (W2SG) occurs when a student trained on a teacher's predictions outperforms that teacher. We study when this happens under fully converged, ridgeless two-stage learning, with no early stopping, no explicit regularization, and no assumption that the student is more expressive than the teacher. In two-stage linear regression, a teacher is fit from $n$ labeled examples and a student is trained solely on the teacher's predictions on $m$ fresh, unlabeled inputs. Although both stages share the same hypothesis class and the same training rule, we show that the student outperforms the teacher exactly when $m$ lies in an explicit intermediate range: too few pseudo-labels leave the student without enough signal, too many let it inherit the teacher's noise. Under power-law covariance, we derive this range in closed form as a function of the spectral decay and noise level, including regimes where the improving region splits into two disjoint intervals of $m$. We then study a random-feature model in which the student has strictly more features than the teacher, and identify two regimes, again given by explicit thresholds: one where improvement occurs only for $m$ in a bounded interval, and one where it occurs only once the student width $N_S$ exceeds an explicit threshold. Both regimes are governed by a single "active-bottleneck" principle: whichever of $m$ or $N_S$ is scarcer controls how much teacher error is filtered out, while increasing the other resource only reduces estimation noise. Together, these results show that finite data and finite width can themselves regularize a two-stage learner, with no explicit mechanism doing so.