The Inference Report

Large language models can interpret natural lan- guage, yet robust decisions remain challenging. Jev-like models expose structured choices, but these interfaces do not directly provide numeri- cal values at a requested precision. We propose NUMERICJEV, a training-free numerical decod- ing algorithm that enables numerical output from any LLM with a Jev-like structured-choice in- terface. Surprisingly, on our arithmetic bench- mark, it outperforms direct selection from a can- didate list containing the correct answer by 2.93 percentage points (Figure 1). Our motivation comes from the observation that numerical range selection is itself a decision problem that Jev- like LLMs can address. NUMERICJEV recur- sively refines a range through a multiway deci- sion tree while retaining the original question in context, without parameter updates or hidden- state access. On a 100-value grid, a ten-way tree requires only two decision rounds. Range- normalized MAE is 1.84% versus 5.18% for di- rect choice. A separate three-date historical- index study yields 4.58% mean relative recall er- ror and 0% readout error when the value is sup- plied. Code is available at https://github. com/Bring-AI/jev-numeric.

We study Bayesian linear bandits with an isotropic Gaussian parameter, independent Gaussian candidate arms, and Gaussian reward noise when the horizon is proportional to the dimension. The normalized posterior uncertainty has an explicit limit that is uniform over all causal policies. Gaussian posterior identities then determine the limiting parameter overlaps without an assumed closure of the adaptive recursion. These results yield exact regret curves for Thompson sampling, posterior-mean greedy selection, and a family of policies that scale the posterior sampling covariance. The normalized realized cumulative regret converges in L1, uniformly on compact proportional-time intervals. A policy-uniform lower bound identifies the limiting optimal Bayes regret and proves that posterior-mean greedy selection attains it. Thompson sampling incurs a strictly larger leading regret; its instantaneous regret ratio relative to greedy selection lies between one and two and approaches two at long proportional horizons. Closed-form cumulative curves also identify a different comparison in the vanishing-noise limit. Finally, the instantaneous regret converges to a nondegenerate Gaussian decision-loss distribution, rather than to its mean. The analysis separates the amount of information acquired by a bandit policy from the quality of the decisions made using that information.

Deep clustering is a powerful approach for discovering meaningful structures in high-dimensional data by learning a low-dimensional latent representation prior to clustering. Despite its empirical success, assessing the statistical reliability of the resulting clusters remains challenging. Testing discovered clusters on the same data induces selection bias and invalidates classical $p$-values. Selective inference (SI) provides a principled framework for correcting this bias, but existing methods focus on clustering performed directly on the observed features. In this work, we develop an SI framework for deep clustering with a fixed pretrained encoder. The key challenge is that cluster assignments are determined through a nonlinear transformation from the original data space to the latent space, resulting in a substantially more complex selection process than in conventional clustering. Our method provides a computationally tractable way to account for this process and enables valid statistical testing of differences between clusters identified in the latent space. Synthetic experiments demonstrate that the proposed method controls the Type I error rate while achieving higher power than valid but conservative baselines, and genomic applications show that it can identify significant cluster differences while appropriately accounting for selection bias. Our framework provides a principled approach to quantifying the statistical reliability of structures discovered by deep clustering.

Federated learning (FL) lets EEG decoders learn from recordings of several subjects without pooling them. We consider two light EEG decoders, the Riemannian SPDNet and the Euclidean EEGNet. Both split into a trunk, which builds a latent representation, and a head, which classifies it. Inter-subject variability, however, makes a single shared FL model a poor fit for each subject. Personalised FL addresses this: all subjects learn a common trunk, and each subject keeps its own head. We adapt it for SPDNet and study its effects against standard FL and centralised training, with EEGNet as a Euclidean baseline. Experiments cover three motor-imagery datasets that span diverse regimes in channels, subjects and classes. We observe that personalised SPDNet reaches higher accuracy than both standard FL and centralised training, while converging in fewer rounds and communicating fewer parameters than standard FL. It also outperforms every EEGNet configuration on two of the three datasets, although centralised EEGNet outperforms centralised SPDNet.

Transformers achieve remarkable performance by jointly learning broad families of tasks during pretraining and adapting to unseen tasks from only a short prompt. Yet a rigorous mathematical and statistical understanding of this phenomenon remains limited. This paper aims to study how Transformers exploit shared cross-task structure and how this structure affects the sample complexity of in-context learning (ICL). Specifically, we characterize task-space complexity through covering numbers under a prescribed metric, thereby quantifying the low-dimensional cross-task structure without requiring an explicit parametric representation. The resulting cover provides a set of anchor functions, which we use to introduce a task-identification-and-evaluation procedure: context observations localize an unseen task among the anchor functions, and the response at a query is predicted by aggregating the corresponding anchor function query evaluations. For approximation, we explicitly construct a Transformer with Softmax attention to approximate this procedure. For generalization, we derive an error bound that separates the effects of the number of pretraining tasks and the prompt length. The scaling with respect to the number of pretraining tasks is governed by the intrinsic dimensions of the task space and input domain; once sufficiently many tasks are available, the dependence on the prompt context length becomes dimension-free. To the best of our knowledge, this is the first work to quantify cross-task complexity for general nonlinear task families and explicitly construct a Transformer that exploits their low-dimensional structure to perform ICL. Our theory provides a quantitative explanation of how joint pretraining across related tasks improves in-context generalization.

Predicting blood-brain barrier (BBB) permeability is critical for central nervous system drug discovery. Using the MoleculeNet BBBP dataset (n = 2039), this study systematically ablates molecular feature spaces to isolate featurisation from model architecture. We evaluate three feature families (Morgan fingerprints, RDKit physicochemical descriptors, SMILES bigrams) across four learning algorithms. Results demonstrate that predictive performance depends jointly on feature representation and algorithm. Dynamic Random Forest using combined features achieved the highest mean AUC (0.970, 95% CI: 0.963-0.977). Second, this optimal representation enables exploratory estimation of heterogeneous associations between molecular structure and BBB permeability using Generalized Random Forests. Constructing a pseudo-treatment from a LogP median split, we applied double/debiased machine learning to account for confounding. Orthogonalization substantially attenuates the heterogeneity detected by naive causal forests; no conditional effects remained significant after false discovery rate correction (smallest adjusted p = 0.082). Furthermore, orthogonalized feature importance shifted toward residual structural information in SMILES bigrams. Ultimately, once observed confounding is properly accounted for, evidence that LogP-BBB associations vary systematically across chemical space is insufficient. This underscores that feature representation and model architecture are coupled design choices, and that unorthogonalized causal forests risk overstating genuine treatment effect heterogeneity.

Selective classification enables a model to abstain from predictions on uncertain instances, but existing approaches typically reject them through confidence scores, predefined coverage constraints or instance-level distance measures. These approaches may overlook the collective geometric structure of difficult samples in learned representation spaces. We propose Guided Clustering-based Uncertain Learning (GCUL), a geometric-guided selective classification framework that identifies misclassified and ambiguous instances as a potential confusion attractor in the representation space. GCUL uses a three-phase procedure to initialize, cluster, and explicitly relabel this uncertain region, allowing the rejection boundary to emerge from the underlying representation geometry rather than from a prescribed rejection rate. We further derive a selectivity score and a geometric sufficient condition that characterizes when rejection can provide positive operational utility, enabling pre-deployment feasibility assessment. GCUL improves DistilBERT accuracy from 89.37 percent to 94.98 percent with less than 9 percent rejection. Beyond accuracy, our selectivity score correctly pre-detects the only dataset (GoEmotion) where all baselines fail, and controlled simulations yield 6.1 percent Type-I and 0 percent Type-II errors, validating the sufficient condition's conservatism. These results suggest that collective representation geometry provides a useful alternative perspective for selective prediction.

In this paper, a method for achieving exact unlearning for Gibbs supervised learning algorithms is proposed using a variational formulation inspired by empirical risk minimization subject to relative entropy regularization (ERM-RER). Such a method consists of maximizing the expected empirical risk over the dataset to be unlearned subject to a regularization by relative entropy with respect to the original algorithm. The optimization variable is a probability measure on the models; and the solution is another Gibbs probability measure that represents a new Gibbs supervised learning algorithm. The method guarantees exact unlearning in the sense that the new Gibbs algorithm coincides in distribution with the algorithm that would have been obtained by retraining from scratch on the dataset to be retained. As a byproduct, a framework for reweighting data points in ERM-RER by strategically choosing both the reference measure and the regularization factor is obtained. In this framework, exact unlearning is the special case in which zero-weight is assigned to the contribution of the data points to be unlearned. More generally, depending on the choice of certain parameters, data points can be up-weighted or down-weighted in ERM-RER problems for particular purposes, e.g., controlling the generalization error of Gibbs algorithms. This paves the way for new constructive or adversarial views on classical reweighting data points in ERM-RER.

We study the detection of LLM-generated text under editing and contamination. Modeling human and machine text as finite-order Markov processes with Huber contamination, we characterize an exact boundary for reliable detection under our assumptions. Detection is impossible when contamination is sufficiently large relative to clean-source separation. Below this boundary, a collection of clipped likelihood-ratio tests achieves vanishing worst-case errors. This construction motivates clipping as a simple modification of existing statistical detectors. For a broad class of additive scores, we identify conditions under which the clipped test is consistent while the raw test's worst-case power tends to zero. We evaluate seven detectors across three datasets and three generation models, and on the RAID benchmark. Clipping improves robustness in both studies, with gains varying across detectors and contamination settings. For example, at a target false-positive rate of 5\%, clipping improves the log-likelihood--log-rank ratio (LRR) detector's true-positive rate by a median of 8.3 percentage points in the controlled study and 2.1 and 4.3 points in rate- and attack-specific RAID evaluations, respectively.

A central goal when designing treatment policies is often to "do no harm", that is, to avoid interventions that improve average outcomes while worsening outcomes for some individuals. A widely used notion for harm is the fraction of negatively affected (FNA), defined as the probability that an intervention decreases an individual's outcome. However, in many applications, treatments operate through mediators, and a single "total" FNA can obscure whether harm arises primarily through direct pathways or indirect (mediator-induced) pathways. In this work, we introduce a path-specific analogue of the FNA. For this, we disentangle total harm into direct and indirect harm in causal mediation settings. However, these quantities depend on joint distributions of potential outcomes that are not point-identified even in randomised controlled trials. As a remedy, we develop a novel partial identification framework for direct and indirect FNA. In our framework, we (i) derive sharp Makarov bounds for the FNA, and (ii) propose a semiparametrically efficient estimator with valid confidence intervals for these bounds under mild margin conditions. We demonstrate our framework across various numerical experiments. To the best of our knowledge, we are the first to study path-specific decomposition of causal harm and to develop an orthogonal inference framework for its analysis.

Matrix completion, the problem of estimating missing entries in a matrix from noisily observed ones, underlies a diverse array of problems such as recommender systems and counterfactual outcome estimation in panel data. Many algorithms address the problem using regularized least squares, often with the nuclear norm as a regularizer, but this method yields a point estimate with no built-in uncertainty quantification. A Bayesian formulation is a natural alternative, and if the noise variance is known, the nuclear norm-based prior yields a log-concave posterior. Unfortunately, in practice, the noise variance will not be known a priori, so for a fully Bayesian approach, a prior must be imposed on it. We give the first sampler for this model with an explicit non-asymptotic guarantee: polynomial in the matrix dimensions and in the reciprocal of the target accuracy. Our technique is to discretize the distribution of the noise precision onto a grid and build a categorical posterior via thermodynamic integration. This extension is not specific to matrix completion and may be useful in other non-log-concave sampling problems where the non-log-concavity is restricted to a single variable and the joint distribution of the remaining variables is nonsmooth. Our contribution is a feasibility result: we show that a polynomial-time Bayesian sampler for this model exists at all, and the resulting complexity, while polynomial, is not intended as a deployable algorithm at current problem scales.

Sparse Gaussian processes achieve $O(N)$ inference by replacing the kernel with an appropriate expansion in a fixed basis $\{φ_j\}$ on the input space. Given a compute budget $M \ll N$, practitioners conventionally truncate the basis to its first $M$ entries. Nothing in the formalism, however, prevents one from selecting only those $M$ basis functions that matter for the data at hand. This would avoid spending budget on basis functions where there is no signal, but it requires a criterion for ranking the candidates. We propose three such criteria derived from an information-theoretic view of the basis-function selection problem. Each criterion matches a different state of knowledge at selection time: a no-data state, a no-prior state, and an in-between state. We then study the performance of truncation versus selection strategies on six UCI regression benchmarks across three basis families: Hilbert-space Gaussian processes (HSGP), variational Fourier features (VFF), and variational inducing spherical harmonics (VISH). We observe that the no-data criterion is a safe default, matching or improving on truncation for HSGP, VFF and VISH, with substantial gains for VISH and improvements over a recently developed selection heuristic for that basis family. The data-aware no-prior and in-between criteria provide substantial gains over truncation specifically for HSGP, which is the most broadly used of the three families in practice.

We study online inverse linear optimization with a fixed unknown linear utility: in each round, an environment presents a compact action set, the learner recommends an action from it, and the environment returns an action that maximizes the utility over the same set. When the utility vector and the actions lie in the $d$-dimensional Euclidean unit ball, we give a randomized algorithm whose regret---the cumulative utility shortfall relative to optimal actions---is $O(\sqrt d)$ in expectation for every time horizon, without knowledge of the horizon. The dependence on $d$ is optimal up to a constant factor by the known $Ω(\sqrt d)$ lower bound for horizons $T\ge d$. Our algorithm maintains matrix multiplicative weights on polynomial feature spaces at geometrically spaced scales. It selects a recommendation distribution by solving a linear program and updates its score matrices by comparing the available actions with the feedback action. With rational oracle outputs and feedback actions, an implementation computable relative to a linear-optimization oracle preserves the $O(\sqrt d)$ regret bound. Whether the same rate is attainable with running time polynomial in the dimension, horizon, and input length remains open.

Estimating the causal effects of medical treatments is difficult when clinically important outcomes are costly to measure or require long follow-up. Short-term or inexpensive surrogate outcomes offer a potential alternative, but surrogate biomarkers may be unavailable or difficult to identify. Advances in artificial intelligence (AI) have enabled increasingly accurate prediction of clinical outcomes from inexpensive, high-dimensional measurements, which creates an opportunity to use AI predictions themselves as surrogates. To this end, we develop a framework for estimating treatment effects from paired measurements obtained before and after treatment for each treated individual. A pretrained AI model is applied to the before and after measurements, and our estimator compares the resulting outcome predictions. We characterize the technical assumptions under which this within-person contrast identifies the average treatment effect on the treated, even when clinical outcomes are never observed for treated individuals. When these assumptions cannot be justified, we use prediction-powered inference to correct bias using a small number of observed clinical outcomes and obtain valid inference. Synthetic and real-world cardio-oncology experiments demonstrate the validity and accuracy of the approach.

Prediction with expert advice is a fundamental problem in online learning. When the time horizon $T$ is known in advance, the minimax cumulative regret over $n$ experts is asymptotically $\sqrt{\frac{T \ln n}{2}}$. This is achieved by the Multiplicative Weights Update algorithm with a learning rate tuned to $T$, and is known to be tight. If instead the regret bound is required to hold simultaneously at every time $t$, the best known guarantee has been $\sqrt{t \ln n}$---a factor of $\sqrt{2}$ worse---and it has remained unknown whether this factor of $\sqrt{2}$ is necessary. We show that it is not. We give an algorithm, requiring no knowledge of the horizon, whose cumulative regret satisfies $R_t \le \bigl(1 + O(\sqrt{\ln \ln n / \ln n})\bigr)\sqrt{t \ln n / 2}$ simultaneously for every $t \ge 1$.

This work revisits a fundamental question in active learning: how powerful is the ability to synthesize arbitrary queries? Compared to pool-based active learning, where the learner only selects queries from a given unlabeled pool, we find that this seemingly mild change in query ability may dramatically alter the difficulty of statistical learning. In particular, some hypothesis classes that are inherently slow to learn in the pool-based setting, achieving only polynomial error decay in the number of samples, become exponentially learnable once synthesized queries are allowed. This striking gap suggests that membership query synthesis induces a fundamentally different mode of learning, one that is not adequately captured by existing active learning theory and calls for new analytical tools to characterize its complexity. Motivated by this phenomenon, we develop several sufficient conditions, present intriguing examples, and propose a conjectural perspective toward understanding which hypothesis classes admit efficient learning through synthesized queries.

We study multitask regression when coefficient sharing can differ by predictor. For a given predictor, many tasks may have the same coefficient while a few differ, and the exceptional tasks need not be the same for another predictor. We describe this structure by two quantities: the number of active predictors and the total number of task coefficients that differ from the most common value for their predictor. We estimate the coefficient matrix by penalizing all pairwise coefficient differences across tasks, with an additional group penalty when predictor selection is needed. The resulting upper and lower bounds have the same dependence on these two quantities. We also consider the stronger setting in which a large set of tasks shares one entire coefficient vector. Under explicit sample-size conditions, the same pairwise estimator pools those tasks exactly, while allowing the remaining tasks to differ. Simulations and household energy data illustrate the transition between broad sharing and task-specific coefficients.

Verified income is often unavailable in digital loan applications, forcing lenders to rely on reported income and potentially leading to over-lending, overly conservative offers, or rejection of creditworthy applicants. Cross-institutional data-sharing constraints make this problem especially difficult for smaller lenders with limited training data. We introduce FedIncome, a federated learning framework for income estimation that enables institutions to train a shared model without pooling raw borrower records. Using more than one million LendingClub loans partitioned into $50$ state-level clients, we simulate a heterogeneous lending consortium. The best federated model achieves out-of-time $R^2=0.608$, compared with $0.619$ for a pooled centralised benchmark. Small-sample clients obtain an average out-of-time $R^2$ improvement of $3.8$ percentage points relative to the pooled centralised benchmark, while the fitted client-level relationship places the empirical crossover at approximately $4,790$ training observations in this setting. When pooling is infeasible and the relevant alternative is local-only training, federation improves out-of-time performance across all sample-size groups, with the largest gains for data-scarce clients. We also combine federated income estimates with state- and income-specific debt-to-income thresholds. In a retrospective decision analysis, replacing reported income with the federated estimate increases simulated approval rates with only modest changes in observed default rates. FedIncome supports collaborative learning under data-locality constraints with little aggregate loss relative to pooled training and larger gains relative to local-only estimation.

Data assimilation estimates a dynamical state from partial and noisy observations. Classical ensemble filters are efficient but restrict analysis updates through finite sample covariance and affine Gaussian distribution. We introduce the Flow Ensemble Filter (FlowEF), which uses conditional flow matching to transport the forecast ensemble from a classical baseline filter to an analysis ensemble. FlowEF uses a localized Gaussian source during training, transports forecast ensemble members from a baseline filter at deployment, and conditions its velocity field on ensembles from that baseline filter and the observation. The proposed model therefore learns a nonlinear update while mapping each baseline ensemble independently. For sparsely observed dynamical systems, FlowEF improves both deterministic and probabilistic metrics over all four classical ensemble filters. It also achieves the best performance among the state-of-the-art generative data assimilation models.

Neural Processes (NPs) are model-based meta-learners that implicitly learn a stochastic process and adapt to a new task from a small context set. Most extensions of NPs focus on improving the neural network architecture. We instead develop an extension motivated by the shared hierarchical interpretation of meta-learning and mixed-effects models. Specifically, we introduce Neural Process Boosting (NPBoost), which decomposes structured response variability into tree-boosted fixed effects shared across tasks and NP random effects that capture stochastic task-to-task variation. We propose to train the two components jointly using a boosting algorithm in which an NP learns residual task-specific structure and a tree ensemble estimates common patterns across tasks. Across synthetic and real-world tabular meta-learning problems, this decomposition improves over a standard NP when the shared structure contains discontinuities or other irregular patterns that boosted trees can represent effectively.

LLM leaderboard gains can reflect selection among privately evaluated model variants, yet neither the number of variants nor their dependence is public. We ask how many hidden variants a published margin can support while retaining statistical evidence of a provider's advantage over a fixed comparator. For a fixed candidate family under a Gaussian margin model, we derive a sensitivity curve that reports this maximum count as a function of a lower bound on within-family correlation. The relevant correlation must match the score used for ranking and the sampling model: in a controlled family, pooled item correlation is 0.90, whereas composite-score correlation is 0.46 under item resampling and 0.92 when MMLU subjects are resampled. An item-based audit of 394 adjacent-rank claims on the Open LLM Leaderboard finds that 391 lack statistical support even before accounting for selection. Among claims that pass the uncorrected test, certification can depend on assumptions about the hidden family's correlation. The resulting curves make these assumptions explicit without estimating the unobserved search size.

We empirically audit privacy leakage by testing whether outputs from edge-neighbouring inputs remain distinguishable, using statistically valid lower bounds on the privacy loss witnessed by our attacks. Our framework compares direct-edge, local-structural, and GNN-based attacks through the geometry surrounding a target edge. Experiments across two generators and two networks show that privacy leakage is both mechanism- and network-dependent, with learned representations revealing information not captured by conventional local statistics.

We propose penalized nonreversible Langevin algorithms for sampling from $π(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)$, where $\mathcal C\subset\mathbb R^d$ is a compact convex set. The algorithms combine a squared distance penalty with constant or compatible state dependent skew symmetric perturbations that preserve the penalized Gibbs distribution. For smooth, possibly nonconvex $f$, we derive nonasymptotic total variation bounds for the full gradient algorithm under a log Sobolev inequality. When unbiased stochastic gradients are available, we establish $2$-Wasserstein bounds under global contraction and Lipschitz conditions on the full drift in an adapted quadratic metric. For a fixed penalty parameter, the error relative to the penalized Gibbs distribution decays exponentially to an $\mathcal{O}(\sqrtη)$ neighborhood, where $η$ is the stepsize. We also bound the discrepancy between the penalized Gibbs distribution and the constrained target. In a two dimensional quadratic model, we establish nonreversible acceleration by tuning the skew perturbation to the curvature imbalance induced by penalization. With the target accuracy and smaller curvature fixed and initial Wasserstein distances uniformly bounded, tuning the skew perturbation improves the sufficient Euler iteration bound from linear to logarithmic in the curvature ratio. Numerical experiments evaluate the algorithms on constrained Bayesian regression, classification, neural networks, and truncated sampling, and examine the acceleration mechanism in a stochastic quadratic model.

A classical question in statistics is which observable quantities to condition on when drawing inferences about unobservable targets. For conformal prediction in nonparametric uncertainty quantification, standard marginal validity offers limited resolution at the prediction values on which decisions are based, and fully conditional guarantees with respect to the covariates are provably unattainable. We address this gap by introducing a prediction-based conditioning framework that we refer to as Prediction-Interval-Conditional Prediction Intervals (PICPIs). Formally, a PICPI is an interval $I$ satisfying a self-consistency condition: $$\mathbb{E} [Y \mid p(X) \in I] \in I,$$ for predictive model $p$, contextual covariate $X$, and outcome $Y$. Thus, an interval simultaneously defines a stratum of prediction values and certifies that the mean outcome in that stratum lies in the same interval. This self-consistency condition yields data-adaptive strata without altering the original prediction. Such intervals can be constructed using practical algorithms. Under regularity of the prediction distribution, the constructed intervals cover all but an arbitrarily small fraction of prediction values and have widths that decrease at rate $n^{-1/3}$, up to logarithmic factors and the prediction error. Moreover, identifying these locally calibrated intervals can, in turn, inform downstream decision-making. We derive inference procedures for PICPIs in probabilistic prediction and multi-class classification, accompanied by theoretical guarantees. Empirical results are provided that compare PICPIs with existing interval-based baselines.

We study the estimation of a $K$-dimensional simplex from $N$ i.i.d.\ points sampled uniformly from its interior; the observations are convex combinations of $K+1$ unknown prototypes. Existing polynomial-time estimators need cubic per-sample work or $O(NK)$ storage and are impractical at $N\sim 10^6$--$10^8$. We propose DeepMVSA, which re-expresses the minimum-volume principle in neural implicit form: a lightweight coordinate network generates the mixing weights and a triangular LU-type parameterization the dual simplex matrix, reducing the trainable-state memory to $O(K^2)$, independent of $N$, and the cost per data pass to $O(NK^2)$. We prove a non-asymptotic sample-complexity bound of the polynomial-time benchmark order for a localized surrogate estimator; an oracle inequality for every global minimizer of the neural objective, with volume-inflation control and an explicit shrinkage bias; a conditional end-to-end error budget separating statistical, approximation, optimization, and enclosure-residual terms on an explicit envelope event; and two-point lower bounds: at any noise level $σ>0$ fixed independently of $N$, the $N^{-1/2}$ scaling is unimprovable in its $N$-exponent. Experiments with up to $N=10^8$ synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of $\sim 10^7$ pixels is demonstrated.

In this paper, we study the estimation of a marginal regression function from independent units with repeated binary, count, or continuous responses using ReLU deep neural networks. In the model, we assume that the dependence is generated by an unobserved random mean function within each unit. We then fit a neural network with a convex generalized regression loss. We show an oracle inequality by separating conditional measurement variation from between-unit variation. In addition, we prove that with $n$ units and $m$ measurements per unit, ReLU networks can attain an integrated mean squared error of order $n^{-1}+(nm)^{-2β/(2β+d)}$, up to logarithmic factors, over $β$-Hölder classes. We also derive a weighted oracle inequality for unequal cluster sizes and a rate for compositionally smooth functions. For pointwise ensemble inference, we give a projection central limit theorem and prove infinitesimal jackknife consistency under an explicit asymptotic linearity condition. Simulations and real data examples are provided to support our theoretical findings and practical implications.

Federated learning (FL) is intrinsically heterogeneous: honest clients may have different data-generating models. On top of that, adversarial clients can make heterogeneity even more pronounced by sharing arbitrary updates. Existing analyses typically control the interaction between statistical heterogeneity and adversarial behavior through gradient-dissimilarity conditions. However, the underlying bound is imposed a priori and may yield conservative guarantees even for least-squares regression. We instead derive the gradient heterogeneity from the statistical model of linear and nonlinear regression with fresh data samples at every round. Our bounds separate heterogeneity among the honest clients' ground-truth model parameters, finite-sample label noise, and initialization. We then demonstrate that, for any $(f,κ)$-robust aggregator with coefficient $κ= O(f/n)$, where $f$ is the number of adversarial clients and $n$ the total number of clients (with $f/n < 1/2$), convergence holds after an explicit sample burn-in.

The statistical accuracy of neural networks depends on both their approximation power and the complexity of the class fitted from data. While increasing network size is a natural way to improve approximation, parameter magnitude provides another resource whose role must be quantified in both respects. We establish a sharp width--magnitude tradeoff at fixed depth using one elementary bounded $1$-Lipschitz Dyadic--Triangular Activation. For the unit $β$-Hölder ball on $[0,1]^d$ with $0<β\leq1$, the optimal $L^p$ approximation error for $0<p<\infty$ is of order $[N^2\log(eNT)]^{-β/d}$ when the network width satisfies $N\geq2d+3$ and the parameter magnitudes are bounded by $T\geq1$. Matching lower bounds hold for every fixed globally Hölder activation; its Hölder exponent affects the constants but not the rate. Under bounded design densities and independent centered sub-Gaussian noise, approximate least squares over the full clipped class at depth $23$ attains the classical Hölder minimax risk $\mathcal{O}(M^{-\frac{2β}{2β+d}})$ without logarithmic loss whenever $N^2\log(eNT)\asymp M^{\frac{d}{2β+d}}$, where $M$ is the sample size. This yields a continuum of statistically optimal choices, ranging from unit parameter radius to fixed network size. At fixed size, four hidden layers with at most $8d+7$ nonzero parameters give a near-optimal radius, while six layers with at most $8d+27$ attain the optimal order $\log T=\mathcal{O}(η^{-d/β})$ at approximation error $η$. The same decoding method also yields fixed-size Transformer approximation.

Growing memory demands in artificial intelligence motivate learning with fewer trainable parameters. We ask whether a looped estimator, which repeatedly applies one fitted operator with parameters shared across iterations, can improve statistical accuracy under a common parameter budget. Its conventional untied counterpart uses separate parameters at each iteration. For general likelihood models, we establish an upper bound on squared Hellinger risk for looped sieve maximum likelihood and a minimax lower bound over the tuned untied family. These bounds reveal a parameter--iteration--accuracy tradeoff: repeated computation can improve approximation without adding parameters, while increasing computational cost and fitted-class complexity. For targets of known Hölder smoothness, looped residual feedforward networks and a specified post-layer-normalized Transformer attain the minimax polynomial rate up to logarithmic factors with a fixed number of bounded real parameters. At sufficiently large fixed budgets, looped worst-case risk vanishes as sample size grows, whereas optimal worst-case untied risk remains bounded away from zero. Under specified growing-budget conditions, the loop-to-untied risk ratio also tends to zero. Gaussian and Laplace regression, binary response, and energy-based density estimation illustrate the theory.

Causal inference guides operational and managerial decisions but remains challenging in high-dimensional panel or tensor settings, where decisions may depend on the joint conditional distribution of missing control outcomes. We develop \emph{Counterfactual Tucker Diffusion} (\CFTDiff), which integrates the treatment mask and latent Tucker structure into conditional diffusion to recover this distribution given observed control outcomes through efficient nonlinear score learning in a low-dimensional core. The masked Tucker score preserves dependence across tensor modes while reducing the dimension of nonlinear score learning from the product of mode dimensions to the much smaller product of Tucker ranks. We establish high-probability error bounds for conditional score estimation that depend on the Tucker ranks, largest mode dimension, and the factor-strength-adjusted number of missing outcomes, and show how these bounds translate into recovery guaranties for the conditional distribution of the missing control outcomes. Across missing rates, simulations show more accurate point recovery than common causal panel and matrix/tensor completion methods; comparisons with nested diffusion specifications further demonstrate the gains from masked conditioning and Tucker dimension reduction. In Norway's iFlex experiment, \CFTDiff recovers missing outcomes more accurately than competing methods; when applied to causal analysis, its estimated conditional distributions yield counterfactual prediction intervals and target-attainment probabilities, allowing pricing interventions to be evaluated by demand-reduction magnitude and reliability.