Estimators

How should an agent's performance in a multiagent environment be evaluated when there is a limited sample size or a high cost of running a trial? The AIVAT family of variance reduction techniques was proposed to address this challenge by introducing unbiased low-variance estimators of agents' expected payoffs. An important component of AIVAT is a heuristic value function that discriminates between potentially low- and high-value counterfactual histories. A notable gap in the literature is that there is little to no constraint or guideline on how the heuristic value function should be chosen or how uncertainty in its output should be handled. In our first contribution, we parameterize the heuristic value function to highlight AIVAT's potential vulnerabilities: a) the sample variance can be set pathologically low by directly applying gradient descent on the sample variance, and b) one can p-hack to draw a desired statistical conclusion via gradient descent/ascent on the test statistic. The main takeaway is that the heuristic value function should be fixed prior to observing the evaluation data! In our second contribution, we show how the heuristic uncertainty can be propagated to quantify the uncertainty of AIVAT estimates. It is then possible to further reduce the variance using inverse-variance weighted averaging, but AIVAT's unbiasedness guarantee may have to be sacrificed. In our experiments, we use a dataset of 10,000 poker hands to demonstrate our heuristic pathology and uncertainty results, with the latter yielding a 43.0% reduction in the number of samples (poker hands) needed to draw statistical conclusions.

We revisit Byzantine robust distributed estimation for high-dimensional sparse linear models. By combining local $\ell_1$-regularized robust estimation with robust aggregation at the server, the framework applies to pseudo-Huber regression, quantile regression, and sparse SVM. We show that the resulting estimators yield non-asymptotic guarantees and attain near-optimal statistical rates under mild conditions, while remaining communication-efficient. Simulations confirm strong robustness in estimation, support recovery and classification accuracy under various Byzantine attacks.

Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on Gaussian processes, kernel machines, linear models, or linearized neural approximations, leaving a gap between theory and the nonlinear models used in practice. We develop a kernel based framework for analyzing regularized nonlinear parametric models trained on adaptively collected data. Our approach uses kernels over the parameter space to induce reproducing kernel Hilbert space structures over the corresponding model class, yielding confidence bounds for models trained with broad classes of regularized convex losses. We show how these bounds can support convergence guarantees for nonlinear acquisition and surrogate models, including randomized regularized policies that select points by maximizing a trained random model. These results provide a unified route to analyzing nonlinear parametric models in Bayesian optimization and related adaptive optimization settings.

We propose and analyze a model-based bootstrap for transition kernels in finite controlled Markov chains (CMCs) with possibly nonstationary or history-dependent control policies, a setting that arises naturally in offline reinforcement learning (RL) when the behavior policy generating the data is unknown. We establish distributional consistency of the bootstrap transition estimator in both a single long-chain regime and the episodic offline RL regime. The key technical tools are a novel bootstrap law of large numbers (LLN) for the visitation counts and a novel use of the martingale central limit theorem (CLT) for the bootstrap transition increments. We extend bootstrap distributional consistency to the downstream targets of offline policy evaluation (OPE) and optimal policy recovery (OPR) via the delta method by verifying Hadamard differentiability of the Bellman operators, yielding asymptotically valid confidence intervals for value and $Q$-functions. Experiments on the RiverSwim problem show that the proposed bootstrap confidence intervals (CIs), especially the percentile CIs, outperform the episodic bootstrap and plug-in CLT CIs, and are often close to nominal ($50\%$, $90\%$, $95\%$) coverage, while the baselines are poorly calibrated at small sample sizes and short episode lengths.

Reversible instance normalization (RevIN) and its successors (Dish-TS, SAN, FAN) have become the de facto plug-in for time-series forecasting, yet the map they apply to each data point is strictly affine, $x \mapsto ax+b$, so they cannot reshape the underlying distribution -- heavy tails remain heavy and skewness remains uncorrected. We propose NoRIN, a non-linear reversible normalization based on the arcsinh-form Johnson $S_U$ transform with two shape parameters $(δ,\varepsilon)$ that control tailedness and skewness; the linear $Z$-score used by RevIN is recovered only in the limit $δ\to \infty$. Training $(δ,\varepsilon)$ jointly with the backbone via gradient descent reliably pushes them toward this linear limit within a few epochs -- a phenomenon we name the degeneration problem: the forecasting loss is locally indifferent to shape, and the high-capacity backbone compensates for any monotone reparameterization of its input. NoRIN escapes the degeneration by decoupling shape selection from gradient training: $(δ,\varepsilon)$ are initialized by a closed-form Slifker-Shapiro quantile fit and refined by Bayesian optimization on the validation objective, while the inner training loop is identical to standard RevIN-style training. Across six representative backbones x five real-world datasets x three prediction horizons (90 configurations), decoupled shape optimization recovers $(δ^\star,\varepsilon^\star)$ that sit systematically far from the linear limit, with values that vary in a backbone-dependent way. This empirically supports the central thesis: different backbones genuinely require different normalization parameters to reach their best performance.

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

Unobserved confounding prevents standard covariate adjustment from identifying causal response functions in observational studies. Proxy causal learning addresses this problem through bridge equations involving treatment- and outcome-inducing proxies, avoiding direct recovery of the latent confounder. Existing doubly robust proxy estimators combine outcome and treatment bridges, but typically rely on fixed kernels, sieves, or low-dimensional semiparametric models; existing neural proxy methods are more flexible, but are largely single-bridge estimators. We develop a neural doubly robust framework for proxy causal learning with continuous and structured treatments. Our method introduces a neural mean-embedding estimator for the treatment bridge, combines it with a neural outcome bridge, and estimates the doubly robust correction through a final regression stage. The framework covers population, heterogeneous, and conditional dose-response functions, yielding full response-curve estimators rather than binary-treatment effects. The algorithms use two stages for each bridge and history-aware updates of the final linear layers to stabilize stochastic multi-stage training. We prove consistency of the algorithms showing that the doubly robust error is controlled by the final averaging and regression errors together with the smaller of the outcome- and treatment-side weak-norm bridge errors. Across synthetic and image-valued benchmarks, the proposed estimators outperform existing baselines and single-bridge neural estimators, showing the benefit of combining learned outcome and treatment bridges in a doubly robust construction. Our implementation is available at https://github.com/BariscanBozkurt/DRPCL-Neural-Mean-Embedding.

The problem of predicting unobserved entries in a binary matrix, known as 1-bit matrix completion, has found diverse applications in fields such as recommendation systems. In this study, we develop an empirical Bayes method for 1-bit matrix completion motivated by the Efron--Morris estimator, a matrix generalization of the James--Stein estimator that shrinks singular values toward zero. The proposed method exploits the underlying low-rank structure of binary matrices, drawing parallels with multidimensional item response theory. Simulation studies and real-data applications demonstrate that the proposed method achieves a superior balance of predictive accuracy, calibration reliability (uncertainty quantification), and computational efficiency compared to existing methods.

We show that the core components of the Transformer---attention, residual connections, and normalization---arise naturally from a single geometric state estimation problem. Modeling the latent state in polar coordinates naturally separates radial and hyperspherical dynamics, yielding a precision-weighted filtering procedure in which normalization enforces the hyperspherical constraint, attention aggregates directional evidence, and the residual connection implements an incremental state update. The standard Transformer block with rotary positional encodings is recovered by discarding the geometric correction terms of the resulting state estimator, showing that its architecture follows from the underlying estimation problem rather than from independent design choices. The proposed \textit{Polar Transformer} retains these geometric corrections.

Continuous-time state estimation is gaining in popularity due to its abilities to provide smooth solutions, handle asynchronous sensors, and interpolate between data points. While there are two main paradigms, parametric (e.g., temporal basis functions, splines) and nonparametric (Gaussian processes), the latter has seen less adoption despite its technical advantages and relative ease of implementation. In this article, we seek to rectify this situation by providing a new simplified explanation of GP continuous-time estimation rooted in the language of factor graphs, which have become the de facto estimation paradigm in much of robotics. To simplify onboarding, we also provide three working examples implemented in the popular GTSAM estimation framework.

A series of papers has introduced the Heuristic Rating Estimation method, which evaluates a set of alternatives based on pairwise comparisons and the weights of reference alternatives. We formulate the conditions under which the HRE method can be applied correctly. The research considers both arithmetic and geometric algorithms for complete and incomplete pairwise comparison methods. The illustrative examples show that the estimations of inconsistency in the arithmetic variant are optimal.

We study online estimation in latent-variable models by recasting the problem as tracking a moving empirical equilibrium. Standard online EM and stochastic approximation analyses primarily study convergence toward the population parameter and typically do not isolate the empirical batch optimum from the online tracking error at finite horizon. Our framework decomposes the online estimate into the frozen batch equilibrium at the current running statistic and a tracking lag that captures the algorithm's delay behind this moving target. We prove a batch-to-online transfer theorem: provided $\lVert e_T \rVert_{L^{2}} = o(T^{-1/2})$, the online estimator inherits the batch central limit theorem and the sharp first-order risk constant. Our key observation is that the empirical optimum evolves on a smooth equilibrium manifold indexed by the running statistic. An $m$-th order equilibrium-jet predictor combined with an order-$ν$ frozen corrector yields localized tracking rates $O(T^{-ν(m+1)})$. We formalize EM-compressibility and EM-jet$^R$-compressibility as the structural conditions that make the equilibrium response and the Newton corrector evaluable from a retained streaming statistic. The theory is instantiated in latent linear Gaussian covariance estimation, where the first-order scheme operates on a compressed $d \times d$ statistic with explicit finite-sample risk envelopes and a certified restart rule.

Deep vision models degrade sharply in low-data regimes, particularly in medical imaging where labeled samples are scarce. We show this arises not merely from overfitting but from a geometric failure: finite-sample noise corrupts the embedding covariance, collapsing the eigengap and limiting the number of recoverable signal-bearing modes. We develop a spectral theory of finite-sample representation learning that quantifies the recoverable dimension K(N), the number of eigenmodes that can be stably estimated from N samples. Using perturbation theory and concentration bounds, we show that only modes with eigenvalues above the noise floor $\|\hatΣ - Σ\|_{\mathrm{op}} \sim \sqrt{D/N}$ are reliable, yielding a truncated Mahalanobis energy that governs classification performance. Under a power-law spectral model, this energy can be approximated by a truncated Riemann zeta function, linking eigenvalue decay to data efficiency and AUC. Within this framework, multimodal learning acts as spectral stabilization: vision-language models impose low-rank constraints that suppress noise-dominated directions and preserve the eigengap, increasing K(N) under data scarcity. Across MNIST and multi-disease neuroimaging, we show that multimodal training maintains more stable modes and improves class separation, even when unimodal models achieve comparable few-shot accuracy. These results identify spectral collapse as a fundamental bottleneck in low-data learning. We use truncated Mahalanobis energy and K(N) to diagnose encoder quality, and introduce zeta-based spectral filtering as a principled approach to improve data efficiency.

Convex functionals are ubiquitous in applied analysis, appearing as value functions, risk measures, super-hedging prices, and loss functionals in machine learning. In many applications, however, the functional is only observed through finitely many exact pointwise evaluations. We ask whether a convex functional on a separable Hilbert space $H$ can be reconstructed, up to arbitrary uniform accuracy, by an explicit formula which preserves convexity and Lipschitz regularity and is finitely computable. We answer this affirmatively. For every compact convex $C\subseteq H$, every $L$-Lipschitz convex functional $ρ:C\to\mathbb{R}$, and every $\varepsilon>0$, we construct an explicit finite-sample reconstruction which is convex, $L$-Lipschitz, and uniformly $\varepsilon$-accurate on $C$. The construction uses only finitely many linear measurements $\langle b,\cdot\rangle_H$, with $b$ lying in a finite-dimensional subspace of $H$, and is exactly implementable by a $\operatorname{ReLU}$-MLP. Building on this, we introduce convex neural functionals (CNFs), a structured trainable architecture class containing our reconstruction, whose every admissible parameter configuration is automatically convex and Lipschitz, providing a principled foundation for learning convex functionals from finite data.

We describe an approach for estimating the statistical risk of deep networks that contain a mix of learned and known operators. Building on the maximal training error bounds previously established for known operator learning, we derive a deep risk estimator that connects the expected error of a layered network to the size of the training sample. The estimator decomposes the total risk into a sum over learned layers; every known operator contributes zero to this sum, while every learned layer adds an approximation term inspired by Barron's classic work and an estimation term that decreases with the number of training samples. We are able to show that the bound shrinks whenever a learned layer is replaced by a known operator and that the corresponding sample requirement scales with the number of trainable parameters of the layer that is replaced. As an application, we use computed tomography as an example and compare an operator-aware filtered backprojection network with a fully connected substitute that collapses the entire reconstruction pipeline into a single learned dense matrix. The predicted parameter ratio coincides with the structural sparsity that the analytic decomposition into a circulant filter and a sparse backprojection exposes. We confirm the predicted scaling on CPU at small image scale and on GPU at medium image scale, all on the same scaling law. Beyond CT reconstruction, the estimator applies to physics-informed neural networks that hardcode a known physical operation in its architecture, and we expect the result to be of interest for a broad community working on operator-aware deep learning. Calibrating the per-layer constants on each sweep yields a bound that tracks the empirical test MSE within a factor of two at every training-set size, so the estimator can be inverted to predict how many training samples are required to reach a target error.

We define susceptibilities as a measure of the response of an observable quantity of a parameterized statistical model to a perturbation of the data for a general class of observables. We define estimators for these susceptibilities as statistics in a sequence of n data-points and prove that these estimators are consistent and asymptotically unbiased in the large n regime.

Grammar-constrained generation is often combined with local vocabulary masking and speculative decoding, but the resulting sampling law is not the grammar-conditional distribution users usually intend. We show that any speculative decoder with local mask access, Leviathan rejection, and rollback soundness samples from the locally projected distribution $μ^{\mathrm{proj}}$ rather than the grammar-conditional distribution $μ^\star$. This extends the GAD impossibility result to speculative decoding; on Dyck grammars with Qwen3-8B, the total-variation gap can reach 0.996. We identify the future-validity function $Φ_t(y)=\Pr_p[\mathrm{valid\ completion}\mid y]$ as the missing correction statistic. The target distribution is a Doob transform of the base model with $h=Φ$, while local masking corresponds to setting $h$ to one. With exact $Φ$, our oracle decoder FVO-Spec samples exactly from $μ^\star$; with approximate $Φ$, we bound the resulting total-variation error. Because exact future validity is hard for general context-free grammars, we evaluate estimator hierarchies on tractable Dyck and finite JSON languages. OneStep reduces Dyck TV by 14% with under 1% throughput overhead, exact dynamic programming reduces it by 97%, and finite-language correction closes JSON gaps to numerical precision. All fidelity claims are scoped to enumerable grammars and token tries.

The Translation Tax is often treated as a scalar: translated benchmarks are assumed to inflate scores by preserving English-source cues. We audit this claim in an English-to-Chinese setting. Three proxy estimators disagree: back-translation gaps are small and parser-fragile; cue-score calibration does not predict item-level gains; and a six-model native-control comparison shows model-family rather than uniform benchmark effects. We add a same-item LLM-naturalization stress test that holds answer, options, and content fixed while rewriting Chinese surface form. After correcting a prompt-construction bug, this contrast no longer supports a model-family interaction, but it preserves a residue dose-response: high-residue items benefit while low-residue items do not. The result is not a single Translation Tax, but a set of estimator- and item-dependent validity risks. We release per-cell evidence, the naturalization protocol, human QC, and a reporting checklist for translated multilingual benchmark papers.

Individual treatment effects are not point-identified from data. The Probability of Necessity and Sufficiency (PNS) circumvents this limitation by characterizing individual-level causality through intersection bounds derived from combined experimental and observational data. In finite samples, however, standard plug-in estimators systematically fail: they violate structural probability constraints and suffer from extremum bias induced by max-min operators, yielding spuriously narrow intervals. We propose a neural framework for finite-sample PNS estimation that resolves both pathologies. We introduce an anchored neural architecture that guarantees structural constraint satisfaction by construction. To correct extremum bias, we employ precision-corrected intersection-bound inference, leveraging Epistemic Neural Networks for scalable, high-dimensional uncertainty quantification. Empirical evaluations confirm that this approach maintains nominal coverage and exact constraint validity in high-dimensional regimes where standard estimators systematically undercover.

This work studies decentralized novelty detection with global false discovery rate (FDR) control across heterogeneous composite null distributions, without sharing the raw data due to privacy and bandwidth considerations. We propose a framework based on the exchange of quantized surrogate models, allowing independent agents to share low-precision representations of locally learned non-conformity score functions. We prove that evaluating data against these quantized composite scores preserves conditional exchangeability, providing rigorous finite-sample guarantees for global FDR control. Empirical studies on synthetic datasets confirm our theoretical results, demonstrating that the proposed approach maintains competitive statistical power while drastically reducing the communication cost.

Estimating causal effects from observational data requires identifying valid adjustment sets. This task is especially challenging in realistic settings where latent confounding and feedback loops are present. Existing approaches typically assume acyclicity or rely on global causal structure learning, limiting applicability and computational efficiency. In this work, we study a local, data-driven method for covariate selection based on conditional independence information. While this method is known to be sound and complete in acyclic causal models, its validity in the presence of cycles has remained unclear. Our main contribution is to show that these guarantees extend to cyclic causal models. In particular, our result relies on the invariance of conditional independence assertions under $σ$-acyclification. These findings establish a unified, cycle-agnostic perspective on covariate selection and causal effect estimation, showing that the method applies across cyclic and acyclic settings without modification. Empirically, we validate this on extensive synthetic data, showing reliable performance in cyclic causal models.

Adaptive prompt and program search makes LLM evaluation selection-sensitive. Once benchmark items are reused inside tuning, the observed winner's score need not estimate the fresh-data performance of the full tune-then-deploy procedure. We study inference for this procedure-level target under explicit tuning budgets. We propose SIREN, a selection-aware repeated-split reporting protocol that freezes the post-search shortlist, separates splitwise selection from held-out evaluation, and uses an item-level Gaussian multiplier bootstrap for uncertainty quantification. In a fixed-shortlist regime with smooth stabilized selection, the estimator admits a first-order item-level representation, and the bootstrap yields valid simultaneous inference on a finite budget grid. This supports confidence intervals for procedure-performance curves and pre-specified equal-budget and cross-budget comparisons. Controlled simulations and MMLU-Pro tuning experiments show that winner-based reporting can be optimistic and can change deployment conclusions, while SIREN remains close to the finite-sample reporting target.

While reinforcement learning (RL) promises to revolutionize the control of complex nonlinear robotic systems, a profound gap persists between the heuristic success of model-free off-policy deep RL and the underlying theory, which remains largely confined to tabular or linearizable settings. We identify the cause of this gap as an emergent isolation of three traditions: (i) measure-theoretic MDP foundations on general spaces limit their analysis to exact dynamic programming and ignore all error sources of a learning process; (ii) deterministic error propagation analysis addresses the approximation error via concentrability coefficients without a finite-sample analysis of the estimation error; and (iii) PAC generalization bounds characterize the estimation errors of simplified topologies. We bridge these traditions with a unified theoretical framework for fitted Q-iteration (FQI) on general measurable Borel spaces. Our main result provides a finite-sample, adaptive-data performance bound by chaining measure-theoretic probability with Bellman-operator contraction in Banach spaces. We prove that sequential Rademacher complexity controls Bellman-regression generalization under policy-dependent data collection. We further extend this analysis to provide the first cumulative, pathwise online regret guarantee for FQI in continuous spaces. These results lay the necessary foundations for the formal analysis of many modern deep RL algorithms.

We study nonparametric estimation of Schrödinger bridge (SB) drifts from i.i.d.\ data observed on a single time interval. Starting from the conditional-ratio form of the Schrödinger bridge time-series (SBTS) drift formula, we analyze a direct Nadaraya--Watson plug-in estimator built from kernelized numerator and denominator terms. Unlike recent SB analyses based on entropic-OT potentials, Sinkhorn iterations, or iterative bridge solvers, our approach works directly at the drift level and isolates \emph{statistical error} from optimization, approximation, and discretization error. Under Hölder regularity, a marginal-density floor, and bounded support, we prove a uniform non-asymptotic bound for admissible bandwidth pairs, a pointwise CLT under genuine undersmoothing, and an adaptive bandwidth selector satisfying an oracle inequality. We also prove a pivot-local minimax lower bound which, through an explicit uniform pivot, yields a global minimax lower bound under transparent compatibility conditions; hence the adaptive selector is minimax-rate optimal up to logarithmic factors. Synthetic experiments provide theorem-targeted diagnostics for finite-sample scaling, Gaussian approximation, and adaptive behavior.

Information theory plays a central role in establishing fundamental limits on what any learning or estimation algorithm can -- and cannot -- achieve, regardless of computational power. In this chapter, we provide an introduction to these connections. End-of-chapter exercises makes the material suitable for both classroom use and self-study. We begin by introducing concentration inequalities along with the notions of covering and packing in metric spaces, and the associated concept of metric entropy. These tools are essential for our analysis. We then introduce the learning-theoretic framework and derive upper bounds on generalization error in terms of metric entropy, Rademacher complexity, and the VC dimension, as well as mutual information and relative entropy. Finally we discuss the minimax estimation framework and establish lower bounds on minimax risk using Fano's inequality, yielding bounds in terms of relative entropy and covering and packing numbers. This manuscript contains preprint of a chapter under consideration for inclusion in the forthcoming third edition of Cover and Thomas's Elements of Information Theory, posted with permission from Wiley. It would follow the chapter posted at arXiv:2605.02989 . The table of contents of the new edition can be found at: https://docs.google.com/document/d/1L-m4oQEJw1PJhoxBeMwrrBD8S_HmvzMEkPbYvS24980/edit?usp=sharing . For feedback, please contact abbas@ee.stanford.edu.

Minimum-fuel low-thrust rendezvous guidance yields bang-bang control structures highly sensitive to estimation errors, sensor anomalies, and solver regularization, making aggressive closed-loop execution brittle for uncooperative proximity operations. This paper proposes a tightly-coupled estimation and guidance architecture where navigation confidence directly modulates the homotopy parameter of a receding-horizon indirect optimal control solver. Relative motion is modeled in the Clohessy-Wiltshire frame. The translational state is estimated via a linear Kalman filter augmented by a Multiple Tuning Factors (MTF) covariance inflation mechanism that suppresses suspicious innovation directions. A composite score from the normalized innovation and MTF activity is mapped online to the homotopy parameter, allowing the controller to relax toward a smoother, conservative regime when confidence degrades, and recover fuel-efficient bang-bang control as sensing improves. Numerical results under severe measurement degradation show fixed bang-bang guidance remains brittle; both plain-KF and MTF-KF fixed-epsilon controllers yield large terminal miss distances. Conversely, the proposed MTF-adaptive homotopy controller reduces terminal miss by roughly two orders of magnitude, from hundreds of meters to sub-meter levels, requiring only a moderate increase in control effort versus the open-loop fuel-optimal benchmark. A comparison indicates adaptive homotopy is the dominant robustness mechanism, while MTF provides additional accuracy and efficiency improvements. The receding-horizon implementation exhibits consistently fast and reliable solution times, supporting the practical online viability of the proposed method.

Doubly Robust (DR) estimation of treatment effect relies on an untestable assumption that is the absence of unobserved confounding. This assumption is par- ticularly problematic in the context of healthcare research, where variables like pre- scription refill rates serve as proxies for unobserved behaviors such as medication adherence. These proxy variables are often endogenous, exhibiting correlation with the regression error term due to unmeasured confounding or measurement error. We propose a copula-corrected doubly robust estimator that addresses endogeneity in both the treatment and outcome models without requiring instrumental variables. Gaussian copulas model the joint distribution of endogenous covariates and the error term, enabling consistent estimation while preserving the doubly robust property that requires correct specification of either the treatment or outcome model, not both. Monte Carlo simulations demonstrate that naive DR estimation exhibits substantial bias under endogeneity, whereas our corrected estimator recovers unbiased treatment effects across different data-generating processes. We apply our method to examine the effect of nutritional counseling on blood pressure using the National Health and Nutrition Examination Survey (NHANES) data. Naive DR estimation suggests counseling is associated with increased blood pressure. After copula correction, this effect becomes statistically insignificant, consistent with literature showing modest effects of nutri- Counseling in reducing blood pressure. Our methodology provides researchers with a practical tool for obtaining treatment effects in the presence of endogeneity.

Conformal prediction provides distribution-free predictive intervals with finite-sample marginal coverage. However, achieving conditional validity and interval efficiency (in terms of short interval length) remains challenging, particularly in complex settings with heteroskedasticity, skewed responses, or estimation errors. We propose a conformal-style calibration method for responses obtained by the probability integral transform (PIT) of the conditional cumulative distribution function (CDF) estimated via neural networks to construct a finite-sample-adjusted percentile interval with the shortest length determined by the estimated conditional CDF. Calibrating in PIT space is effective because PIT values are asymptotically feature-independent when the CDF estimator is accurate, which mitigates feature-dependent miscoverage and improves conditional calibration. On the other hand, our percentile calibration adapts to the empirical PIT distribution, which is robust against a possibly imperfect estimation of the conditional CDF. We prove the finite-sample marginal coverage property of the proposed method and show its asymptotic conditional coverage under mild consistency conditions. Experiments on diverse synthetic and real-world benchmarks demonstrate better conditional calibration and substantially shorter intervals than existing methods.

Probabilistic values, including Shapley values and semivalues, provide a model-agnostic framework to attribute the behavior of a black-box model to data points or features, with a wide range of applications including explainable artificial intelligence and data valuation. However, their exact computation requires utility evaluations over exponentially many coalitions, making Monte Carlo approximation essential in modern machine learning applications. Existing estimators are often developed through different representation strategies, including weighted averages, self-normalized weighting, regression adjustment, and weighted least squares. Our key observation is that these seemingly distinct constructions share a common first-order expansion, in which the leading term is determined by the sampling law and a working surrogate function. This first-order representation yields an explicit expression for the leading mean squared error (MSE), which characterizes how the sampling law and the surrogate jointly determine statistical efficiency. Guided by this criterion, we propose an Efficiency-Aware Surrogate-adjusted Estimator (EASE) that directly chooses the sampling law and surrogate to minimize the first-order MSE. We demonstrate that EASE consistently outperforms existing estimators for various probabilistic values.

The proliferation of large-scale and structurally complex data has spurred the integration of machine learning methods into statistical modeling. Recurrent neural networks (RNNs), a foundational class of models for time-dependent data, can be viewed as nonlinear extensions of classical autoregressive moving average models. Despite their flexibility and empirical success in machine learning, RNNs often suffer from limited interpretability and slow training, which hinders their use in statistics. This paper proposes the Parallelized RNN (ParaRNN), a novel model composed of multiple small recurrent units. ParaRNN admits an additive representation that decouples recurrent dynamics into interpretable components, whose behavior can be characterized through recurrence features. This interpretability enables its applications in nonparametric regression for time-dependent data, while the design also allows efficient parallelization. The approximation capacity and non-asymptotic prediction error bounds in a nonparametric regression setting are established for ParaRNN. Empirical results on three sequential modeling tasks further demonstrate that ParaRNN achieves performance comparable to vanilla RNNs while offering improved interpretability and efficiency.

We study the problem of denoising observations $Y_i=X_i+Z_i$, where the latent variables $X_i$ are sampled from a low-dimensional manifold in $\mathbb{R}^n$ and the noise variables $Z_i$ are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

A fixed-confidence, finite-sample problem of active hypothesis testing arises in many safety-critical applications. Situated in the context of sequential hypothesis testing, this paper studies the effect of hypothesis elimination on the stopping time. We introduce an elimination-augmented Track-and-Stop algorithm, in which champion-specific active-opponent sets are progressively pruned, and sensing effort is reallocated toward the surviving alternatives. Our analysis derives a non-asymptotic upper bound on the expected stopping time. The gain in finite-sample from elimination appears on the scale of the non-leading term, resulting from tighter tracking and concentration constants on the reduced hypothesis set. Furthermore, we introduce an aggressiveness parameter to modulate the trade-off between faster elimination and weaker confidence guarantee. An experimental study on synthetic Gaussian instances confirms the theoretical predictions.

We study $d$-dimensional unbiased mean estimation in the single-message shuffle model, where each user sends a single privatized message and the analyzer only observes the shuffled multiset of reports. While minimax-optimal mechanisms are well understood in the local differential privacy setting, the corresponding notion of optimality after shuffling has remained largely unexplored. To address this gap, we introduce the recently proposed shuffle index and use it to formulate the post-shuffling mechanism design problem as an explicit optimization problem. We then establish a minimax lower bound on the achievable mean squared error in terms of the shuffle index, which implies that mechanisms that are optimal under LDP can become suboptimal once shuffling is applied. Finally, we construct an asymptotically minimax optimal mechanism in the high privacy regime, which as a consequence achieves a privacy-utility trade-off nearly identical to that of the central Gaussian mechanism.

Recent advances in large language models (LLMs) have increasingly relied on reinforcement learning (RL) to improve their reasoning capabilities. Three types of approaches have been widely adopted: The first relies on a deep neural network to estimate the value function of the learning policy in order to reduce the variance of the policy gradient. However, estimating and maintaining such a value network incurs substantial computational and memory overhead. The second avoids training a value network by approximating the value function using sample averages. However, it samples a large number of reasoning traces per prompt for accurate value function approximation, making it computationally expensive. The third samples only a single reasoning trajectory per prompt, which reduces computational cost but suffers from poor sample efficiency. This paper focuses on a practical, resource-constrained setting in which only a small number of reasoning traces can be sampled per prompt, while low-variance gradient estimation remains essential for high-quality policy learning. To address this challenge, we bring classical nonparametric statistical methods, which are both computationally and statistically efficient, to LLM reasoning. We employ kernel smoothing as a concrete example for value function estimation and the subsequent policy optimization. Numerical and theoretical results demonstrate that our proposal achieves accurate value and gradient estimation, leading to improved policy optimization.

The rapidly expanding artificial intelligence (AI) industry has produced diverse yet powerful prediction tools, each with its own network architecture, training strategy, data-processing pipeline, and domain-specific strengths. These tools create new opportunities for semi-supervised inference, in which labeled data are limited and expensive to obtain, whereas unlabeled data are abundant and widely available. Given a collection of predictors, we treat them as a mixture of experts (MOE) and introduce an MOE-powered semi-supervised inference framework built upon prediction-powered inference (PPI). Motivated by the variance reduction principle underlying PPI, the proposed framework seeks the mixture of experts that achieves the smallest possible variance. Compared with standard PPI, the MOE-powered inference framework adapts to the unknown performance of individual predictors, benefits from their collective predictive power, and enjoys a best-expert guarantee. The framework is flexible and applies to mean estimation, linear regression, quantile estimation, and general M-estimation. We develop non-asymptotic theory for the MOE-powered inference framework and establish upper bounds on the coverage error of the resulting confidence intervals. Numerical experiments demonstrate the practical effectiveness of MOE-powered inference and corroborate our theoretical findings.

Face recognition from a single image per person is a challenging problem because the training sample is extremely small. We consider a variation of this problem. In our problem, we recognize only one person, and there are no labeled data for any other person. This setting naturally arises in authentication on personal computers and mobile devices, and poses additional challenges because it lacks negative examples. We formalize our problem as one-class classification, and propose and analyze an algorithm that learns a non-parametric model of the face from a single labeled image and a stream of unlabeled data. In many domains, for instance when a person interacts with a computer with a camera, unlabeled data are abundant and easy to utilize. This is the first paper that investigates how these data can help in learning better models in the single-image-per-person setting. Our method is evaluated on a dataset of 43 people and we show that these people can be recognized 90% of time at nearly zero false positives. This recall is 25+% higher than the recall of our best performing baseline. Finally, we conduct a comprehensive sensitivity analysis of our algorithm and provide a guideline for setting its parameters in practice.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

Many sign language translation (SLT) systems operate on pose sequences instead of raw video to reduce input dimensionality, improve portability, and partially anonymize signers. The choice of pose estimator is often treated as an implementation detail, with systems defaulting to widely available tools such as MediaPipe Holistic or OpenPose. We present a systematic comparison of pose estimators for pose-based SLT, covering widely used baselines (MediaPipe Holistic, OpenPose) and newer whole-body/high-capacity models (MMPose WholeBody, OpenPifPaf, AlphaPose, SDPose, Sapiens, SMPLest-X). We quantify downstream impact by training a controlled SLT pipeline on RWTH-PHOENIX-Weather 2014 where only the pose representation varies, evaluating with BLEU and BLEURT. To contextualize translation outcomes, we analyze temporal stability, missing hand keypoints, and robustness to occlusion using higher-resolution videos from the Signsuisse dataset. SDPose and Sapiens achieve the best translation performance (BLEU ~11.5), outperforming the common MediaPipe baseline (BLEU ~10). In occlusion cases, Sapiens is correct in all tested instances (15/15), while OpenPifPaf fails in nearly all (1/15) and also yields the weakest translation scores. Estimators that frequently leave out hand keypoints are associated with lower BLEU/BLEURT. We release code that can be used not only to reproduce our experiments, but also considerably lowers the barrier for other researchers to use alternative pose estimators.

We study the $K$-armed logistic bandit problem, where at each round, the agent observes $K$ feature vectors associated with $K$ actions. Existing approaches that achieve a rate-optimal $\tilde{\mathcal{O}}(\sqrt{dT})$ regret bound rely heavily on context diversity assumptions, such as strict positivity of the minimum eigenvalue of a context covariance matrix. These assumptions, however, impose strong restrictions on the context process, as they rule out the situation where the context vectors are concentrated in a low-dimensional subspace. In this paper, we propose SupSplitLog, which, to the best of our knowledge, is the first algorithm for logistic bandits that achieves $\tilde{\mathcal{O}}(\sqrt{dT})$ regret without any context diversity assumption. The key idea is to split the collected samples into two disjoint subsets when constructing estimators; one is used to compute an initial-point estimator, while the other is used to apply a Newton-type one-step correction procedure. The splitting rule is carefully designed to balance the accuracy requirements of the initial-point estimator and the one-step correction procedure. Moreover, SupSplitLog strictly improves on the existing algorithms in terms of the dependence on dimension $d$ in the regret upper bound. Furthermore, SupSplitLog can be adapted simply to deduce a regret bound that grows with a data-dependent complexity measure, avoiding a direct dependence on $d$, which is favorable when the context vectors are concentrated in a low-dimensional subspace. We also provide experimental results that demonstrate numerically the superiority of our algorithm, validating the theoretical results.

In machine learning, privacy requirements at inference or deployment time often evolve due to changing policies, regulations, or user preferences. In this work, we aim to construct a magnitude of models to satisfy any target differential privacy (DP) requirement without additional training, given a set of existing models trained on the same dataset with different privacy/utility tradeoffs. We propose two post-processing techniques, namely random selection and linear combination, to generate final private models satisfying any target privacy parameter. We provide privacy accounting of these approaches from the lens of R'enyi DP and privacy loss distributions on general problems, as well as on private mean estimation, where we precisely characterize the privacy/utility tradeoffs and compare the two mechanisms. Empirically, we demonstrate the effectiveness of our approaches and validate our analyses on several models and both synthetic and real-world datasets.

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

Neural operators are widely used to approximate solution maps of complex physical systems. In many applications, however, the goal is not to recover the full solution trajectory, but to summarize the solution trajectory via a scalar target quantity (e.g., a functional such as time spent in a target range, time above a threshold, accumulated cost, or total energy). In this paper, we introduce DOPE (debiased neural operator): a semiparametric estimator for such target quantities of solution trajectories obtained from neural operators. DOPE is broadly applicable to settings with both partial and irregular observations and can be combined with arbitrary neural operator architectures. We make three main contributions. (1) We show that, in contrast to DOPE, naive plug-in estimation can suffer from first-order bias. (2) To address this, we derive a novel one-step, Neyman-orthogonal estimator that treats the neural operator as a high-dimensional nuisance mapping between function spaces, and removes the leading bias term. For this, DOPE uses a weighting mechanism that simultaneously accounts for irregular observation designs and for how sensitive the target quantity is to perturbations of the underlying trajectory. (3) To learn the weights, we extend automatic debiased machine learning to operator-valued nuisances via Riesz regression. We demonstrate the benefits of DOPE across various numerical experiments.

Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric structure of a Riemannian manifold. Typically, the Laplace-Beltrami operator-based manifold regularization can be approximated empirically by the Laplacian regularization associated with the entire training data and its corresponding graph Laplacian matrix. However, the graph Laplacian matrix depends heavily on the prespecified similarity metric and may lead to inappropriate penalties when dealing with redundant or noisy input variables. To address the above issues, this paper proposes a new Semi-Supervised Meta Additive Model (S$^2$MAM) based on a bilevel optimization scheme that automatically identifies informative variables, updates the similarity matrix, and simultaneously achieves interpretable predictions. Theoretical guarantees are provided for S$^2$MAM, including the computing convergence and the statistical generalization bound. Experimental assessments across 4 synthetic and 12 real-world datasets, with varying levels and categories of corruption, validate the robustness and interpretability of the proposed approach.

We propose a distributionally robust approach to risk-sensitive estimation of an unknown signal x from an observed signal y. The observation and unknown signal are modeled as random vectors whose joint probability distribution is unknown, but assumed to belong to a given type-2 Wasserstein ball of distributions, termed the ambiguity set. The performance of an estimator is measured according to the conditional value-at-risk (CVaR) of the squared estimation error. Within this framework, we study the problem of computing affine estimators that minimize the worst-case CVaR over all distributions in the given ambiguity set. As our main result, we show that, when the nominal distribution at the center of the Wasserstein ball is finitely supported, such estimators can be exactly computed by solving a tractable semidefinite program. We evaluate the proposed estimators on a wholesale electricity price forecasting task using real market data and show that they deliver lower out-of-sample CVaR of squared error compared to existing methods.

In policy gradient reinforcement learning, access to a differentiable model enables 1st-order gradient estimation that accelerates learning compared to relying solely on derivative-free 0th-order estimators. However, discontinuous dynamics cause bias and undermine the effectiveness of 1st-order estimators. Prior work addressed this bias by constructing a confidence interval around the REINFORCE 0th-order gradient estimator and using these bounds to detect discontinuities. However, the REINFORCE estimator is notoriously noisy, and we find that this method requires task-specific hyperparameter tuning and has low sample efficiency. This paper asks whether such bias is the primary obstacle and what minimal fixes suffice. First, we re-examine standard discontinuous settings from prior work and introduce DDCG, a lightweight test that switches estimators in nonsmooth regions; with a single hyperparameter, DDCG achieves robust performance and remains reliable with small samples. Second, on differentiable robotics control tasks, we present IVW-H, a per-step inverse-variance implementation that stabilizes variance without explicit discontinuity detection and yields strong results. Together, these findings indicate that while estimator switching improves robustness in controlled studies, careful variance control often dominates in practical deployments.

Factor-based Structural Equation Modeling (SEM) relies on likelihood-based estimation assuming a nonsingular sample covariance matrix, which breaks down in small-sample settings with $p>n$. To address this, we propose a novel estimation principle that reformulates the covariance structure into self-covariance and cross-covariance components. The resulting framework defines a likelihood-based feasible set combined with a relative error constraint, enabling stable estimation in small-sample settings where $p>n$ for sign and direction. Experiments on synthetic and real-world data show improved stability, particularly in recovering the sign and direction of structural parameters. These results extend covariance-based SEM to small-sample settings and provide practically useful directional information for decision-making.

Data attribution and valuation are critical for understanding data-model synergy for Large Language Models (LLMs), yet existing gradient-based methods suffer from scalability challenges on LLMs. Inspired by human cognition, where decision making relies on a focused readout of relevant memories rather than replaying all pathways, we introduce RISE (Readout Influence Sketching Estimator). Instead of computing and indexing gradients across the entire LLM, RISE focuses on influence hotspots at the output layer, where influence signals concentrate, and the gradient admits a decomposed outer-product form. This enables a dual-channel representation combining a lexical residual channel (RH) and a semantic projected-error channel (GH). Applying CountSketch projections to these channels achieves strong compression while maintaining accurate attribution. Across the OLMo (1B-32B) and Pythia (14M-6.9B) families, RISE reduces index storage by up to 112$\times$ compared to RapidIn and scales to 32B parameters LLM, where gradient-based baselines such as RapidIn and ZO-Inf become memory-infeasible. We evaluate RISE on two paradigms: (1) retrospective attribution, retrieving influential training examples for specific predictions, and (2) prospective valuation, scoring candidate data utility zero-shot. We validate RISE on three tasks: Howdy backdoor data detection, Finance-Medical domain separation, and Brain Rot high-quality data selection. In a closed-loop Brain Rot study, continued pretraining on RISE-selected data yields consistent downstream improvements. Overall, RISE provides a practical and scalable primitive for influence analysis and training-data selection in modern large language models.

Synthetic augmentation is increasingly used to mitigate data scarcity in financial machine learning, yet its statistical role remains poorly understood. We formalize synthetic augmentation as a modification of the effective training distribution and show that it induces a structural bias--variance trade-off: while additional samples may reduce estimation error, they may also shift the population objective whenever the synthetic distribution deviates from regions relevant under evaluation. To isolate informational gains from mechanical sample-size effects, we introduce a size-matched null augmentation and a finite-sample, non-parametric block permutation test that remains valid under weak temporal dependence. We evaluate this framework in both controlled Markov-switching environments and real financial datasets, including high-frequency option trade data and a daily equity panel. Across generators spanning bootstrap, copula-based models, variational autoencoders, diffusion models, and TimeGAN, we vary augmentation ratio, model capacity, task type, regime rarity, and signal-to-noise. We show that synthetic augmentation is beneficial only in variance-dominant regimes, such as persistent volatility forecasting-while it deteriorates performance in bias-dominant settings, including near-efficient directional prediction. Rare-regime targeting can improve domain-specific metrics but may conflict with unconditional permutation inference. Our results provide a structural perspective on when synthetic data improves financial learning performance and when it induces persistent distributional distortion.

Power System State Estimation (PSSE) converts geographically distributed measurements into the voltage magnitudes and phase angles needed for grid monitoring and control. Learned estimators can perform this mapping rapidly, but model-aware False Data Injection Attacks (FDIAs) may corrupt their inputs while retaining AC plausibility and residual-based stealth. Physics-Informed Neural Networks (PINNs) limit candidate states through power-flow consistency; however, their robustness depends on balancing supervised and physics losses whose scales and gradient contributions evolve during training. This paper proposes a PINN that jointly learns homoscedastic uncertainty parameters and uses them to adapt the two objectives. The formulation assigns trainable log-uncertainties to the active-power, reactive-power, voltage, and angle losses while safeguarding against an underweighted physics objective. The estimator is trained only on clean steady-state data and is evaluated, without adversarial retraining, under baseline state-distortion and stricter residual-profile-matching regimes on the IEEE~118-bus system. Accuracy is measured against the uncompromised system state. Relative to a fixed-weight PINN, dynamic weighting reduces overall Mean Absolute Error (MAE) by $55\%$ while also improving voltage- and angle-estimation accuracy. The results show that learning the physics/data balance from clean data improves robustness to unseen FDIAs.

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

In contextual bilateral trade under full feedback, the posted price does not affect which valuations are observed. We show that in this model such action-independent feedback removes the polynomial adaptation penalty familiar from heavy-tailed bandits: fully parameter-free algorithms attain the oracle minimax $T$-exponents up to logarithmic factors, with no knowledge of the moment order $p \in (1,2)$ or its scale $σ_p$, and -- in the nonparametric case -- none of the effective Hölder smoothness $β\in (0,1]$. The statistic that makes model selection possible is a paired squared-loss difference, whose noise-square term cancels exactly, leaving noise damped by the candidate gap. The resulting bilateral-trade regret rates are new. Trader valuations have bounded conditional densities and heavy tails -- finite $p$-th moments for some $p \in (1,2)$, with possibly infinite variance. An epoch-based algorithm with truncated means achieves regret $\widetilde{O}(T^{(2-p)/p})$ in the parametric model and $\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$ when the market value function is $β$-Hölder, with matching $Ω(\cdot)$ lower bounds -- under a mild nondegeneracy condition -- via Assouad's method and a fixed-support mixture construction -- characterizing the minimax rate in $T$ up to logarithmic factors over the effective smoothness range $β\in (0,1]$, interpolating between the classical nonparametric rate at $p{=}2$ and the trivial linear rate as $p \to 1^+$. The enabling structural step extends the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations: within our conditionally independent, conditionally centered noise model, bounded conditional densities and finite first moments suffice for the expected regret of any price $π$ to satisfy $\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$ -- no second moment is needed.

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is known to be thorny, and a recent paper proposes a notably simple and effective solution. Additionally, it opens an remarkable flexibility when it comes to estimating observation parameters. The proposed strategy enables to define an optimal estimator for both observation parameters and image of interest. Furthermore, the strategy provides a means for uncertainty quantification. In addition, MCMC algorithms allow for the computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainty quantification.

The problem of corrupted data, missing features, or missing modalities continues to plague the modern machine learning landscape. To address this issue, a class of regularization methods that enforce consistency between imputed and fully observed data has emerged as a promising approach for improving model generalization, particularly in partially observed settings. We refer to this class of methods as Measure Consistency Regularization (MCR). Despite its empirical success in various applications, such as image inpainting, data imputation and semi-supervised learning, a fundamental understanding of the theoretical underpinnings of MCR remains limited. This paper bridges this gap by offering theoretical insights into when MCR yields a more favorable finite-sample estimation-error upper bound, viewed through the lens of neural network distance. Under ideal interpolation and compatibility conditions, we show that the MCR estimation-error upper bound is no larger than vanilla supervised training and becomes strictly smaller when the mixed-sample alternative is favorable. We then extend the analysis to the non-ideal regime, where optimization and compatibility residuals can potentially offset this finite-sample advantage. Guided by these insights, we propose a novel practical diagnostic that leverages the duality gap and a calibrated drift estimator to infer the potential benefit of MCR training. We present detailed empirical evidence to support our theoretical claims and to show the effectiveness and accuracy of our practical diagnostic. We further provide simulations on real-world datasets to show the versatility of MCR under different model architectures designed for different data sources.

Maximum likelihood estimators (MLE) and control variate estimators (CVE) have been used in conjunction with known information across sketching algorithms and applications in machine learning. We prove that under certain conditions in an exponential family, an optimal CVE will achieve the same asymptotic variance as the MLE, giving a fixed point algorithm for the MLE. Experiments show the fixed point algorithm is faster and numerically stable compared to other root finding algorithms for the MLE for the bivariate Normal distribution, and we expect this to hold across distributions satisfying these conditions. We show how this algorithm leads to reproducibility for algorithms using MLE / CVE, and demonstrate how the algorithm leads to finding the MLE when the CV weights are known.

Accurately and efficiently estimating the variance of the Maximum Mean Discrepancy (MMD) remains challenging, particularly for unbalanced sample sizes. In this paper, we derive a finite-sample unbiased estimator of the MMD variance. To overcome the traditional $\mathcal{O}(N^2)$ computational bottleneck, we develop a recursive prefix-suffix accumulation scheme for the Laplace kernel, reducing the computational complexity to $\mathcal{O}(N \log N)$ while requiring $\mathcal{O}(N)$ memory. Experimental results verify the theoretical exactness and numerical stability of the proposed estimator and demonstrate its scalability on large datasets. Furthermore, the method proves effective for monitoring distributional convergence during the training of Time-series Generative Adversarial Networks (TimeGAN).

Full conformal prediction is a framework that implicitly formulates distribution-free confidence prediction regions for a wide range of estimators. However, a classical limitation of the full conformal framework is the computation of the confidence prediction regions, which is usually impossible since it requires training infinitely many estimators (for real-valued prediction for instance). The main purpose of the present work is to describe a generic strategy for designing a tight approximation to the full conformal prediction region that can be efficiently computed. Along with this approximate confidence region, a theoretical quantification of the tightness of this approximation is developed, depending on the smoothness assumptions on the loss and score functions. The new notion of thickness is introduced for quantifying the discrepancy between the approximate confidence region and the full conformal one.

Sampling from unnormalized probability densities is a pervasive challenge across the computational and physical sciences. Diffusion models provide a powerful generative framework for this task, but their success relies on accurately estimating the score of the perturbed target distribution. Current approaches face a dichotomy between two standard estimation methods: the Denoising Score Identity (DSI) requires data samples and exhibits high variance at low noise levels, whereas the Target Score Identity (TSI) relies on the energy function and suffers from diverging variance at high noise levels. In this work, we reconcile both approaches by introducing the Control Variate Score Identity (CVSI), an unbiased estimator with an analytically optimal, state- and time-dependent control coefficient that theoretically minimizes variance over the entire diffusion process. CVSI serves as a robust plug-in estimator that significantly enhances performance and efficiency in data-free sampler learning and training-free diffusion sampling. These gains scale to complex, high-dimensional energy-based models.

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

We consider the problem of learning the parameters of a $N$-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time $T$. We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only $T=\mathcal{O}(\log N)$ observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.