SEARCH · Search arXiv

Results for “stat.AP”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

528 records · Page 10Linked to original sources

arXiv2026

Non-Adaptive 1-Bit Mean Estimation: Minimax Rates and the Sample-Interval Tradeoff

We study distributed one-dimensional mean estimation under a 1-bit communication constraint. Each agent observes one sample, drawn independently from an unknown distribution, and returns a single bit in response to a query $Q: \mathbb{R}\to\{0,1\}$ chosen by a central learner. The distribution has mean in $[-λ,λ]$ and $k$-th central moment at most $σ^k$, for a fixed $k>1$. The order-optimal two-stage protocol of Lau and Scarlett uses responses from the first batch to choose the second-batch queries, motivating the question of whether this single round of interaction is necessary. We answer this negatively: for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate (and concurrent works reached the same conclusion via different strategies). We further determine the minimax sample complexity among non-adaptive 1-bit estimators when every one-set $Q^{-1}(1)$ is restricted to a union of at most $s$ intervals. Relative to unrestricted non-adaptive 1-bit querying, this constraint adds a term of order $(λσ/(s\varepsilon^2))\log(1/δ)$, giving the full tradeoff between sample complexity and interval complexity to within $k$-dependent constant factors. As a corollary, we identify, order-wise, the minimum interval budget needed to retain the unrestricted 1-bit minimax sample rate.

stat.ML↗

arXiv2026

Ideal Observer for Segmentation of Dead Leaves Images

The visible parts of a scene are determined by occlusion among overlapping surfaces. Here we consider "dead leaves" models, which replicate this by independently sampling objects ("leaves") with position, shape, color, and texture and layering them until the image is covered. Building on prior theory, we present a self-contained framework that rigorously defines the dead leaves model and derives an analytical Bayesian ideal observer for partitioning finite pixel sets. The longest part of the paper spans the derivation of the prior probability, which elevates the observer beyond pixel-similarity methods by incorporating geometric information. These computations are practical only for small pixel sets (up to 9-10 pixels). We emphasize accessibility through step-by-step derivations, extensive visualizations, and examples. We empirically evaluate three tractable observers (prior-only, likelihood-only, and the full ideal observer), plus a random baseline on 108 dead leaves image datasets varying in texture intensity, leaf size, and image size. Likelihood-only performance falls with increasing texture intensity and image size. Prior-only performance falls with decreasing leaf size and increasing maximal image dimension. All model-based observers strongly outperform the random baseline, and the ideal observer consistently outperforms the others by combining both information sources. The model provides a principled upper bound on segmentation performance for limited pixel sets, enabling comparisons with human observers and algorithms.

cs.CV↗

arXiv2026

Multi-Task Learning with Covariate-Overlap Regularization

Multi-task learning improves data efficiency by sharing information across related tasks, but indiscriminate sharing can be harmful when their covariate distributions and response relationships differ. We propose COVariate-ovERlap regularized multi-task learning (COVER) to address covariate and posterior heterogeneity. The model combines a common component function with a shared neural representation and low-dimensional task-specific coefficients. Taskwise second-moment matrices summarize covariate heterogeneity and determine the strength of coefficient integration in each representation direction. We derive a covariate-overlap penalty by minimizing the total squared change in two task predictors when their coefficients are replaced by one auxiliary coefficient. An equivalent auxiliary formulation supports end-to-end training without matrix inversion. An exact fixed-representation bias--variance decomposition quantifies how covariate overlap controls variance reduction and how posterior heterogeneity determines shrinkage bias. Global and localized end-to-end oracle inequalities account for jointly learning the neural functions and estimating the overlap matrices from the same observations. We give explicit neural-network rates and sharpen the stochastic prediction term when the regularized oracle risk and overlap-estimation error are small. Simulations across diverse heterogeneity settings show competitive performance against deep-learning and statistical data-integration methods, with the largest gains under joint heterogeneity. In a GTEx central-nervous-system analysis, COVER achieves the lowest response-averaged prediction error among the compared methods and reveals tissue-pair integration patterns.

stat.ML↗

arXiv2026

On the Sequential Test and Distributed Detection

We present a simple definition of stopping time and its role in the formulation of sequential tests for both centralized and distributed detection, providing a straightforward procedure for obtaining optimal decision rules. Upper bounds for optimal stopping time are derived and numerically shown to possess certain qualitative features expected of the optimal stopping time. The results are extended to any distributed detection network in the form of an acyclic directed graph.

cs.IT↗

arXiv2026

Bellman-sufficient Information Complexity

We introduce Bellman-sufficient information complexity for minimax analysis of sequential decision problems. A Bellman-sufficient state retains enough of the history to close the controlled recursion, while an index $Y=χ(Ω)$ specifies the decision-relevant information being charged. The upper bound is a log-penalized Bellman program; the lower bound is a Bellman--Fano comparison along an algorithm-dependent reference trajectory. If the two values match at a common localization scale and the stated admissibility, calibration, and growth conditions hold, they form an information-risk sandwich. UCB, E2D, and AMS/EBO control or relax the upper Bellman bracket in different ways. For the main application, we give a negative answer to a widely studied form of the GP--UCB minimax-optimality question. For every $0<α<1/4$, we construct one bounded continuous kernel whose minimax regret is $Θ(T^{1-α})$ along an infinite sequence of horizons, while two globally calibrated GP--UCB rules incur linear regret under one fixed truth. An epochwise finite-marginal action-index AIR Bellman policy, implemented through robust AIR/AMS/EBO control, attains the minimax order. The construction separates realized information from the cost of uniform optimism: many low-value directions inflate the exploration multiplier and change the trajectory. Through the canonical RKHS feature map, it also yields a finite-horizon polynomial minimax separation for the specified maximal-information-calibrated LinUCB rule. A reproducible experiment illustrates the mechanism.

cs.LG↗

arXiv2026

Decision-Centered Abstractions via Orthogonal Estimation of Difference-of-Q Functions

Offline reinforcement learning enables evaluation and optimization of sequential decisions from historical data, when it is not possible to deploy new policies online due to safety, cost, and other concerns. Big data advances enable rich state information, but may naively include reward- and action- irrelevant dynamics that are ultimately unnecessary for learning optimal actions. We introduce state abstractions that target preservation of the difference-of-Q functions, and we propose to learn these abstractions via causal machine learning of the difference-of-Q function and standard statistical sparse learning. Under a nonparametric additive-rewards model, we characterize when decision-centered abstractions are simpler than the full state space, motivating our estimation procedure. We develop a dynamic generalization of the R learner (Nie et al. 2021, Lewis and Syrgkanis 2021) for estimating difference of Q-functions, for discrete-valued actions a, a0. We leverage orthogonal estimation to improve convergence rates, even if the required estimates of Q and behavior policy converge at slower rates and prove consistency of policy optimization under a margin condition. The method can leverage black-box estimators of the Q-function and behavior policy to target estimation of a more structured Q-function contrast, and uses simple squared-loss minimization. We demonstrate variance improvements from our estimator and how our approach enables us to isolate the information needed for sequential decision-making, which can be less than that for state prediction, in simulated data and simulator-augmented real data.

stat.ML↗

arXiv2026

On Weighted Mathai-Haubold Entropy Measures

In this paper, we propose weighted Mathai-Haubold entropy along with their residual and past versions and study their properties. We develop aging classes based on the weighted Mathai-Haubold residual and past entropy measures. Also, some inequalities related to the three proposed measures are discussed. Non-parametric estimators of the proposed measures are introduced and their properties are investigated. Performance of this estimator is evaluated by means of bias and mean squared error using Monte-Carlo simulations.

cs.IT↗

arXiv2026

Robust Assortment Optimization from Observational Data

Assortment optimization is a fundamental challenge in modern retail and recommendation systems, where the goal is to select a subset of products that maximizes expected revenue under complex customer choice behaviors. While recent advances in data-driven methods have leveraged historical data to learn and optimize assortments, these approaches typically rely on strong assumptions -- namely, the stability of customer preferences and the correctness of the underlying choice models. However, such assumptions frequently break in real-world scenarios due to preference shifts and model misspecification, leading to poor generalization and revenue loss. Motivated by this limitation, we propose a robust framework for data-driven assortment optimization that accounts for potential distributional shifts in customer choice behavior. Our approach models potential preference shift from a nominal choice model that generates data and seeks to maximize worst-case expected revenue. We first establish the computational tractability of robust assortment planning when the nominal model is known, then advance to the data-driven setting, where we design statistically optimal algorithms that minimize the data requirements while maintaining robustness. Our theoretical analysis provides both upper bounds and matching lower bounds on the sample complexity, offering theoretical guarantees for robust generalization. Notably, we uncover and identify the notion of ``robust item-wise coverage'' as the minimal data requirement to enable sample-efficient robust assortment learning. Our work bridges the gap between robustness and statistical efficiency in assortment learning, contributing new insights and tools for reliable assortment optimization under uncertainty.

stat.ML↗

arXiv2026

One-Layer Transformer Provably Learns Multiclass One-Nearest Neighbor in Context

We extend recent work establishing an equivalence between one-layer transformers and nearest-neighbor classifiers in the binary setting to the multiclass case. By leveraging the simplex encoding, we show that one-layer transformers with an argmax classification head behave identically to a one-nearest-neighbor classifier in the multiclass setting. This closes a gap left by prior work, whose multiclass result relied on a non-standard rounding-based approach rather than the typical argmax head used in practice.

cs.LG↗

arXiv2026

Risk-Aware Goal-Oriented Bayesian Optimal Experimental Design

Traditional Bayesian optimal experimental design (OED) selects measurements that best inform a model's parameters. However, such measurements can be suboptimal for downstream predictions. Goal-oriented OED targets the prediction directly. However, the existing goal-oriented criteria value all reductions in predictive uncertainty equally, with no way to prioritize rare, high-consequence outcomes. In this article, we develop a risk-aware framework that composes risk at three levels, each generalizing an ingredient of classical $I$- and $G$-optimal design: a deviation measure of the posterior predictive uncertainty (generalizing the predictive variance), a risk measure across the prediction domain (interpolating $I$-optimal averaging and $G$-optimal worst-case selection), and a risk measure over datasets (generalizing the expectation). We generate each level from a regret function in the risk quadrangle, so that one triple specifies a practitioner's risk preference. We relax the design to continuous weights on the unit simplex and construct a nested-quadrature estimator that is differentiable in the design variable. This enables solving the optimal design problem with gradient-based methods, avoiding a combinatorial search over candidate designs. For a linear-Gaussian lognormal model and a nonlinear extension, we derive closed-form objectives. These give exact references against which we verify that the estimator converges. We demonstrate this framework for finding optimal sensor placements in an inverse problem governed by an advection-diffusion equation. We find that the risk-aware designs substantially outperform the expected-information-gain baseline, which is statistically indistinguishable from a random allocation.

stat.ME↗

arXiv2026

Generalized Regret Analysis of Thompson Sampling using Fractional Posteriors

Thompson sampling (TS) is one of the most popular and earliest algorithms to solve stochastic multi-armed bandit problems. We consider a variant of TS, named $α$-TS, where we use a fractional or $α$-posterior ($α\in(0,1)$) instead of the standard posterior distribution. To compute an $α$-posterior, the likelihood in the definition of the standard posterior is tempered with a factor $α$. For $α$-TS we obtain both instance-dependent $\mathcal{O}\left(\sum_{k \neq i^*} Δ_k\left(\frac{\log(T)}{C(α)Δ_k^2} + \frac{1}{2} \right)\right)$ and instance-independent $\mathcal{O}(\sqrt{KT\log K})$ frequentist regret bounds under very mild conditions on the prior and reward distributions, where $Δ_k$ is the gap between the true mean rewards of the $k^{th}$ and the best arms, and $C(α)$ is a known constant. Both the sub-Gaussian and exponential family models satisfy our general conditions on the reward distribution. Our conditions on the prior distribution can be easily satisfied by a density that is positive, continuous, and bounded. We also establish another instance-dependent regret upper bound that matches (up to constants) to that of improved UCB [Auer and Ortner, 2010]. Our regret analysis carefully adapts and combines recent theoretical developments in the non-asymptotic concentration analysis and Bernstein-von Mises type results for the $α$-posterior distribution. Moreover, our analysis does not require additional structural properties such as closed-form posteriors or conjugate priors.

stat.ML↗

arXiv2026

Debiased Inference for AI-Generated Data without Gold-Standard Labels: Identification via Multiple Imperfect Measurements

An increasing number of scholars use AI to measure variables they subsequently include in downstream analyses. Although AI-measured variables are often analyzed as if observed without error, ignoring prediction errors in automated measurement leads to substantial bias and invalid confidence intervals in downstream analyses, even if AI measurement accuracy is high, e.g., above 90%. Existing solutions, such as design-based supervised learning and prediction-powered inference, combine error-prone AI-based measurements with gold-standard labels, which may be costly and difficult to obtain in some application areas. In this paper, we propose debiased inference with multiple imperfect measurements (DMM), a framework that combines multiple error-prone AI measurements to enable valid downstream inference without gold-standard labels. Building on the established results on CP decomposition, DMM assumes that these measurements are independent conditional on the latent true label and observed unit-level features, such as text features represented by embeddings. This framework allows for unknown misclassification rates to vary across annotation methods (e.g., large language models) and across units of annotation (e.g., texts). Under this assumption, we use semiparametric inference theory to prove that the DMM estimator is consistent and asymptotically normal, enabling valid inference for a wide range of downstream statistical analyses common in the social sciences. Our simulation results show that DMM yields valid inference and that adding accurate, though imperfect, measurements can improve efficiency. Focusing on common applications of large language model annotations, we also develop diagnostics to assess the conditional independence assumption.

stat.ME↗

arXiv2026

On the Equality of the ELBO to a Sum of Entropies at Stationary Points of Learning

The variational lower bound (a.k.a. ELBO or free energy) is the central objective for many established as well as for many novel algorithms for unsupervised learning. Such algorithms usually increase the bound until parameters have converged to values close to a stationary point of the learning dynamics. Here we show that (for a very large class of generative models) the variational lower bound is at all stationary points of learning equal to a sum of entropies. Concretely, for standard generative models with one set of latents and one set of observed variables, the sum consists of three entropies: (A) the (average) entropy of the variational distributions, (B) the negative entropy of the model's prior distribution, and (C) the (expected) negative entropy of the observable distribution. The obtained result applies under realistic conditions including: finite numbers of data points, at any stationary point (including saddle points) and for any family of (well behaved) variational distributions. The class of generative models for which we show the equality to entropy sums contains many standard as well as novel generative models including standard (Gaussian) variational autoencoders. The prerequisites we use to show equality to entropy sums are relatively mild. Concretely, the distributions defining a given generative model have to be of the exponential family, and the model has to satisfy a parameterization criterion (which is usually fulfilled). Proving equality of the ELBO to entropy sums at stationary points (under the stated conditions) is the main contribution of this work.

stat.ML↗

arXiv2026

Common-Center Geometry and Certified Radial Reconstruction for Energy-Form Full Conformal Regions

This note studies the geometry of full conformal prediction (FullCP) regions generated by an empirical energy-form pairwise score. Candidate-score convexity alone does not guarantee connected FullCP regions, even for empirical averages of losses convex in the candidate argument. For the energy-form score, each leave-one-out training comparison reduces exactly to a pairwise-dissimilarity sublevel condition. Under symmetry, a constant diagonal, a diagonal lower bound, and attainment of the associated Fréchet-type objective, all comparison regions share a minimizer; if they are convex, every nontrivial exact conformal region is star-shaped about that point. For power distances $ρ_β(x,y)=\|x-y\|^β$, this geometry holds for $β\ge1$, while the conventional energy score is strictly proper for $0<β<2$. For $d=1,β=1$, every nontrivial empirical-CRPS FullCP region is a nonempty closed interval (possibly $\mathbb R$ when $m=1$). For $1<β<2$ and $m\ge2$, explicit data-checkable derivative bounds give Lipschitz control of the radial exits and exact conformal radial function. Combined with directional root search and classical Lipschitz extensions, they yield certified inner and outer radial envelopes of width at most $δ+2\widetilde Lh_{\mathcal U}$ and same-ray Hausdorff guarantees. An analytic two-dimensional example shows why preserving star-shaped but nonconvex geometry can matter. A staged two-dimensional study finds modest but systematic tightening of the generic certificate and frequent robust nonconvexity witnesses, with detected normalized radial departures typically small. The method is intended for low-dimensional multivariate outputs rather than high-dimensional scaling or runtime improvement.

stat.ML↗

arXiv2026

Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space

Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the vocabulary size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes vocabulary-size-independent convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our results can extend existing step complexity guarantees to any IPM. Also, our theory relies only on a single standard rate-matrix regularity assumption and applies to general priors. Five novel techniques drive our improvements: 1. working in the space of observables via adjoint equations rather than directly with probability measures; 2. a regularity analysis that yields bounds on any IPM; 3. a coupling argument that removes $S$-dependence under uniform transitions; and 4. score-marginal cancellation and 5. exit-routing techniques that remove $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models, including principled choices of loss functions and vocabulary-size-independent step complexity.

cs.LG↗

arXiv2026

When the Martingale Never Stops Firing: Anytime-Valid Gating on Real Forecast Streams

Machine learning systems are increasingly corrected while they run, and the decision of when to intervene is increasingly delegated to statistical monitors. Anytime-valid inference promises evidence that can be acted on at any moment, exactly the guarantee this setting needs, and it is moving from theory into deployed monitoring. Conformal test martingales are the change-detection instrument, and Ville's inequality caps their false-alarm probability on exchangeable data. The guarantee is conditional. A deployment inherits it only if the stream it monitors behaves exchangeably. The premise is hardest to satisfy where these monitors are most useful, on dependent data and inside loops where the monitor modifies the learner whose scores it reads. It is also rarely measured. We measure it in a pre-specified case study, where such a monitor gates the online updates of a Kalman adapter correcting frozen time-series foundation models on five forecasting streams. On exchangeable synthetic streams, the same implementation fires in at most 1 of 60 runs. On the real streams, at alpha = 0.05, 135 of 135 clean-stream runs fired. The construction does not explain the firing; the failure comes from the deployed score stream itself. Repeated fires hold the gate's drift response active, and the gated filter amplifies the very transient it was designed to prevent. The component worth keeping makes no validity claim. Huber-style gating of the filter's own updates cuts isolated-spike degradation by an order of magnitude with no dataset specific tuning. Anytime-valid methods proposed for dependent data should therefore be accompanied by null-calibration controls and mechanism traces.

cs.LG↗

arXiv2026

Low-Rank Tensor Estimation from Nonlinear Observations: A Unified Framework

We consider the estimation of a $d_1\times d_2\times d_3$ tensor $X^\star$ of Tucker rank $(r_1,r_2,r_3)$ from the nonlinear observations $\{y_i=f_i(\langle A_i,X^\star\rangle)\}_{i=1}^n$. We develop a unified approach that first constructs a gradient map from the data and then establishes the tensor restricted approximate invertibility condition (T-RAIC), a condition that quantifies how well the gradient map aligns with the ideal descent step under a low-rank tensor dual norm. We show that T-RAIC yields local linear convergence guarantees for a Riemannian gradient descent (RGD) algorithm, which may incorporate a normalization step if $\|X^\star\|_{\rm F}$ is known a priori. Under $O(r_1r_2r_3+\sum_{1\le i\le 3}r_id_i)$ Gaussian measurements, we establish T-RAICs for single-index models, logistic regression, phase retrieval, ReLU regression, and one-bit compressed sensing. The RAICs imply that RGD locally converges to $X^\star$ exactly in phase retrieval and ReLU regression, and up to near-optimal estimation errors in the remaining models. We further show that, in all these models except for phase retrieval, a simple spectral initialization yields the desired initialization from $O(d^{3/2})$ measurements under $d_1=d_2=d_3=d$ (ignoring dependence on the Tucker rank and condition number of $X^\star$). This is also the best known sample complexity for polynomial-time and end-to-end algorithms in tensor linear regression and tensor completion. Numerical simulations are provided to corroborate our theoretical findings.

math.ST↗

arXiv2026

Accountable and uncertainty-aware evaluation of sensor-based AI under distribution shift: devices, subjects, and nearly three years underground

Sensor-based AI systems are rarely operated under the conditions under which they were trained: devices, personnel and recording epochs change, and each change degrades performance in ways a random train-test split cannot reveal. We propose a staged, accountable evaluation protocol that treats the evaluation of a deployed model as a measurement with declared reference levels and a quantified uncertainty. Four cumulative generalisation stages hold out devices, subjects and time. Each stage is judged on quantiles of repeated trainings against chance references with the correct class count, an out-of-present-scope rate exposes silent misdirection towards classes that are no longer present in deployment relative to training, and an explicit decision rule ties roll-out decisions not to means but to 5% quantiles. We demonstrate the protocol on infrastructure-free geomagnetic localisation with smartphone-based recurrent classifiers in two real underground mines, including a replication of the scheme's training stages at the second site. Unchanged models are re-evaluated on data recorded 34 months after the training campaign, on a device generation unknown at training time and with a held-out surveyor. The 5% quantile of their present-conditioned precision there is 0.39 over 299 repeated trainings, 16.5 times the chance level; across the composition of the 42 reachable location classes the figure varies by +/-0.08, several times the spread between repeated runs. Repeated trainings of a single configuration show why means mislead: a bimodal configuration passes a mean-based test decisively while its 5% quantile lies more than an order of magnitude below chance.

cs.LG↗