measure concentration

This work addresses the limitations of traditional generalization analyses, which rely on the often unverifiable assumption of independent and identically distributed (i.i.d.) data and thus struggle to accurately characterize model performance on unseen data. The paper proposes a deterministic generalization analysis framework that dispenses with any prior probabilistic assumptions. By examining the sensitivity of optimization solutions to data perturbations, it decomposes the generalization error into geometric and probabilistic components, achieving their first-ever decoupling. The framework expresses generalization bounds via a variational principle, leveraging deterministic perturbation analysis and optimization sensitivity theory to capture the discrepancy between in-sample and out-of-sample performance. Error terms are evaluated through posterior statistical hypotheses, enabling the recovery of conventional high-probability or expected generalization guarantees—all without requiring distributional assumptions.

This paper addresses fundamental challenges in optimization and data science: streaming convex polyhedral approximation, sparse robust least-squares regression, adversarial optimization, modeling backdoor data poisoning attacks, and robustness analysis of graph clustering under “benign misspecification.” Methodologically, it unifies these problems through a high-dimensional geometric lens, integrating random projections, streaming algorithm design, statistical learning theory, and graph signal processing. Key contributions are: (1) the first provably secure theoretical model for backdoor attacks, yielding explicit safety thresholds; (2) novel algorithms with rigorous approximation and statistical guarantees, significantly improving streaming approximation accuracy and noise resilience; and (3) a characterization of strong consistency—despite model misspecification—for several classical graph clustering methods, precisely delineating their robustness boundaries.

The PAC-Bayes theory lacks systematic, accessible introductory resources for newcomers. Method: This paper introduces a pedagogical framework for beginners: (i) it unifies and clarifies the equivalence between Catoni’s localization technique and mutual information bounds, proposing a simpler, reproducible derivation paradigm; (ii) it explicitly positions PAC-Bayes as a theoretical bridge connecting Bayesian inference and frequentist generalization analysis; and (iii) it integrates core methodological components—including distributional modeling, KL-divergence constraints, randomized predictor construction, and information-theoretic bound derivation—into a coherent, understandable, and reproducible teaching system. Contribution/Results: The work fills a critical gap in foundational PAC-Bayes pedagogy and provides both theoretical grounding and practical methodology for applications such as posterior compression of neural networks in deep learning. It is poised to become a standard introductory reference for the PAC-Bayes community.

In high-dimensional, small-sample regimes, conventional machine learning generalization bounds suffer from excessive looseness due to the “curse of dimensionality.” Method: This paper proposes an adaptive family of generalization bounds tailored to discretized Euclidean spaces. We first derive a non-asymptotic concentration inequality for finite metric spaces; introduce geometric representation dimension (m) as a pivotal parameter; and construct bounds with constant factor (c_m) scaling as (sqrt{m}). Tight analysis is achieved via metric embedding combined with discretization-based modeling. Contribution/Results: The proposed bounds yield significant tightening under practical sample sizes; retain the optimal (O(1/sqrt{N})) convergence rate; and break the exponential or polynomial dependence on ambient dimension inherent in classical bounds—achieving constant-factor improvement even in high-dimensional, low-precision settings.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

This paper addresses the challenge of proving inequalities in high-dimensional spaces by introducing a systematic dimension-reduction framework: it structurally reduces inequalities over high-dimensional sets or probability distributions to well-understood, tractable one-dimensional problems. Methodologically, it unifies and extends the Lovász–Simonovits deterministic localization and Eldan’s stochastic localization theories—revealing their universal mechanisms across isoperimetric inequalities, concentration phenomena, convex optimization, and Markov chain mixing times. By integrating geometric probability, convex analysis, and measure concentration theory, the work establishes the first unified proof paradigm applicable to broad classes of strongly log-concave distributions. Key contributions include tight new bounds on isoperimetric constants, Lipschitz function concentration, convergence rates in convex optimization, and mixing time upper bounds—significantly advancing the state of the art in high-dimensional inequality analysis.

This study extends Lee’s (2009) bounds approach for treatment effects to settings with selective nonresponse and outcomes taking values in general metric spaces—such as compositional or distributional data. By embedding the Fréchet means of potential outcomes into Euclidean or Hilbert spaces, the authors characterize the identified set for the average treatment effect and construct corresponding estimators along with bootstrap-based confidence regions. This work represents the first extension of Lee’s bounds to non-Euclidean random objects, thereby overcoming the conventional restriction to real-valued outcomes. Numerical experiments demonstrate that the proposed method yields valid bounded estimates for both compositional and distributional data, confirming its feasibility and practical utility in complex data environments.

This work addresses the unclear theoretical mechanism by which data augmentation improves generalization through invariance to label-irrelevant transformations. The authors propose an information-theoretic framework that models augmentation as a mixture of the original and transformed distributions, deriving a mutual information–based generalization bound. This bound is innovatively decomposed into fidelity, stability, and sensitivity components, revealing an inherent trade-off between generalization and invariance learning under augmentation. To unify the geometric characterization of augmentations, they introduce the notion of “group diameter” and integrate orbit-averaged loss, sub-Gaussian assumptions, and geometric metrics under group actions into their analysis. Empirical results demonstrate that the proposed bound effectively tracks and predicts the true generalization gap, confirming the reliability and practical utility of the theoretical framework.

Traditional PAC-Bayesian frameworks provide only expected risk guarantees for randomized hypotheses, limiting their direct applicability to real-world deployment scenarios requiring a single deterministic classifier. Method: We propose the first general-purpose framework that systematically transforms randomized PAC-Bayesian bounds into provable risk upper bounds for a single deterministic hypothesis. Our approach unifies PAC-Bayesian theory, numerical optimization, and statistical learning to derive computationally tractable deterministic generalization bounds. Crucially, we establish a unified oracle bound and specialize it to weighted majority voting, significantly improving bound tightness. Contribution/Results: Theoretical analysis yields tighter, certifiable risk bounds for deterministic classifiers. Empirically, our method consistently outperforms state-of-the-art baselines across multiple benchmarks, achieving up to a two-fold improvement in bound tightness for deterministic risk. This work bridges the gap between PAC-Bayesian theory and practical deterministic model deployment, providing both theoretical foundations and an effective computational tool.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.