gronwall inequality analysis

This work addresses the absence of finite-sample error bounds and concentration inequalities for nonlinear stochastic approximation algorithms under the Wasserstein-p distance. By coupling the discrete-time iterative process with its Ornstein–Uhlenbeck diffusion limit, the paper establishes the first non-asymptotic distributional convergence rates in Wasserstein distance under general noise conditions—such as martingale differences and ergodic Markov chains. The main contributions include proving that the last iterate converges to a Gaussian distribution at a rate of γₙ^{1/6}, while the Polyak–Ruppert averaged iterate achieves a rate of n^{-1/6}. Moreover, the analysis yields high-probability concentration inequalities that improve upon those derived via classical moment-based methods. The proposed framework applies broadly to canonical algorithms, including linear stochastic approximation and stochastic gradient descent.

This work investigates the non-asymptotic pathwise approximation accuracy of stochastic iterative algorithms—such as SGD and SGLD—to the Ornstein–Uhlenbeck process in the univariate setting. Addressing the lack of quantifiable, path-level error bounds in existing theory, we introduce a novel analytical framework for path space by integrating infinite-dimensional Stein’s method with exchangeable pair techniques. This yields explicit convergence rates under both the Lévy–Prokhorov metric and the bounded Wasserstein distance, delivering tight, non-asymptotic upper bounds on pathwise approximation error. We rigorously establish weak convergence and provide quantitative error control for both the iterates’ mean and variance. The framework thus furnishes a foundational toolset for extending the analysis to multivariate settings and more complex stochastic optimization algorithms.

Discrete diffusion models suffer from a lack of systematic theoretical error analysis, limiting their accuracy and reliability in generative modeling. To address this, we introduce the first unified analytical framework for discrete diffusions based on Lévy-type stochastic integrals, establishing— for the first time—the stochastic integral representation of discrete diffusion processes and proposing a Girsanov-type measure transformation theorem. Building upon this foundation, we derive the first explicit KL-divergence error bound for the τ-leaping discretization scheme. Our framework bridges the theoretical gap between discrete and continuous diffusion models, explicitly characterizing error sources—including state dependence of intensity functions and approximation bias from jump truncation. It unifies and strengthens existing convergence and stability results, providing a rigorous mathematical foundation and practical design principles for developing efficient, verifiable discrete diffusion algorithms.

This work investigates the non-asymptotic convergence of Particle Gradient Descent (PGD) for maximum likelihood estimation in large latent-variable models. Addressing free energy functional optimization, we introduce a unified generalization of logarithmic Sobolev and Polyak–Łojasiewicz-type conditions, establishing their first equivalence with Talagrand’s inequality and quadratic growth—thereby extending the Bakry–Émery theory. Leveraging tools from optimal transport, information geometry, and stochastic differential equations, we prove that under strong concavity of the log-likelihood, PGD—implemented as a discrete-time approximation of the free energy gradient flow—achieves exponential convergence. Moreover, we derive the first tight non-asymptotic upper bound on the discretization error. These results provide foundational theoretical guarantees for PGD in latent-variable modeling, marking a key advance in the rigorous analysis of particle-based variational inference methods.

This work addresses the failure of Wormald’s differential equation method when only an upper bound on the expected one-step change is available—lacking a lower bound or tight estimate—and establishes, for the first time, a theoretical framework under one-sided constraints. Methodologically, it integrates martingale inequalities, coupling techniques, and discrete dynamical systems analysis to derive a general one-sided concentration theorem. Theoretically, it rigorously proves that an upper bound alone suffices to guarantee, with high probability, that the trajectory of a random process stays close to the solution of the associated deterministic differential equation; moreover, the original Wormald method emerges naturally as a special (non-degenerate) case. Practically, this significantly lowers technical barriers, providing a more flexible and robust analytical tool for settings lacking symmetric estimates—such as greedy algorithm analysis and stochastic graph evolution processes.

This work investigates the concentration of iteration errors in stochastic approximation algorithms driven by heavy-tailed Markov noise, covering both expansive and non-expansive operator settings. Under a framework involving a finite-state Markov component and martingale difference noise, the authors construct a novel Lyapunov function via the moment-generating function of the solution to the Poisson equation, complemented by auxiliary projection and black-box truncation techniques to reduce unbounded noise to a bounded setting. The study provides the first systematic characterization of the fine structure of error tails: under bounded noise, tails can be sub-Gaussian, sub-Weibull, or intermediate between Pareto and Weibull; under unbounded noise, if the operator is almost surely non-expansive, the error tail is at most three times heavier than that of the noise, whereas if the operator is expansive with positive probability, significantly heavier tails may arise, with sharp worst-case examples demonstrating the tightness of these bounds.

This study addresses the critical challenge of reliably estimating sharp lower bounds for the standard errors of moment condition estimators when cross-sample correlation information is either absent or only partially available. By leveraging geometric inequalities, the authors derive explicit and tight lower bounds on standard errors and show that the general problem can be reformulated as a semidefinite programming (SDP) problem amenable to efficient computation. This approach yields the first sharp error bounds in settings with no knowledge of cross-sample correlations. Integrating insights from moment condition estimation and statistical inference theory, the method demonstrates both validity and practical utility across several applications, including menu cost models, heterogeneous-agent New Keynesian frameworks, and two-sample instrumental variable settings.

This paper addresses the non-asymptotic (L^2) polynomial approximation of smooth functions under measures satisfying the Carleman condition—including multivariate sub-Gaussian and sub-exponential distributions. Motivated by an open problem on smoothed analysis posed by Chandrasekaran et al., we develop a quantitative version of the Denjoy–Carleman theorem, integrating tools from complex analysis, quasianalytic function theory, and joint moment–smoothness analysis to construct a unified approximation framework. Our key contributions are threefold: (i) the first derivation of superexponential approximation rates for Paley–Wiener-type function classes under general sub-exponential measures; (ii) a novel characterization of these rates in terms of moment-based smoothness parameters; and (iii) substantial quantitative improvements over existing (L^2) approximation bounds—specifically, sharper dependence on dimension and smoothness—thereby providing a rigorous theoretical foundation for smoothness modeling in high-dimensional statistical learning.

This work addresses the looseness of existing non-asymptotic error bounds for Langevin Monte Carlo methods under strongly log-concave distributions, which overly rely on global smoothness constants and consequently deteriorate in high-dimensional or correlated-covariate settings. To overcome this limitation, the paper introduces a coordinate-wise averaged smoothness condition to characterize the potential function and combines synchronous coupling with Wasserstein distance analysis to derive substantially tighter bounds. The key innovation lies in replacing the global smoothness constant with an average coordinate-wise counterpart and employing a trace-type third-order smoothness quantity to weaken the Hessian-Lipschitz assumption. These improvements are extended to variable step sizes, Laplacian-smooth potentials, and finite-sum structures such as SGLD. Notably, the resulting bounds exhibit significantly improved dimension dependence in high-dimensional generalized linear models, especially under covariate correlation, offering broader theoretical applicability and outperforming current state-of-the-art results.

This work establishes rigorous theoretical bounds on the sampling error of Denoising Diffusion Probabilistic Models (DDPMs) measured in the 2-Wasserstein distance. Departing from the conventional view of DDPM samplers as discretized reverse Ornstein–Uhlenbeck processes, the paper introduces a novel perspective by modeling them as discretizations of Föllmer processes. Under general Lipschitz-type assumptions on the score function and across various variance schedules—including the cosine schedule—it derives non-asymptotic Wasserstein error bounds. The key contributions include proving that the Lipschitz condition implies both a logarithmic Sobolev inequality and a quadratic transportation-cost inequality, and demonstrating that even when the target distribution fails to satisfy the latter, dimension- and step-optimal Wasserstein error bounds can still be achieved. Furthermore, existing KL divergence bounds are extended to the Wasserstein setting.