derive nonasymptotic mse bounds

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.

This paper studies sequential mean-squared error (MSE) estimation and optimal $m$-dimensional subset identification for a $K$-dimensional Gaussian vector, under a feedback-constrained setting where only $m < K$ components are observable per round. To address this problem, we propose the first feedback-aware estimation framework tailored for MSE-optimal subset identification. We design an adaptive regression-based estimator and an enhanced successive elimination algorithm, substantially improving both estimation accuracy and subset identification reliability. Leveraging concentration inequalities and minimax theory, we derive a tight lower bound on sample complexity. Theoretically, our estimator exhibits superior concentration properties; the algorithm identifies the MSE-optimal $m$-dimensional subset with high probability; and we establish the fundamental sample-efficiency limit for this task.

This paper addresses the finite-sample error and computational inefficiency of sequential Monte Carlo (SMC) methods in static spaces. We propose a theoretical analysis framework grounded in interpolation distribution design. First, we establish finite-sample convergence guarantees for SMC without assuming bounded importance weights—an unprecedented result. Second, via $L_2$-error analysis and Markov chain mixing time theory, we derive explicit theoretical bounds quantifying how interpolation distribution choice affects estimation error. Third, we provide rigorous theoretical justification for adaptive path selection based on relative effective sample size (REES), overcoming the bias and degeneracy inherent in conventional data-tempering approaches. Experiments demonstrate that our adaptive method automatically approximates the optimal interpolation sequence, achieving both significantly reduced computational complexity and improved approximation accuracy.

This paper addresses the optimal nonparametric estimation of the covariance kernel under supremum-norm loss for synchronously sampled functional data. We propose a kernel-based estimator constructed directly from discrete synchronous observations, without requiring prior estimation or smoothness assumptions on the mean function; it accommodates high-order smoothness off the diagonal while allowing low regularity on the diagonal. Theoretically, we establish, for the first time under dense sampling, a $sqrt{n}$-rate convergence without logarithmic penalties; under sparse sampling, the rate degrades to that of one-dimensional mean estimation, breaking the two-dimensional structural barrier. We derive an information-theoretic lower bound and achieve matching upper bounds, yielding tight minimax-optimal rates. Moreover, in the dense regime, we prove a supremum-norm central limit theorem, enabling construction of uniform confidence sets. Simulation studies and real-data analysis demonstrate the method’s effectiveness and robustness.

This work addresses the absence of explicit error bounds for expectation estimators in adaptive rare-event Markov chain Monte Carlo (MCMC) by establishing the first explicit mean squared error upper bound for the time-averaged estimator of adaptive incremental rare-event MCMC under a simultaneous Wasserstein contraction condition. The proposed method integrates normalizing flows, adaptive stereographic projection, and the Metropolis–Hastings algorithm, and is extended to a general adaptive framework applicable to doubly intractable problems. The theoretical analysis combines Wasserstein contraction properties with complexity assessments, yielding an error bound that informs computational resource allocation for achieving a desired target accuracy. Experimental evaluations across multiple adaptive schemes confirm both the tightness of the derived theoretical bound and the scalability of the algorithm.

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 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 study addresses the joint optimization of experimental design and estimation to minimize worst-case mean squared error (MSE) in finite populations with bounded potential outcomes, such as binary outcomes. By analyzing all assignment mechanisms within the class of affine estimators, the authors demonstrate that independent randomization coupled with an intercept-free regression using midpoint-centered covariates achieves optimal performance. This approach reduces the worst-case MSE by 50% compared to classical paired randomization with fixed-effects regression. Moreover, the optimality of this method is shown to extend beyond the affine class, establishing the theoretical superiority of independent random assignment in minimizing worst-case estimation error under bounded potential outcomes.