non-asymptotic error analysis

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 work addresses the long-standing challenge of establishing finite-time convergence guarantees for GTD-class off-policy reinforcement learning algorithms (e.g., TDC, GTD2) under Markovian noise. We derive the first tight finite-time mean-square error bound for two-timescale linear stochastic approximation with Markovian noise. Methodologically, we integrate Lyapunov function analysis, matrix perturbation theory, and spectral analysis of Markov chains to precisely characterize the coupled dynamics of the two-timescale iterates. Our key contribution is an error upper bound whose dominant term is $mathrm{trace}(Sigma^y)/k$, which exactly matches the asymptotic covariance from the central limit theorem—thereby unifying finite-time bounds with asymptotic distributional characterization. Furthermore, we provide the first sample-complexity-optimal guarantees for TDC, GTD, GTD2, and Polyak–Ruppert averaged TD algorithms under Markovian sampling.

This work investigates the long-term approximation accuracy of stochastic gradient Langevin dynamics (SGLD) to continuous Langevin diffusion, focusing on uniform-in-time error bounds for the Kullback–Leibler (KL) divergence and Wasserstein/total variation distances between their invariant measures. Leveraging a synthesis of stochastic differential equation analysis, information-theoretic entropy estimation, and diffusion approximation theory under non-convex potentials, we establish, for the first time, a sharp, uniform-in-time $O(eta^2)$ upper bound on the KL divergence for step size $eta$. This directly implies $O(eta)$ bounds on the Wasserstein and total variation distances between the invariant measures. The results hold for general non-convex potentials and accommodate variable step sizes—significantly improving upon prior $O(eta)$ KL bounds. To date, this provides the strongest theoretical guarantee for the stability and statistical fidelity of SGLD in Bayesian inference and sampling.

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 convergence guarantees of stochastic line search optimization for over-parameterized models under interpolation conditions. We establish a necessary and sufficient condition on the search direction—applicable to a broad class of methods—that ensures finite termination and bounded backtracking steps, and rigorously prove linear convergence under the Polyak–Łojasiewicz (PL) assumption. The condition unifies major first-order strategies—including momentum, conjugate gradient, and adaptive preconditioning—providing a verifiable theoretical foundation for their principled integration with stochastic line search. Our analysis fills a critical gap in the convergence theory of stochastic line search methods and significantly extends both the applicability and reliability of efficient first-order optimization in interpolation learning regimes.

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference 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 addresses the lack of finite-time convergence guarantees in zeroth-order multi-timescale stochastic optimization by studying two-timescale gradient and three-timescale Newton methods that rely solely on function-value feedback. By employing smoothing functionals to estimate gradients and Hessians, the paper establishes the first non-asymptotic convergence analysis for zeroth-order multi-timescale algorithms, explicitly characterizing the coupling between timescales and the associated error propagation mechanisms. Key contributions include deriving mean-squared error bounds for Hessian estimation, providing a finite-time upper bound on the norm of the objective gradient, proving convergence to a first-order stationary point, and proposing a stepsize strategy that balances dominant error sources to achieve a near-optimal convergence rate. The theoretical findings are validated in the Continuous Mountain Car environment.

This study addresses the absence of globally reliable and computable neural network loss functions for parametric monotone nonlinear partial differential equations. To overcome this limitation, we propose error estimators based on operator splitting and discrete dual norms to construct variationally correct training objectives. Methodologically, the approach integrates first-order system least squares, discontinuous Petrov–Galerkin formulations, and computable surrogate dual norm techniques. Theoretically, monotonicity is leveraged to establish the two-sided reliability of the estimators over global trial spaces, thereby transcending the constraints of conventional local error estimation. This work achieves strict computability for arbitrary inputs, providing rigorous theoretical guarantees and an efficient training paradigm for neural network approximations of parameter-to-solution maps.

This work investigates the finite-time convergence of projected linear two-timescale stochastic approximation algorithms. For the constant-stepsize variant combined with Polyak–Ruppert averaging, it establishes the first explicit mean-squared error bound, cleanly decomposing it into an approximation error dictated by the constraint subspace and a statistical error that decays at a sublinear rate. This decomposition hinges on a restricted stability margin and a coupling invertibility condition, effectively disentangling the influence of subspace selection from that of the averaging window. The theoretical findings are validated through experiments on both synthetic data and reinforcement learning tasks, confirming the accuracy of the error decomposition and the algorithm’s superior performance.