Nonasymptotic Convergence Rates

• Nonasymptotic convergence rates are explicit error bounds defined as functions of sample size, iterations, and problem parameters without relying on asymptotic approximations.
• They are applied across stochastic optimization, statistical learning, and signal processing to offer actionable performance estimates in high-dimensional and computationally constrained settings.
• Methodologies include proximal algorithms, operator splitting, and stochastic sampling, with analyses that provide concrete constants and rate dependencies critical for robust algorithm design.

Nonasymptotic convergence rates quantify a statistical or optimization algorithm’s finite-sample or finite-iteration error, measured explicitly as a function of problem parameters and sample size or computational budget, without relying on asymptotic (infinite-sample/iteration) regimes. Such guarantees are essential in modern machine learning, signal processing, and computational mathematics, as they provide actionable performance estimates for high-dimensional or computationally constrained regimes and often fundamentally improve understanding beyond classic asymptotic analyses.

1. Mathematical Characterization and Frameworks

Nonasymptotic convergence rates provide explicit finite-$n$ (or $k$) control on algorithmic or estimator error, with constants and dependencies made explicit. A nonasymptotic rate typically takes the form

$$E_n \leq C(\text{problem parameters}, \text{confidence level}) \cdot R(n, d, \text{others})$$

where $E_n$ is a risk, optimization gap, or distance metric; $n$ is the number of samples or iterations; and $R(\cdot)$ is an explicit rate (e.g., $O(1/\sqrt{n})$, $O(1/k)$), with $C$ capturing dependencies on other parameters such as dimension, smoothness, or covering numbers.

Key frameworks in which nonasymptotic rates are analyzed include:

• Stochastic approximation and stochastic optimization (e.g., SGD, SPP, EM-type algorithms)
• Plug-and-play and variational methods (including nonconvex or composite settings)
• Empirical process and statistical learning theory
• Sampling and Monte Carlo methods (MCMC, Langevin, Quasi-Monte Carlo, etc.)
• Optimal transport and measure estimation

Rates are often established for risks such as normed errors, Wasserstein distances, KL divergence, suboptimality, or residuals, under explicit conditions on data, regularity, or algorithmic structure.

2. Proximal and Operator-Splitting Methods

Nonasymptotic rates have been rigorously developed for proximal-type algorithms and operator splitting frameworks, including stochastic proximal point (SPP), ADMM variants, and plug-and-play (PnP) methods with nonconvex/nonsmooth regularization.

• Stochastic Proximal Point & Linear Regularity: For problems with weak linear regularity, SPP achieves $\mathcal{O}(1/k)$ rates for the distance to optimality in convex optimization, even in the absence of strong convexity or smoothness. In shared minimizer (interpolation) regimes, rates improve to linear (geometric) convergence (Patrascu, 2019).
  • For constrained settings with potentially infinite set intersections, SPP with suitable step-size decay matches the optimal nonasymptotic rate for the expected squared error to the optimum, provided the linear regularity constant is controlled (Patrascu et al., 2017).
• Variable Metric ADMM: The variable-metric ADMM (VM-PADMM) achieves $\mathcal{O}(1/\sqrt{k})$ pointwise and $\mathcal{O}(1/k)$ ergodic nonasymptotic rates for KKT residuals, with explicit complexity depending on initial gap and target tolerance, even when the proximal metric is allowed to be degenerate and variable between iterations (Goncalves et al., 2017).
• Plug-and-Play Iterative Schemes: For plug-and-play optimization with MMSE denoisers, the implicit regularizer is characterized as an upper Moreau envelope of the negative log-marginal. This regularizer is 1-weakly convex, allowing one to prove (for data-fidelity with $L$-Lipschitz gradient, $L<1$) that the stationary residuals decrease at a rate $\mathcal{O}(1/\sqrt{k})$. This is (as of the cited work) the first such nonasymptotic sublinear guarantee for general PnP-PGD with Gaussian MMSE denoisers (Pritchard et al., 31 Oct 2025).

3. Stochastic Algorithms, Averaging, and Acceleration

The domain of stochastic approximation and learning exploits nonasymptotic analyses both for expected error and distributional convergence, often leveraging martingale and concentration techniques.

• Polyak-Ruppert Averaging (PR): For single timescale and linear two-time-scale stochastic approximation (TSA) algorithms, nonasymptotic central limit theorems (CLTs) in Wasserstein-1 can be established. With PR averaging, the mean error achieves the optimal $O(1/\sqrt{n})$ rate with a constant given by the expected norm of the limiting Gaussian. For unaveraged TSA, the best known rates are suboptimal, e.g., $O(n^{-a/2})$ for a step-size $\alpha_t\sim t^{-a}$ (Kong et al., 14 Feb 2025). This closes the nonasymptotic gap between single and two-time-scale methods.
• Incremental EM and Variants: For large-scale finite-sum problems, fast incremental expectation-maximization (FIEM) demonstrates, via stochastic approximation and uniform random-stopping analyses, nonasymptotic rates such as $O(n^{2/3}/\epsilon)$ or $O(\sqrt{n}/\epsilon^{3/2})$ for reaching an $\epsilon$-accurate stationary point, with less conservative step sizes than prior work (Fort et al., 2020).
• Accelerated (Inexact) First-Order Methods: Nonasymptotic PEP-based analysis for inexact OGM and FGM yields decoupled convergence rates: a rate term scaling as $O(1/K^2)$ (e.g., for $$f(\mathbf{x}_K) - f_* - \frac{1}{2L}\|\nabla f(\mathbf{x}_K)\|^2$$), and an accumulation of error determined by the magnitude and localization of gradient errors. The rate/error tradeoff is characterized, revealing that acceleration amplifies the impact of inexactness compared to non-accelerated schemes (Liu et al., 1 Aug 2024).

4. Stochastic Sampling and Markov Chain Algorithms

Explicit nonasymptotic convergence rates for Markov chain Monte Carlo (MCMC), Langevin, and Hamiltonian Monte Carlo methods are crucial for Bayesian inference and nonconvex optimization:

• SGLD/SGHMC and Langevin Schemes: Under local, rather than global, regularity assumptions, Wasserstein-$2$ convergence can be ensured uniformly in time at rate $O(h^{1/4})$ in the step size $h$, with independence from iteration count after mixing ($h$ being the discretization parameter); this improves upon previous bounds, which grew with $n$ or required global log-concavity (Akyildiz et al., 2020). Weak convergence for discretized SDEs is shown to be $O(h)$ and unaffected by randomization in the drift (Majka et al., 2018).
• Proximal/Stochastic Gradient Langevin Algorithms: For composite potentials with both smooth and nonsmooth terms, nonasymptotic rates for sampling algorithms such as SPLA are established. Sublinear rates ($O(1/\varepsilon^2)$ complexity for KL divergence $\le \varepsilon$) are obtained for convex settings, and linear rates (up to step size and variance-dependent bias) hold in strongly convex regimes (Salim et al., 2019).

5. Statistical Learning, Empirical Processes, and Optimal Transport

Nonasymptotic statistical rates are central for quantifying the efficiency of algorithms in empirical risk minimization, density estimation, and optimal transport.

• Empirical Measure and Wasserstein Distance: Explicit dimension-dependent nonasymptotic upper bounds for the expected $p$-Wasserstein distance between an empirical measure and its target in $\mathbb{R}^d$ are given, typically of the form $N^{-1/2}$ when $p > d/2$ and with moment/rate adjustments for other regimes. Explicit constants are provided, matching quantization rates (Fournier, 2022, Boissard et al., 2011).
• Optimal Transport Map Estimation: For the plug-in estimator of the Brenier map and their sieve variant, nonasymptotic $L^2$ rates $O(n^{-1/\gamma})$ for function class entropy exponent $\gamma$ are proved without requiring global compactness, convexity, or density lower-boundedness. When Poincaré-type inequalities (of a new, local form) hold and the function class is Donsker, rates improve to $O(n^{-2/(\gamma+2)})$. These results bridge the gap between classical theory—often demanding restrictive assumptions—and practical estimators, including those using neural network parameterizations (Ding et al., 11 Dec 2024).
• Nonparametric Density Estimation with GANs: For GANs with sufficiently expressive generator/discriminator classes (e.g., deep ReQU networks), JS divergence between true and generated distribution obeys the minimax-optimal rate (up to log factors), $(\log n / n)^{2\beta/(2\beta+d)}$ for smoothness $\beta$ and dimension $d$ (Puchkin et al., 2021).

6. Distributed Learning, Consensus, and Networks

Nonasymptotic analysis has enabled clear finite-time characterization for distributed and federated learning over networks:

• Distributed Belief Aggregation & Learning: Under time-varying, potentially directed graphs, beliefs on incorrect hypotheses decay geometrically, with explicit rates governed by the minimal collective KL divergence gap and the mixing parameters of the communication network, and explicit transient periods (Nedić et al., 2014). Protocol acceleration in static graphs enables mixing times that scale linearly (rather than quadratically) in the number of agents, optimizing network-wide learning (Nedić et al., 2015).

7. Impact and Methodological Directions

The development and application of nonasymptotic convergence rates has had several far-reaching implications:

• Robust algorithm design: Nonasymptotic rates enable algorithmic choices (e.g., step sizes, batch schedules, sample sizes) that are robust rather than optimistic, as they hold for all regimes, including high dimension, small data, or strong nonstationarity.
• Performance tuning without constants: Parameter-agnostic and adaptive algorithms, such as LeAP-SSN, achieve rates matching best-known local rates while not requiring knowledge of Lipschitz/Hessian/PL constants, making them well-suited for practical large-scale problems (Alphonse et al., 22 Aug 2025).
• Theory-practice gap closure: With advances such as local Poincaré-type inequalities and new function-class admissibility concepts (e.g., in OT map estimation), rates have been extended to heavy-tailed, unbounded, or geometrically complex distributions (Ding et al., 11 Dec 2024).
• Explicit constants and dimension-dependence: Recent work emphasizes explicit, computable rates (not simply asymptotic order), key for risk assessment in statistical learning, probability, or UQ.

Method/Setting	Metric/Residual	Nonasymptotic Rate
SPP (weak reg.)	Distance to optimal set	$\mathcal{O}(1/k)$
Variable-metric ADMM	KKT residuals (ergodic)	$\mathcal{O}(1/k)$
PnP-PGD + MMSE denoiser	Stationary subgrad norm	$\mathcal{O}(1/\sqrt{k})$
PR-Averaged (TSA, SGD)	Expected error	$\mathcal{O}(1/\sqrt{n})$
Empirical measure in $W_p$	Wasserstein distance	$N^{-1/d}$ (typical $d$)
OT map estimation (plug-in)	$L^2$ risk	$O(n^{-1/\gamma})$
GANs (JS, smooth class)	JS risk	$(\log n/n)^{2\beta/(2\beta+d)}$
SGHMC/SGLD (general data)	$W_2$ distance	$O(\eta^{1/4})$ ($\eta$: step size)

These advances enable practitioners and theorists to develop, implement, and analyze statistical or optimization algorithms with rigorous, finite-sample guarantees, in a robust, practical, and scalable fashion across domains.