Non-Asymptotic Error Bounds

• Non-asymptotic error bounds are explicit finite-sample guarantees that quantify estimation or prediction errors in terms of model, dimension, and algorithm parameters.
• They employ spectral, algebraic, and probabilistic techniques to provide rigorous performance metrics in settings where asymptotic theory falls short.
• These bounds guide practical algorithm tuning by balancing bias, noise, and computational complexity in regression, tracking, and summation tasks.

Non-asymptotic error bounds provide explicit, finite-sample guarantees on estimation or prediction error for statistical and computational procedures, rather than asymptotic rates or limiting distributions. Unlike classical asymptotic results, these bounds hold for any sample size, iteration count, or computational budget, and their constants and dependencies are fully specified in terms of model, problem dimension, and algorithm parameters. The move toward non-asymptotic analysis allows rigorous performance assessment in practical, high-dimensional, or complex settings where traditional asymptotic theory fails to capture operational guarantees.

1. Core Methodologies and Representative Models

Non-asymptotic error analysis spans a wide range of settings, each requiring tailored techniques depending on the model structure and error metrics.

• Linear Regression on Time Series (LTI Models): For least-squares regression of the system matrix $A$ in $x_{t+1}=A x_t$, deterministic non-asymptotic error bounds are derived for the estimator $\hat A_k=Y_k X_k^\dagger$, where $X_k=[x_0,\dots,x_k]$ and $Y_k=[x_1,\dots,x_{k+1}]$ (Alaeddini et al., 2018). The key object controlling bias in the under-determined case ($k+1<n$) is the projector $S_k=I_n-U_{k-1} U_{k-1}^\top$ onto the nullspace of $X_{k-1}$, with instantaneous covariance $P_k=x_k x_k^\top$. The error is

$$\|\hat A_k-A\|_2 \leq \left\|I_n-(S_k P_k)/\operatorname{Tr}(S_k P_k)\right\|_2$$

and the analysis incorporates the effect of symmetry and eigenvalue multiplicity in $A$.

• Stochastic Approximation and Tracking: For constant-stepsize SA schemes in $d$ dimensions,

$$x_{n+1} = x_n + a\left(h(x_n,y_n) + M_{n+1} + \varepsilon_{n+1}\right)$$

with slowly moving target $y_n$, the non-asymptotic tracking error is given by (Kumar et al., 2018)

$$\mathbb{E}\|x_n-\lambda(y_n)\|^{1/2} \leq \frac{\varepsilon^*}{\beta} + \frac{K_\gamma L_\lambda}{\beta}\varepsilon + O\Big(\max\{a^{1.5}d^{3.25},a^{0.5}d^{2.5}\}\Big) + e^{-\beta(t_n-t_0)}\|x_0-\lambda(y_0)\|$$

where $\beta$ is the exponential contraction rate of the underlying ODE.

• Floating Point Summation: Deterministic and probabilistic non-asymptotic error bounds for forward error in summation algorithms are derived via martingale-on-a-tree recurrences, controlling both "first order" and "higher order" rounding terms in terms of unit roundoff $u$, tree height $h$, and the structure of the summation (general, shifted, compensated, or mixed-precision algorithms) (Hallman et al., 2022).
• Sequential MCMC and Feynman–Kac Propagators: Particle-based approximations to target measures $\mu_n$ are analyzed using norm-stability of the propagators $q_{j,k}$. Under suitable mixing and density bounds, for $N$ particles,

$$N\mathbb{E}|\mu_N(f)-\mu_n(f)|^2 \leq \sum_j \operatorname{Var}_{\mu_j}(q_{j,n}(f)) + C_n \|f\|_n^2 \varepsilon_N$$

with $C_n$ explicit in terms of spectral gaps, density ratios, and dimension scaling (Schweizer, 2012).

2. Spectral, Algebraic, and Structural Dependence in Error Bounds

Non-asymptotic error rates are fundamentally governed by algebraic properties and spectral characteristics of underlying operators, system matrices, or data distributions:

• Role of Eigenvalue Multiplicity: In the symmetric case for $A$, repeated eigenvalues $m(\lambda)>1$ induce an error plateau (Alaeddini et al., 2018). After a number of snapshots equal to the number of distinct eigenvalues $s$, the error does not decay further and is given in spectral norm by

$$\|\hat A_k-A\|_2 = \lambda^* := \max_{j: m(\lambda_{t_j})>1} |\lambda_{t_j}|$$

and in Frobenius norm by

$$\|\hat A_k-A\|_F^2 = \sum_{j: m(\lambda_{t_j})>1} (m(\lambda_{t_j})-1)\lambda_{t_j}^2$$

This reflects identifiability limitations inherent to data snapshots.

• Dimensional Scaling: Many bounds scale polynomially or exponentially in problem dimension $d$. In stochastic approximation, the martingale error term is $O(a^{1.5}d^{3.25})$ or $O(a^{0.5}d^{2.5})$, controlling the effect of noise and discretization (Kumar et al., 2018). For floating-point summation, error grows as $O(u\sqrt{\ln n})$ for balanced trees, and the probabilistic bounds explicitly account for $h$ and $n$ (Hallman et al., 2022).
• Spectral Gap and Mixing: For sequential MCMC, spectral gaps $\lambda_k$ and bridging density ratios $\gamma$ appear directly in the stability constants $C_{j,k}$, dictating how quickly variance is dissipated and sampling accuracy achieved (Schweizer, 2012).

3. Algorithmic Implications and Practical Tuning

Non-asymptotic bounds encode actionable insights for algorithm design, sample complexity, and hyperparameter selection:

• Linear Model Regression: Number of snapshots $k$ required for zero estimation error in $A$ is precisely $n$, given no repeated eigenvalue and full support in $x_0$. Otherwise, a nonzero plateau emerges, which is analytically predictable (Alaeddini et al., 2018).
• Stochastic Approximation Tracking: To keep tracking error below $\Delta$, step-size $a$ must be small enough that stochastic and drift terms $\sim a^{1.5}d^{3.25}$ and $(K_\gamma L_\lambda/\beta)\varepsilon$ are each controlled. The dimension $d$ strongly inflates required computational effort or forces adaptation via noise-structuring or reduction (Kumar et al., 2018).
• Floating Point Summation: Best accuracy is achieved for compensated summation algorithms; mixed-precision summation benefits from deferring high-precision arithmetic until intermediate partial sums are large. Probabilistic error bands (e.g., $1-\delta$ confidence) are quantitatively validated in numerical experiments (Hallman et al., 2022).
• Sequential MCMC: Leading mean-square error term scales as $O(1/N)$ provided mixing and density conditions hold; poor mixing or large density ratios inflate variance. In high dimensions, the total cost scales as $O(d^3)$ for product measures (Schweizer, 2012).

4. Distinctions from Asymptotic Theory and Key Technical Features

The shift from asymptotic to non-asymptotic analysis demands explicit control of finite-sample deviations and exposes phenomena invisible in limiting theory:

• Sharpness and Oracle Inequalities: Deterministic matrix regression bounds offer exact worst-case rates under rank-deficiency and modal multiplicity, not merely limiting behavior (Alaeddini et al., 2018). In stochastic approximation, uniform-in-time bounds augment classical fixed-horizon sample-complexity results (Kumar et al., 2018).
• Explicit Constants and Error Decomposition: Non-asymptotic bounds specify all scaling constants and provide error decompositions (e.g., initialization bias, deterministic perturbation, stochastic fluctuation). This facilitates practical selection and balancing of algorithmic parameters, such as stepsize, number of iterations, and window size in spectrum estimation.
• Proof Methodologies: Techniques include explicit rank-one Sherman–Morrison formulae, SVD and eigenbasis decompositions (linear regression), martingale difference concentration (stochastic approximation and floating-point summation), stability estimates via Feynman–Kac propagators and norm inequalities (sequential MCMC), and detailed matrix concentration for spectral methods.

5. Impact, Extensions, and Open Problems

Non-asymptotic error bounds have reshaped rigorous performance guarantees across estimation, learning, and simulation in high dimension, system identification, numerical computation, and sampling:

• Unified Frameworks: The explicit dependence on spectral structure, rank-deficiency, and data geometry enables unification and generalization of prior results, including oracle inequalities in functional PCA and Reynolds-type confidence bounds in MCMC.
• Limitations and Extensions: Bounds can deteriorate under poor mixing, highly multimodal distributions, or extreme dimensions. Open questions include tightening constants, developing frequency-resolved error bounds (as in spectrum estimation), and adaptive or structure-exploiting non-asymptotic theory for sequential and particle-based methods.
• Comparative Benchmarks: Numerical validation confirms the predictive accuracy of probabilistic error bounds and exposes trade-offs between algorithmic sophistication and achievable error rates under operational constraints.

In summary, non-asymptotic error bounds offer precise, operational control over the performance of regression, tracking, computation, and sampling algorithms, explicitly reflecting model properties, spectral structure, and dimensional effects, and are foundational in practical settings where asymptotic results are inadequate (Alaeddini et al., 2018, Kumar et al., 2018, Hallman et al., 2022, Schweizer, 2012).