Finite-Sample L^p Consistency

• Finite-sample L^p consistency is a framework that offers explicit, non-asymptotic guarantees for approximating L^p quantities with high probability using finite samples.
• The methodology employs tools like concentration bounds, VC theory, and moment inequalities to derive clear sample size requirements and uniform error control.
• These techniques translate theoretical models into robust, practical algorithms for applications in regression, convex geometry, and system identification.

Finite-sample $L^{p}$ consistency refers to rigorous, non-asymptotic guarantees that estimators or random constructions approximate population $L^p$ quantities (such as norms, risks, or solutions to estimation problems) to specified precision with high probability, using only a finite number of samples. In high-dimensional analysis, regression, convex geometry, and system identification, finite-sample $L^p$ consistency is central to translating theoretical models into robust algorithmic procedures.

1. Definition and Foundational Examples

Let $F$ denote a family of functions, sets, or parameters associated with an $L^p$ norm or risk. A procedure achieves finite-sample $L^{p}$ consistency if, given $N$ samples, it provides an explicit, high-probability bound of the form: $$\sup_{f\in F} \left| \widehat{\Psi}_N(f) - \mathbb{E}|f|^p \right| \leq \varepsilon,$$ where $\widehat{\Psi}_N$ is a data-driven estimator or solution, and the sample count $N$ is polynomial or near-optimal in intrinsic parameters (e.g., dimension $d$, error tolerance $\varepsilon$).

Notable settings include:

• Random geometric approximation: Empirical approximations of the $L_p$ unit ball using random vectors in $\mathbb{R}^d$ (Mendelson, 2020).
• System identification: Instrumental variables (IV) estimators for dynamical systems whose parameter estimation error is controlled in all $L^p$ norms, including non-asymptotic rates (Kuang et al., 12 Nov 2025).
• Polynomial regression: Algorithms for $L_p$ polynomial regression yielding $(1+\varepsilon)$-optimal fits with near-linear sample complexity for all $p \ge 1$ (Meyer et al., 2022).

2. Key Theoretical Results in $L^p$ Consistency

Finite-sample $L^p$ consistency theorems often exhibit the following features:

• Explicit sample size bounds scaling linearly or near-linearly in critical problem parameters (e.g., $d$ for dimension, $n$ for sample size, or $1/\varepsilon^2$).
• Uniform approximation over broad families (e.g., all $v\in\mathbb{R}^d$, all degree-$d$ polynomials, or all admissible parameters in a system).
• High-probability guarantees with subgaussian or exponential tails in the error probability, frequently leveraging VC-theory, Talagrand-type inequalities, or concentration for subgaussian processes.

Table: Sample Complexity Scaling in Three Model Settings

Domain	Sample Size $N$	$L^p$ Consistency Guarantee
$L_p$ unit balls (Mendelson, 2020)	$O(d\,\varepsilon^{-2} \log(1/\varepsilon))$	Uniformly $\pm\varepsilon$ in $\mathbb{E}|\langle X, v\rangle|^p$ for all $v$
IV system ID (Kuang et al., 12 Nov 2025)	$n$ obs (with $h$ step)	$$\mathbb{E}\|\widehat{\theta}-\theta_0\|^{p} \lesssim C n^{-p/2}$$ up to bias
$L_p$ poly. regression (Meyer et al., 2022)	$\tilde O(d\, \varepsilon^{-O(p)})$	$$\|\widehat{q} - f\|_{L_p} \le (1+\varepsilon) \cdot \min_q\|q-f\|_{L_p}$$

The table summarizes the sample complexity and forms of $L^p$ consistency in recent results.

3. Methodological Frameworks

Random Sampling and Empirical Process Analysis

In the context of $L_p$ unit balls, the construction of an empirical estimator

$\Psi_{p,\theta}(v) = \frac{1}{N}\sum_{j \geq \theta N} |\langle X_j, v\rangle|^p_j^*$

yields a $(1\pm\varepsilon)$-approximation to the population $L_p$ norm over all $v$. The proof utilizes:

• Truncation or tail integral techniques to manage heavy-tailed distributions,
• VC-type uniform deviation bounds for empirical probabilities,
• Moment inequalities (e.g., control over $L_q$ moments with $q \geq 2p$) to obtain sharp sample complexity (Mendelson, 2020).

$L^p$-Consistent Regression via Structured Sampling

For polynomial regression, a two-phase Chebyshev-sampling algorithm achieves $L_p$-optimality with linear-in-$d$ sample count:

• Sampling points according to Chebyshev density $v(t) = 1/(\pi\sqrt{1-t^2})$,
• Solving a weighted $\ell_p$ minimization using the sampled design,
• Applying operator Lewis-weight analysis and polynomial structure to extend subspace embedding results from $p \leq 2$ to all $p$ (Meyer et al., 2022).

IV Estimation with Synthesized Instruments

Instrumental variables system identification constructs independent regressors via local polynomial filtering and sample splitting. This ensures that instrumental variable estimators are unbiased and their parameter estimation error decays at the nonparametric $\sqrt{n}$-rate in $L^p$ for all $p \ge 1$:

• Singular-value clipping controls the influence of near-collinearities,
• Subgaussian concentration bounds and operator norms provide explicit high-probability control on the estimator,
• Bias-variance decomposition and balancing yields the optimal rate up to a negligible bias term (Kuang et al., 12 Nov 2025).

4. Principal Assumptions and Limitations

Consistency results rest on specific structural and probabilistic assumptions:

• Isotropy: For random vector models, isotropy (identity covariance) is essential to establish uniform norm control (Mendelson, 2020).
• Moment conditions: Existence of finite $q$-th moments for some $q \geq 2p$ underlies many concentration inequalities.
• Log-concavity: For log-concave $X$, stronger norm-equivalence and moment comparison inequalities enable improved sample complexity.
• Regularity and persistence: In dynamical systems, signal regularity (smoothness), noise subgaussianity, and persistence of excitation are imposed (Kuang et al., 12 Nov 2025).
• Sampling measures: For polynomial regression, Chebyshev-density sampling is essential—uniform or adversarial sampling can require far larger samples for $L_p$, $p>2$ (Meyer et al., 2022).
• Optimality gaps: Lower bounds demonstrate that polynomial or even exponential dependence on $1/\varepsilon$ in sample size is information-theoretically necessary for some $p$.

5. Comparative Insights Across Domains

Finite-sample $L^{p}$ consistency unifies estimation in high-dimensional geometry, machine learning, and system identification:

• For $L_p$ norm approximation, the $O(d)$ sample complexity generalizes classical covariance estimation ($p=2$), and the results for log-concave $X$ achieve dramatic improvements over previous $d^{p/2}$ bounds (Mendelson, 2020).
• In polynomial regression, Chebyshev-weighted sampling is identified as an essentially optimal Lewis-weight surrogate for all $p$, closing the gap between structured and unstructured regression (Meyer et al., 2022).
• IV identification with $L^p$ finite-sample error bounds allows for robust system learning with minimal distributional assumptions, extending finite-sample risk control from classical $L_2$ to all $L^p$ risks and recovering classical $\sqrt{n}$ rates even in the nonparametric regime (Kuang et al., 12 Nov 2025).

A plausible implication is that these methodologies establish general templates for constructing estimators and geometric approximations with dimension- or complexity-optimal sample counts and non-asymptotic control in all $L^p$ norms, even under heavy-tailed or non-Gaussian scenarios.

6. Implications for Algorithms and Applications

These developments carry significant algorithmic and practical consequences:

• Sample-efficient geometric learning: Reliable recovery of norm balls and support functions in convex geometry and compressed sensing is achievable with near-minimal sample sizes.
• High-dimensional regression: Regression in arbitrary $L_p$ norms admits polynomial-time, sample-efficient algorithms with explicit performance guarantees, crucial for robustness in signal and data analysis.
• System identification and control: IV estimators with finite-sample $L^p$ consistency enable bias reduction and fast convergence for learning dynamical systems from noisy data, with rigorous nonasymptotic guarantees.
• Modularity: Techniques such as sample splitting, operator Lewis-weight sampling, and singular-value clipping are adaptable across multiple domains.

In summary, finite-sample $L^{p}$ consistency provides a comprehensive quantitative foundation for the design and analysis of randomized approximation algorithms, robust estimators, and system identification methods across a variety of scientific and engineering disciplines (Mendelson, 2020, Kuang et al., 12 Nov 2025, Meyer et al., 2022).