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Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysis

Frederic Koehler, Beining Wu

TL;DR

The paper develops a quantitative Denjoy–Carleman framework under Carleman’s condition to bound L2 polynomial approximation without relying on explicit Fourier bases. It uses a complex-analytic approach to relate the Fourier transform of the target and the μ-weighted residual, yielding nonasymptotic rates that apply to broad sub-Gaussian and sub-exponential distributions and beyond. These bounds enable efficient, polynomial-approximation-based learning across smoothed agnostic learning, learning intersections of halfspaces, and neural-net-inspired targets, with notable improvements over prior sub-exponential results. The work also establishes universality-type results for Paley–Wiener and Gelfand–Shilov classes and situates its contributions within SQ frameworks and cryptographic hardness discussions, highlighting both practical learning implications and theoretical reach in harmonic analysis and approximation theory.

Abstract

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(μ)$ for any $μ$ such that the moments $\int x^k dμ$ do not grow too rapidly as $k \to \infty$. In this work, we develop a fairly tight quantitative analogue of the underlying Denjoy-Carleman theorem via complex analysis, and show that this allows for nonasymptotic control of the rate of approximation by polynomials for any smooth function with polynomial growth at infinity. In many cases, this allows us to establish $L^2$ approximation-theoretic results for functions over general classes of distributions (e.g., multivariate sub-Gaussian or sub-exponential distributions) which were previously known only in special cases. As one application, we show that the Paley--Wiener class of functions bandlimited to $[-Ω,Ω]$ admits superexponential rates of approximation over all strictly sub-exponential distributions, which leads to a new characterization of the class. As another application, we solve an open problem recently posed by Chandrasekaran, Klivans, Kontonis, Meka and Stavropoulos on the smoothed analysis of learning, and also obtain quantitative improvements to their main results and applications.

Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysis

TL;DR

The paper develops a quantitative Denjoy–Carleman framework under Carleman’s condition to bound L2 polynomial approximation without relying on explicit Fourier bases. It uses a complex-analytic approach to relate the Fourier transform of the target and the μ-weighted residual, yielding nonasymptotic rates that apply to broad sub-Gaussian and sub-exponential distributions and beyond. These bounds enable efficient, polynomial-approximation-based learning across smoothed agnostic learning, learning intersections of halfspaces, and neural-net-inspired targets, with notable improvements over prior sub-exponential results. The work also establishes universality-type results for Paley–Wiener and Gelfand–Shilov classes and situates its contributions within SQ frameworks and cryptographic hardness discussions, highlighting both practical learning implications and theoretical reach in harmonic analysis and approximation theory.

Abstract

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in for any such that the moments do not grow too rapidly as . In this work, we develop a fairly tight quantitative analogue of the underlying Denjoy-Carleman theorem via complex analysis, and show that this allows for nonasymptotic control of the rate of approximation by polynomials for any smooth function with polynomial growth at infinity. In many cases, this allows us to establish approximation-theoretic results for functions over general classes of distributions (e.g., multivariate sub-Gaussian or sub-exponential distributions) which were previously known only in special cases. As one application, we show that the Paley--Wiener class of functions bandlimited to admits superexponential rates of approximation over all strictly sub-exponential distributions, which leads to a new characterization of the class. As another application, we solve an open problem recently posed by Chandrasekaran, Klivans, Kontonis, Meka and Stavropoulos on the smoothed analysis of learning, and also obtain quantitative improvements to their main results and applications.

Paper Structure

This paper contains 168 sections, 86 theorems, 451 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Fourier-based $L^2(\mu)$ polynomial approximation bound
  4. Application: Smoothed agnostic learning.
  5. Extension to learning intersections of half-spaces.
  6. Application: Polynomial approximation to smooth functions.
  7. Implications in agnostic learning.
  8. Overview
  9. Key difficulty: lack of an explicit Fourier basis.
  10. The existing strategies for approximation.
  11. New strategy for approximation.
  12. Special case: strictly sub-exponential distributions.
  13. Carleman's condition.
  14. Quantitative Denjoy-Carleman.
  15. Applications.
  16. ...and 153 more sections

Key Result

Theorem 1.1

Let $\mu$ be a strictly sub-exponential distribution, i.e., the moment generating function of $\mu$ is entire with finite order. Then for a fixed $\Omega>0$ and sufficiently large degree $D \gtrsim \mathrm{poly}(\Omega)$, we have that

Theorems & Definitions (131)

  • Theorem 1.1: Strictly sub-exponential bound, informal, see \ref{['thm:pw-strict']}
  • Theorem 1.2: Sub-exponential bound, informal, see \ref{['thm:pw-subexp']}
  • Theorem 1.3: Learning smoothed optimality, informal, see \ref{['thm:smoothed-learning']}
  • Remark 1.4: Comparison to chandrasekaran2024smoothed in strictly sub-exponential case
  • Theorem 1.5: Polynomial approximation to $C^k$ functions
  • Remark 1.6: Comparison to techniques in previous works
  • Lemma 3.1: Section 5.2.3 of katznelson2004introduction
  • Theorem 3.2: Denjoy-Carleman Theorem carleman1926fonctionsrudin1987real, see also katznelson2004introductionhörmander1983analysis
  • Remark 3.3
  • Lemma 3.4
  • ...and 121 more