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Revisiting general source condition in learning over a Hilbert space

Naveen Gupta, S. Sivananthan

TL;DR

This work removes restrictions on the index function and establishes optimal convergence rates for least-square regression over a Hilbert space with general regularization under a general source condition, thereby significantly broadening the scope of existing theoretical results.

Abstract

In Learning Theory, the smoothness assumption on the target function (known as source condition) is a key factor in establishing theoretical convergence rates for an estimator. The existing general form of the source condition, as discussed in learning theory literature, has traditionally been restricted to a class of functions that can be expressed as a product of an operator monotone function and a Lipschitz continuous function. In this note, we remove these restrictions on the index function and establish optimal convergence rates for least-square regression over a Hilbert space with general regularization under a general source condition, thereby significantly broadening the scope of existing theoretical results.

Revisiting general source condition in learning over a Hilbert space

TL;DR

This work removes restrictions on the index function and establishes optimal convergence rates for least-square regression over a Hilbert space with general regularization under a general source condition, thereby significantly broadening the scope of existing theoretical results.

Abstract

In Learning Theory, the smoothness assumption on the target function (known as source condition) is a key factor in establishing theoretical convergence rates for an estimator. The existing general form of the source condition, as discussed in learning theory literature, has traditionally been restricted to a class of functions that can be expressed as a product of an operator monotone function and a Lipschitz continuous function. In this note, we remove these restrictions on the index function and establish optimal convergence rates for least-square regression over a Hilbert space with general regularization under a general source condition, thereby significantly broadening the scope of existing theoretical results.

Paper Structure

This paper contains 11 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Organization
  4. Preliminaries
  5. Reformulating the minimization problem in terms of operator equation
  6. Regularization family
  7. Relation with classical non-parametric regression
  8. Smoothness condition
  9. Main results
  10. Proof of part(1) of Theorem \ref{['mainresult']}
  11. Proof of part(2) of Theorem \ref{['mainresult']}

Key Result

Theorem 3.1

Assume Assumptions as:1-as:3 are satisfied, and $\nu$ represents the qualification of the regularization family $\{g_{\lambda}\}$. Let $0<\eta<1$, $\sqrt{\lambda} \geq 16 \kappa \sqrt{\frac{\mathcal{N}(\lambda)}{n}} \log(4/\eta)$ and $\nu' = \lfloor \nu \rfloor$.

Theorems & Definitions (13)

  • Example 2.1
  • Definition 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Corollary 3.4
  • Lemma 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 3 more