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Standard Transformers Achieve the Minimax Rate in Nonparametric Regression with $C^{s,λ}$ Targets

Yanming Lai, Defeng Sun

TL;DR

This paper is the first work proving that standard Transformers can approximate Holder functions C^{s,\lambda}\left([0,1]^{d\times n}\right) with arbitrary precision, and it is demonstrated that standard Transformers achieve the minimax optimal rate in nonparametric regression for Holder target functions.

Abstract

The tremendous success of Transformer models in fields such as large language models and computer vision necessitates a rigorous theoretical investigation. To the best of our knowledge, this paper is the first work proving that standard Transformers can approximate Hölder functions $ C^{s,λ}\left([0,1]^{d\times n}\right) $$ (s\in\mathbb{N}_{\geq0},0<λ\leq1) $ under the $L^t$ distance ($t \in [1, \infty]$) with arbitrary precision. Building upon this approximation result, we demonstrate that standard Transformers achieve the minimax optimal rate in nonparametric regression for Hölder target functions. It is worth mentioning that, by introducing two metrics: the size tuple and the dimension vector, we provide a fine-grained characterization of Transformer structures, which facilitates future research on the generalization and optimization errors of Transformers with different structures. As intermediate results, we also derive the upper bounds for the Lipschitz constant of standard Transformers and their memorization capacity, which may be of independent interest. These findings provide theoretical justification for the powerful capabilities of Transformer models.

Standard Transformers Achieve the Minimax Rate in Nonparametric Regression with $C^{s,λ}$ Targets

TL;DR

This paper is the first work proving that standard Transformers can approximate Holder functions C^{s,\lambda}\left([0,1]^{d\times n}\right) with arbitrary precision, and it is demonstrated that standard Transformers achieve the minimax optimal rate in nonparametric regression for Holder target functions.

Abstract

The tremendous success of Transformer models in fields such as large language models and computer vision necessitates a rigorous theoretical investigation. To the best of our knowledge, this paper is the first work proving that standard Transformers can approximate Hölder functions under the distance () with arbitrary precision. Building upon this approximation result, we demonstrate that standard Transformers achieve the minimax optimal rate in nonparametric regression for Hölder target functions. It is worth mentioning that, by introducing two metrics: the size tuple and the dimension vector, we provide a fine-grained characterization of Transformer structures, which facilitates future research on the generalization and optimization errors of Transformers with different structures. As intermediate results, we also derive the upper bounds for the Lipschitz constant of standard Transformers and their memorization capacity, which may be of independent interest. These findings provide theoretical justification for the powerful capabilities of Transformer models.
Paper Structure (20 sections, 35 theorems, 267 equations, 1 figure)

This paper contains 20 sections, 35 theorems, 267 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Our Contributions
  5. Organization of This Paper
  6. Proofs of Main Results
  7. Proof of Theorem \ref{['Lr approximation']}
  8. Proof of Theorem \ref{['Linfty']}
  9. Proof of Theorem \ref{['regression bound']}
  10. Proof of Proposition \ref{['basic approximation']}: Construction of Transformers
  11. Proof of Proposition \ref{['basic approximation']}
  12. Proof of Lemma \ref{['FNN to transformer']}
  13. Proof of Lemma \ref{['monomial']}
  14. Proof of Lemma \ref{['Transformer memorization']}: Memorization of Transformers
  15. Proof of Lemma \ref{['Transformer memorization']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $1\leq t<\infty$. For any $0<\epsilon<1$ and any $\boldsymbol{f}:[0,1]^{d\times n}\to\mathbb{R}^{d\times n}$ with components in Hölder space $C^{s,\lambda}\left([0,1]^{d\times n}\right)$, there exists a Tranformer $\boldsymbol{T}:\mathbb{R}^{d\times n}\to\mathbb{R}^{d\times n}$ with size and dimension vector such that for $p\in[d],q\in[n]$, Furthermore, the weight bounds of $\boldsymbol{T}$

Figures (1)

  • Figure 1: Illustration of the proof process

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof : Proof of Theorem \ref{['Lr approximation']}
  • Lemma 1: lu2021deep, Lemma 3.1
  • Lemma 2: lu2021deep, Lemma 3.4
  • Lemma 3
  • proof : Proof of Theorem \ref{['Linfty']}
  • Definition 1: covering number
  • ...and 58 more