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The Exploration of Error Bounds in Classification with Noisy Labels

Haixia Liu, Boxiao Li, Can Yang, Yang Wang

TL;DR

This paper derives error bounds for the excess risk for classification problems with noisy labels within deep learning frameworks by decomposing it into statistical error and approximation error, and defines an independent block construction to bound the error.

Abstract

Numerous studies have shown that label noise can lead to poor generalization performance, negatively affecting classification accuracy. Therefore, understanding the effectiveness of classifiers trained using deep neural networks in the presence of noisy labels is of considerable practical significance. In this paper, we focus on the error bounds of excess risks for classification problems with noisy labels within deep learning frameworks. We derive error bounds for the excess risk, decomposing it into statistical error and approximation error. To handle statistical dependencies (e.g., mixing sequences), we employ an independent block construction to bound the error, leveraging techniques for dependent processes. For the approximation error, we establish these theoretical results to the vector-valued setting, where the output space consists of $K$-dimensional unit vectors. Finally, under the low-dimensional manifold hypothesis, we further refine the approximation error to mitigate the impact of high-dimensional input spaces.

The Exploration of Error Bounds in Classification with Noisy Labels

TL;DR

This paper derives error bounds for the excess risk for classification problems with noisy labels within deep learning frameworks by decomposing it into statistical error and approximation error, and defines an independent block construction to bound the error.

Abstract

Numerous studies have shown that label noise can lead to poor generalization performance, negatively affecting classification accuracy. Therefore, understanding the effectiveness of classifiers trained using deep neural networks in the presence of noisy labels is of considerable practical significance. In this paper, we focus on the error bounds of excess risks for classification problems with noisy labels within deep learning frameworks. We derive error bounds for the excess risk, decomposing it into statistical error and approximation error. To handle statistical dependencies (e.g., mixing sequences), we employ an independent block construction to bound the error, leveraging techniques for dependent processes. For the approximation error, we establish these theoretical results to the vector-valued setting, where the output space consists of -dimensional unit vectors. Finally, under the low-dimensional manifold hypothesis, we further refine the approximation error to mitigate the impact of high-dimensional input spaces.
Paper Structure (14 sections, 16 theorems, 71 equations)

This paper contains 14 sections, 16 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Related Works
  3. Noisy label
  4. Error estimation
  5. Setup of Classification Problem with Noisy Labels
  6. The Excess Risk under Noisy Labels
  7. Proof of Theorem \ref{['theo:excess_risk']}
  8. Error analysis
  9. Statistical error
  10. Approximate error
  11. The Curse of Dimensionality
  12. Conclusion
  13. Useful Lemmas
  14. The Lipschitz-Type Property

Key Result

Theorem 4.1

Let $f_0=\mathrm{softmax}\circ \kappa:[0,1]^d\to\mathcal{Y}=\{y=[y_1\ \cdots\ y_K]^\top|y_i\ge 0\ \forall i, \sum^K_{i=1}y_i=1\}$ and $\kappa\in\mathcal{C}_{\tau,d,K,V}$. Let $S^\eta=\left \{ z^\eta_i=(x_i,y^\eta_i) \right \}_{i=1}^{n}$, $S=\left \{ z_i=(x_i,y_i) \right \}_{i=1}^{n}$ be strictly sta where $\beta_{a_n}$ is the $a_n$-th $\beta$-mixing coefficient of samples defined in Definition def

Theorems & Definitions (34)

  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Definition 4.2: $\beta$-mixing, Subsection 2.2 in Yu1994
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 24 more