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Harnessing the Power of Vicinity-Informed Analysis for Classification under Covariate Shift

Mitsuhiro Fujikawa, Yohei Akimoto, Jun Sakuma, Kazuto Fukuchi

TL;DR

This work addresses classification under covariate shift with potential non-containment of supports by introducing a vicinity-informed dissimilarity $\Delta_{\mathcal{V}}(P,Q; r)$. It develops the $\Delta_{\mathcal{V}}$-based transfer and self exponents to characterize excess error and proves the existence of a classifier whose excess error decays with the source and target sample sizes, achieving source-sample consistency under finite transfer-exponent $\tau$. The paper also provides a comparative analysis showing that $\Delta_{\mathcal{V}}$ yields faster or competitive rates relative to existing measures, and verifies the theory with synthetic experiments that demonstrate tight bounds and consistency in a non-containment setting. By accounting for vicinity structure, the approach aligns theoretical guarantees with empirical transfer-learning performance observed in practice, particularly when supports differ markedly. Overall, it advances the theory of transfer learning under covariate shift by bridging gaps between non-containment realism and provable convergence behavior.

Abstract

Transfer learning enhances prediction accuracy on a target distribution by leveraging data from a source distribution, demonstrating significant benefits in various applications. This paper introduces a novel dissimilarity measure that utilizes vicinity information, i.e., the local structure of data points, to analyze the excess error in classification under covariate shift, a transfer learning setting where marginal feature distributions differ but conditional label distributions remain the same. We characterize the excess error using the proposed measure and demonstrate faster or competitive convergence rates compared to previous techniques. Notably, our approach is effective in the support non-containment assumption, which often appears in real-world applications, holds. Our theoretical analysis bridges the gap between current theoretical findings and empirical observations in transfer learning, particularly in scenarios with significant differences between source and target distributions.

Harnessing the Power of Vicinity-Informed Analysis for Classification under Covariate Shift

TL;DR

This work addresses classification under covariate shift with potential non-containment of supports by introducing a vicinity-informed dissimilarity . It develops the -based transfer and self exponents to characterize excess error and proves the existence of a classifier whose excess error decays with the source and target sample sizes, achieving source-sample consistency under finite transfer-exponent . The paper also provides a comparative analysis showing that yields faster or competitive rates relative to existing measures, and verifies the theory with synthetic experiments that demonstrate tight bounds and consistency in a non-containment setting. By accounting for vicinity structure, the approach aligns theoretical guarantees with empirical transfer-learning performance observed in practice, particularly when supports differ markedly. Overall, it advances the theory of transfer learning under covariate shift by bridging gaps between non-containment realism and provable convergence behavior.

Abstract

Transfer learning enhances prediction accuracy on a target distribution by leveraging data from a source distribution, demonstrating significant benefits in various applications. This paper introduces a novel dissimilarity measure that utilizes vicinity information, i.e., the local structure of data points, to analyze the excess error in classification under covariate shift, a transfer learning setting where marginal feature distributions differ but conditional label distributions remain the same. We characterize the excess error using the proposed measure and demonstrate faster or competitive convergence rates compared to previous techniques. Notably, our approach is effective in the support non-containment assumption, which often appears in real-world applications, holds. Our theoretical analysis bridges the gap between current theoretical findings and empirical observations in transfer learning, particularly in scenarios with significant differences between source and target distributions.
Paper Structure (37 sections, 8 theorems, 59 equations, 4 figures)

This paper contains 37 sections, 8 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our dissimilarity measure and contributions.
  3. Preliminaries
  4. Notations
  5. Classification under Covariate-shift
  6. Excess Error
  7. Difficulty in Classification under Distribution $Q$
  8. Main Result
  9. Transfer and self exponents
  10. Vicinity set
  11. Our characterization
  12. Comparisons with pathak_new_2022 and kpotufe_marginal_2021.
  13. Example
  14. Analyses
  15. Bounding Excess Error by Vicinity Distance
  16. ...and 22 more sections

Key Result

Theorem 2

Given $\alpha \in (0,1]$, $\beta > 0$, and $\psi\in (0,\infty]$, suppose the target distribution $Q$ is $\mathrm{STN}(\alpha,\beta)$ and has $\Delta_{\mathcal{V}}$-self-exponent of $\psi$. Also, suppose $\left(P,Q\right)$ has $\Delta_{\mathcal{V}}$-transfer-exponent of $\tau$ for some $\tau \in (0,\ where $C > 0$ is some constant independent of $n_P$ and $n_Q$.

Figures (4)

  • Figure 1: $\alpha=\frac{1}{2}, \tau = 1$
  • Figure 2: $\alpha=\frac{1}{4}, \tau = 1$
  • Figure 3: $\alpha=\frac{1}{2}, \tau = 2$
  • Figure 4: $\alpha=\frac{1}{4}, \tau = 2$

Theorems & Definitions (18)

  • Definition 1: Covariate-shift
  • Definition 2: excess error
  • Definition 3: Smoothness
  • Definition 4: Tsybakov's noise condition
  • Definition 5: $\mathrm{STN}(\alpha,\beta)$
  • Definition 6: $\Delta$-transfer-exponent
  • Definition 7: $\Delta$-self-exponent
  • Theorem 2
  • Proposition 1: pathak_new_2022kpotufe_marginal_2021
  • Proposition 2
  • ...and 8 more