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Model-Robust and Adaptive-Optimal Transfer Learning for Tackling Concept Shifts in Nonparametric Regression

Haotian Lin, Matthew Reimherr

TL;DR

This work tackles nonparametric regression under concept shifts by introducing a robust Hypothesis Transfer Learning framework that leverages spectral algorithms with fixed-bandwidth Gaussian kernels. It establishes that such kernels enable minimax-optimal convergence rates for Sobolev-function targets and supports adaptive excess-risk rates up to logarithmic factors, even under misspecification. Building on this, the authors develop RAHTL, a transfer-learning scheme that jointly optimizes pre-training on source data and fine-tuning on target-like shifts, achieving minimax optimality up to logs and revealing a phase-transition governed by the transfer-signal ratio $ ext{ξ}$. Theoretical lower and upper bounds decompose errors into pre-training and fine-tuning contributions, and numerical experiments corroborate the理论, demonstrating transfer gains and the critical influence of similarity between source and target functions. Overall, the paper provides robust, adaptive transfer-learning guarantees for tackling concept shifts in nonparametric regression with RKHS-based methods, with practical implications for settings where labeled target data are scarce.

Abstract

When concept shifts and sample scarcity are present in the target domain of interest, nonparametric regression learners often struggle to generalize effectively. The technique of transfer learning remedies these issues by leveraging data or pre-trained models from similar source domains. While existing generalization analyses of kernel-based transfer learning typically rely on correctly specified models, we present a transfer learning procedure that is robust against model misspecification while adaptively attaining optimality. To facilitate our analysis and avoid the risk of saturation found in classical misspecified results, we establish a novel result in the misspecified single-task learning setting, showing that spectral algorithms with fixed bandwidth Gaussian kernels can attain minimax convergence rates given the true function is in a Sobolev space, which may be of independent interest. Building on this, we derive the adaptive convergence rates of the excess risk for specifying Gaussian kernels in a prevalent class of hypothesis transfer learning algorithms. Our results are minimax optimal up to logarithmic factors and elucidate the key determinants of transfer efficiency.

Model-Robust and Adaptive-Optimal Transfer Learning for Tackling Concept Shifts in Nonparametric Regression

TL;DR

This work tackles nonparametric regression under concept shifts by introducing a robust Hypothesis Transfer Learning framework that leverages spectral algorithms with fixed-bandwidth Gaussian kernels. It establishes that such kernels enable minimax-optimal convergence rates for Sobolev-function targets and supports adaptive excess-risk rates up to logarithmic factors, even under misspecification. Building on this, the authors develop RAHTL, a transfer-learning scheme that jointly optimizes pre-training on source data and fine-tuning on target-like shifts, achieving minimax optimality up to logs and revealing a phase-transition governed by the transfer-signal ratio . Theoretical lower and upper bounds decompose errors into pre-training and fine-tuning contributions, and numerical experiments corroborate the理论, demonstrating transfer gains and the critical influence of similarity between source and target functions. Overall, the paper provides robust, adaptive transfer-learning guarantees for tackling concept shifts in nonparametric regression with RKHS-based methods, with practical implications for settings where labeled target data are scarce.

Abstract

When concept shifts and sample scarcity are present in the target domain of interest, nonparametric regression learners often struggle to generalize effectively. The technique of transfer learning remedies these issues by leveraging data or pre-trained models from similar source domains. While existing generalization analyses of kernel-based transfer learning typically rely on correctly specified models, we present a transfer learning procedure that is robust against model misspecification while adaptively attaining optimality. To facilitate our analysis and avoid the risk of saturation found in classical misspecified results, we establish a novel result in the misspecified single-task learning setting, showing that spectral algorithms with fixed bandwidth Gaussian kernels can attain minimax convergence rates given the true function is in a Sobolev space, which may be of independent interest. Building on this, we derive the adaptive convergence rates of the excess risk for specifying Gaussian kernels in a prevalent class of hypothesis transfer learning algorithms. Our results are minimax optimal up to logarithmic factors and elucidate the key determinants of transfer efficiency.
Paper Structure (35 sections, 19 theorems, 143 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 35 sections, 19 theorems, 143 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulations
  3. Concept Shifts, Learning Framework and Assumptions
  4. Concept Shifts
  5. Learning Framework
  6. Model Assumptions
  7. Spectral Algorithms
  8. Spectral Algorithms with Gaussian Kernels
  9. Warm-up: Misspecification with Optimal Rate and Saturation Effect
  10. Opimal Convergence Rate with Gaussian Kernels
  11. Discussion
  12. Robust and Adaptive Hypothesis Transfer Learning
  13. Methodology
  14. Optimal Convergence Rate Analysis
  15. Discussion
  16. ...and 20 more sections

Key Result

Lemma 1

Let $K(x,x')$ be a stationary kernel. Suppose $\mathcal{X}$ has a Lipschitiz boundary, and the Fourier transform of $K$ has the following spectral density of $m$, for $m > d/2$, for some constant $0 < c_{1} \leq c_{2}$. Then, the associated RKHS of $K$, $\mathcal{H}_{K}(\mathcal{X})$, is norm-equivalent to the Sobolev space $\mathcal{W}^{m,2}(\mathcal{X}):= H^{m}(\mathcal{X})$.

Figures (5)

  • Figure 1: Geometric illustration for how $\xi$ will affect the transfer efficiency. The length of the lines represents the magnitude of $\|f^{P}\|_{H^{m_{P}}}$, $\|f^{Q}\|_{H^{m_{P}}}$ and $\|f^{\delta}\|_{H^{m_{\delta}}}$, respectively. (a) The circle represents a ball centered around the $f^{P}$ with radius $\|f^{\delta}\|_{H^{m_{\delta}}}$. A key observation is $\theta = \arcsin (\|f^{P} - f^{Q}\|_{H^{m_{\delta}}} /\|f^{P}\|_{H^{m_{P}}} )$. (b) $f^{P}$ and $f_{1}^{Q}$ possesses the same magnitude but a rather large angle while $f^{P}$ and $f_{2}^{Q}$ possesses a smaller angle but their magnitude differs.
  • Figure 2: Error decay curves of spectral algorithms with Gaussian kernels under best $C$. Both axes are in log scale. The dashed black lines denote the regression line of $\log \mathcal{E}$ on $\log n$, whose coefficient is $-\frac{2m}{2m+1}$ and is denoted by "True". Blue curves denote the average empirical excess risk, where "Est." means the estimated coefficients of the regression lines.
  • Figure 3: Error decay curves of spectral algorithms with Gaussian kernels under different $C$s. Both axes are in log scale. The constants $C$ presented are those close to best $C$, and the "coef" denotes the corresponding estimated coefficients.
  • Figure 4: Excess risk under different $\xi$ and $m_{\delta}$ with varied $n_{Q}$. The theoretical convergence rate is $n_{Q}^{-\frac{2m_{\delta}}{2m_{\delta}+1}}$ up to some constants.
  • Figure 5: Excess risk under different $\xi$ and $m_{\delta}$ with fixed $n_{Q}$.

Theorems & Definitions (42)

  • Lemma 1
  • Definition 1: Filter function
  • Definition 2: Spectral algorithm
  • Proposition 1: Target-only Learning
  • Theorem 1: Non-Adaptive Rate
  • Remark 1
  • Theorem 2: Adaptive Rate
  • Remark 2
  • Theorem 3: Lower Bound
  • Theorem 4: Upper Bound
  • ...and 32 more