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Transfer Learning for Functional Mean Estimation: Phase Transition and Adaptive Algorithms

T. Tony Cai, Dongwoo Kim, Hongming Pu

TL;DR

Novel data-driven adaptive algorithms are proposed that attain the optimal rates of convergence within a logarithmic factor simultaneously over a large collection of parameter spaces.

Abstract

This paper studies transfer learning for estimating the mean of random functions based on discretely sampled data, where, in addition to observations from the target distribution, auxiliary samples from similar but distinct source distributions are available. The paper considers both common and independent designs and establishes the minimax rates of convergence for both designs. The results reveal an interesting phase transition phenomenon under the two designs and demonstrate the benefits of utilizing the source samples in the low sampling frequency regime. For practical applications, this paper proposes novel data-driven adaptive algorithms that attain the optimal rates of convergence within a logarithmic factor simultaneously over a large collection of parameter spaces. The theoretical findings are complemented by a simulation study that further supports the effectiveness of the proposed algorithms

Transfer Learning for Functional Mean Estimation: Phase Transition and Adaptive Algorithms

TL;DR

Novel data-driven adaptive algorithms are proposed that attain the optimal rates of convergence within a logarithmic factor simultaneously over a large collection of parameter spaces.

Abstract

This paper studies transfer learning for estimating the mean of random functions based on discretely sampled data, where, in addition to observations from the target distribution, auxiliary samples from similar but distinct source distributions are available. The paper considers both common and independent designs and establishes the minimax rates of convergence for both designs. The results reveal an interesting phase transition phenomenon under the two designs and demonstrate the benefits of utilizing the source samples in the low sampling frequency regime. For practical applications, this paper proposes novel data-driven adaptive algorithms that attain the optimal rates of convergence within a logarithmic factor simultaneously over a large collection of parameter spaces. The theoretical findings are complemented by a simulation study that further supports the effectiveness of the proposed algorithms
Paper Structure (17 sections, 14 theorems, 54 equations, 2 figures, 4 algorithms)

This paper contains 17 sections, 14 theorems, 54 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Our contribution
  4. Organization and notation
  5. Transfer learning for functional mean estimation under a common design
  6. Methodology and upper bound
  7. Matching lower bound
  8. Adaptive estimation
  9. Transfer learning for functional mean estimation under an independent design
  10. Methodology and upper bound
  11. Matching lower bound
  12. Adaptive estimation
  13. Numerical experiments
  14. Discussion
  15. Proofs
  16. ...and 2 more sections

Key Result

Theorem 2.1

Suppose no source samples are available, i.e. $n_s = 0$ and the fixed and common design points for the target sample satisfy $\max_{1\leq j\leq m_t+1} \bigl(T_{j}^{[t]}-T_{j-1}^{[t]}\bigr) \leq C_t/m_t$ for some constant $C_t > 0$, where $T_0^{[t]} = 0$ and $T_{m_t+1}^{[t]} = 1$. Then where the infimum is taken over all estimators $\widehat{f}^{[t]} = \widehat{f}^{[t]}(\Dcal^{[t]})$ based on the

Figures (2)

  • Figure 1: The results for numerical experiments under a common design
  • Figure 2: The results for numerical experiments under an independent design

Theorems & Definitions (14)

  • Theorem 2.1: The minimax risk under conventional setup and common design
  • Theorem 2.2: Upper bound under a common design
  • Theorem 2.3: Lower bound under a common design
  • Theorem 2.4: Adaptive estimation under a common design
  • Theorem 3.1: The minimax risk under conventional setup and independent design
  • Theorem 3.2: Upper bound under an independent design
  • Theorem 3.3: Lower bound under an independent design
  • Theorem 3.4: Adaptive estimation under an independent design
  • Proposition 6.1
  • Proposition 6.2
  • ...and 4 more