Expectation Error Bounds for Transfer Learning in Linear Regression and Linear Neural Networks

TL;DR

This paper provides theoretical bounds and conditions under which transfer learning improves generalization in linear regression and neural networks, supported by new mathematical tools and empirical validation.

Contribution

It derives exact error expressions and optimal task weights for linear models, and establishes the first non-vacuous bounds for beneficial transfer in linear neural networks.

Findings

Exact generalization error formulas for linear regression with auxiliary tasks.

Optimal task weights derived from solvable optimization problems.

Non-vacuous sufficient conditions for transfer learning benefits in linear neural networks.

Abstract

In transfer learning, the learner leverages auxiliary data to improve generalization on a main task. However, the precise theoretical understanding of when and how auxiliary data help remains incomplete. We provide new insights on this issue in two canonical linear settings: ordinary least squares regression and under-parameterized linear neural networks. For linear regression, we derive exact closed-form expressions for the expected generalization error with bias-variance decomposition, yielding necessary and sufficient conditions for auxiliary tasks to improve generalization on the main task. We also derive globally optimal task weights as outputs of solvable optimization programs, with consistency guarantees for empirical estimates. For linear neural networks with shared representations of width $q \leq K$, where $K$ is the number of auxiliary tasks, we derive a non-asymptotic expectation bound on the generalization error, yielding the first non-vacuous sufficient condition for beneficial auxiliary learning in this setting, as well as principled directions for task weight curation. We achieve this by proving a new column-wise low-rank perturbation bound for random matrices, which improves upon existing bounds by preserving fine-grained column structures. Our results are verified on synthetic data simulated with controlled parameters.