[Paper Review] A PAC-Bayesian bound for Lifelong Learning

Anastasia Pentina, Christoph H. Lampert|arXiv (Cornell University)|Nov 12, 2013

Domain Adaptation and Few-Shot Learning27 references121 citations

TL;DR

This paper introduces a PAC-Bayesian generalization bound for lifelong learning that unifies parameter transfer and representation transfer as special cases. By optimizing this bound, the authors derive two principled lifelong learning algorithms—one for transferring classifier weights and another for learning low-dimensional feature subspaces—achieving performance comparable to existing methods like ELLA.

ABSTRACT

Transfer learning has received a lot of attention in the machine learning community over the last years, and several effective algorithms have been developed. However, relatively little is known about their theoretical properties, especially in the setting of lifelong learning, where the goal is to transfer information to tasks for which no data have been observed so far. In this work we study lifelong learning from a theoretical perspective. Our main result is a PAC-Bayesian generalization bound that offers a unified view on existing paradigms for transfer learning, such as the transfer of parameters or the transfer of low-dimensional representations. We also use the bound to derive two principled lifelong learning algorithms, and we show that these yield results comparable with existing methods.

Motivation & Objective

• To provide a theoretical foundation for lifelong learning by deriving a generalization bound in the PAC-Bayesian framework.
• To unify existing transfer learning paradigms—parameter transfer and representation transfer—under a single theoretical framework.
• To derive principled lifelong learning algorithms from the bound that are not heuristic but grounded in generalization theory.
• To enable the use of the bound as a quality measure for transferred information, facilitating algorithm design and analysis.
• To explore the implicit assumptions of lifelong learning methods through the lens of the derived bound, particularly regarding task relationships and hyperposterior structure.

Proposed method

• Derives a PAC-Bayesian generalization bound for lifelong learning that depends on the data representation and learning algorithm, quantifying expected loss on future tasks based on average loss on observed tasks.
• Uses the bound to design two algorithms: one for transferring classifier parameters via a weighted combination of previous task weights, and another for identifying a low-dimensional feature subspace for future tasks.
• Applies the bound to linear regression and representation learning settings, showing it recovers known results from prior work under specific assumptions.
• Employs the Kullback-Leibler divergence and Hoeffding’s lemma to derive high-probability bounds on generalization error, leveraging convex duality and expectation inequalities.
• Optimizes the bound with respect to the hyperposterior distribution to produce principled learning rules, avoiding ad hoc regularization.
• Validates the approach empirically on landmine and schools datasets, comparing performance to ELLA and ARR, with hyperparameters tuned via model selection.

Experimental results

Research questions

• RQ1 Can a single theoretical framework unify different transfer learning paradigms in lifelong learning?
• RQ2 How can a generalization bound be constructed that reflects both data representation and learning algorithm choices in lifelong learning?
• RQ3 Can the derived bound be used to design principled lifelong learning algorithms that match or exceed existing heuristic methods?
• RQ4 What are the implicit assumptions encoded in the hyperposterior distribution of the bound, and how do they affect task transfer?
• RQ5 How do different hyperprior choices (e.g., Gaussian variance) impact the performance of the derived algorithms in practice?

Key findings

• The proposed PAC-Bayesian bound successfully unifies parameter transfer and representation transfer as special cases, providing a common theoretical foundation.
• The derived algorithms achieve performance comparable to or slightly better than existing methods like ELLA, particularly when hyperparameters are tuned.
• For the Landmine dataset, the performance of the parameter transfer algorithm is robust to different values of the Gaussian prior variance (σ = 1 and σ = 10).
• For the Schools dataset, the choice of σ significantly affects performance: σ = 10 yields results nearly matching ELLA, while σ = 1 leads to more conservative and less effective learning.
• The results suggest that the lack of sparsity assumptions in the proposed method may explain the performance gap with ELLA, which explicitly enforces sparsity.
• The bound serves as a quality measure for transferred information, enabling principled algorithm design and analysis of implicit assumptions in lifelong learning methods.