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Fast Rate Bounds for Multi-Task and Meta-Learning with Different Sample Sizes

Hossein Zakerinia, Christoph H. Lampert

TL;DR

This work addresses the gap in fast-rate PAC-Bayesian generalization bounds for unbalanced multi-task learning and meta-learning, where tasks have varying numbers of training samples. It develops fast-rate bounds of KL-style and Catoni-style across two risk definitions—task-centric and sample-centric—and extends these results to meta-learning via a hyper-posterior framework. The authors show that unbalanced settings have distinct statistical properties from balanced ones and provide numerically computable bounds, including practical procedures to combine bounds for tighter guarantees. Empirical results on linear models and neural networks demonstrate that the new fast-rate bounds can offer tighter, non-vacuous guarantees than standard-rate bounds, especially when training errors are small.

Abstract

We present new fast-rate PAC-Bayesian generalization bounds for multi-task and meta-learning in the unbalanced setting, i.e. when the tasks have training sets of different sizes, as is typically the case in real-world scenarios. Previously, only standard-rate bounds were known for this situation, while fast-rate bounds were limited to the setting where all training sets are of equal size. Our new bounds are numerically computable as well as interpretable, and we demonstrate their flexibility in handling a number of cases where they give stronger guarantees than previous bounds. Besides the bounds themselves, we also make conceptual contributions: we demonstrate that the unbalanced multi-task setting has different statistical properties than the balanced situation, specifically that proofs from the balanced situation do not carry over to the unbalanced setting. Additionally, we shed light on the fact that the unbalanced situation allows two meaningful definitions of multi-task risk, depending on whether all tasks should be considered equally important or if sample-rich tasks should receive more weight than sample-poor ones.

Fast Rate Bounds for Multi-Task and Meta-Learning with Different Sample Sizes

TL;DR

This work addresses the gap in fast-rate PAC-Bayesian generalization bounds for unbalanced multi-task learning and meta-learning, where tasks have varying numbers of training samples. It develops fast-rate bounds of KL-style and Catoni-style across two risk definitions—task-centric and sample-centric—and extends these results to meta-learning via a hyper-posterior framework. The authors show that unbalanced settings have distinct statistical properties from balanced ones and provide numerically computable bounds, including practical procedures to combine bounds for tighter guarantees. Empirical results on linear models and neural networks demonstrate that the new fast-rate bounds can offer tighter, non-vacuous guarantees than standard-rate bounds, especially when training errors are small.

Abstract

We present new fast-rate PAC-Bayesian generalization bounds for multi-task and meta-learning in the unbalanced setting, i.e. when the tasks have training sets of different sizes, as is typically the case in real-world scenarios. Previously, only standard-rate bounds were known for this situation, while fast-rate bounds were limited to the setting where all training sets are of equal size. Our new bounds are numerically computable as well as interpretable, and we demonstrate their flexibility in handling a number of cases where they give stronger guarantees than previous bounds. Besides the bounds themselves, we also make conceptual contributions: we demonstrate that the unbalanced multi-task setting has different statistical properties than the balanced situation, specifically that proofs from the balanced situation do not carry over to the unbalanced setting. Additionally, we shed light on the fact that the unbalanced situation allows two meaningful definitions of multi-task risk, depending on whether all tasks should be considered equally important or if sample-rich tasks should receive more weight than sample-poor ones.

Paper Structure

This paper contains 32 sections, 22 theorems, 84 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Multi-task and Meta-learning.
  4. PAC-Bayesian bounds.
  5. PAC-Bayesian multi-task learning and meta-learning.
  6. Unbalanced multi-task learning
  7. Existing proofs do not carry over to the unbalanced setting
  8. Discussion.
  9. Task-centric vs sample-centric guarantees for unbalanced multi-task learning
  10. Fast-rate generalization bounds for task-centric multi-task learning
  11. Discussion.
  12. Numerical computation of the bounds
  13. Generalization bounds for sample-centric multi-task learning
  14. Discussion.
  15. Meta-learning with unbalanced tasks
  16. ...and 17 more sections

Key Result

Theorem 2.1

For any fixed $\delta > 0$, and data-independent prior $P$, with probability at least $1-\delta$ over sampling of a dataset $S$ we have

Figures (3)

  • Figure 1: Task-centric MTL: graphical results for linear models on the MDPR dataset
  • Figure 2: Sample-centric MTL: graphical results for linear models on the MDPR dataset
  • Figure 3: Illustration of the constraint and optimal value of bounds

Theorems & Definitions (32)

  • Theorem 2.1: mcallester1998some
  • Theorem 2.2: maurer2004note
  • Theorem 2.3
  • Lemma 3.0
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Theorem 4.1
  • Lemma B.1: Berend2010Efficient Proposition 3.2
  • Lemma B.2: maurer2004note Theorem 1
  • ...and 22 more