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When More Data Doesn't Help: Limits of Adaptation in Multitask Learning

Steve Hanneke, Mingyue Xu

TL;DR

This work extends the no-free-lunch perspective on multitask learning by proving that adaptivity remains impossible even when each source task provides arbitrarily large samples, provided distributional information is not available. Building on prior results, the authors introduce a new two-distribution setting with Fair $P$ and Noisy $Q$ sources and a large number of tasks, leveraging Fano’s method to show fundamental limits on identifying beneficial source aggregations. They also show that ERM over optimally identified datasets can outperform pooling, yet adaptive procedures cannot surpass the derived lower bounds, with pooling attaining near-optimal adaptive rates in their construction. The results highlight deep informational barriers to adaptivity in multisource learning and raise open questions about optimal adaptive strategies, transfer exponents, and the role of task count. The analysis uses information-theoretic tools and KL-divergence bounds on mixture distributions to formalize these limits and to compare adaptive strategies against pooling.

Abstract

Multitask learning and related frameworks have achieved tremendous success in modern applications. In multitask learning problem, we are given a set of heterogeneous datasets collected from related source tasks and hope to enhance the performance above what we could hope to achieve by solving each of them individually. The recent work of arXiv:2006.15785 has showed that, without access to distributional information, no algorithm based on aggregating samples alone can guarantee optimal risk as long as the sample size per task is bounded. In this paper, we focus on understanding the statistical limits of multitask learning. We go beyond the no-free-lunch theorem in arXiv:2006.15785 by establishing a stronger impossibility result of adaptation that holds for arbitrarily large sample size per task. This improvement conveys an important message that the hardness of multitask learning cannot be overcame by having abundant data per task. We also discuss the notion of optimal adaptivity that may be of future interests.

When More Data Doesn't Help: Limits of Adaptation in Multitask Learning

TL;DR

This work extends the no-free-lunch perspective on multitask learning by proving that adaptivity remains impossible even when each source task provides arbitrarily large samples, provided distributional information is not available. Building on prior results, the authors introduce a new two-distribution setting with Fair and Noisy sources and a large number of tasks, leveraging Fano’s method to show fundamental limits on identifying beneficial source aggregations. They also show that ERM over optimally identified datasets can outperform pooling, yet adaptive procedures cannot surpass the derived lower bounds, with pooling attaining near-optimal adaptive rates in their construction. The results highlight deep informational barriers to adaptivity in multisource learning and raise open questions about optimal adaptive strategies, transfer exponents, and the role of task count. The analysis uses information-theoretic tools and KL-divergence bounds on mixture distributions to formalize these limits and to compare adaptive strategies against pooling.

Abstract

Multitask learning and related frameworks have achieved tremendous success in modern applications. In multitask learning problem, we are given a set of heterogeneous datasets collected from related source tasks and hope to enhance the performance above what we could hope to achieve by solving each of them individually. The recent work of arXiv:2006.15785 has showed that, without access to distributional information, no algorithm based on aggregating samples alone can guarantee optimal risk as long as the sample size per task is bounded. In this paper, we focus on understanding the statistical limits of multitask learning. We go beyond the no-free-lunch theorem in arXiv:2006.15785 by establishing a stronger impossibility result of adaptation that holds for arbitrarily large sample size per task. This improvement conveys an important message that the hardness of multitask learning cannot be overcame by having abundant data per task. We also discuss the notion of optimal adaptivity that may be of future interests.
Paper Structure (21 sections, 22 theorems, 138 equations, 1 algorithm)

This paper contains 21 sections, 22 theorems, 138 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Organization
  4. Preliminaries
  5. Main Results
  6. Background
  7. Our New Setting
  8. The Agnostic Case
  9. Impossibility of Adaptation
  10. Formalization
  11. A Stronger No-free-lunch Theorem
  12. Pooling Achieves Optimal Adaptive Rates
  13. Discussion and Future Research
  14. Optimal Adaptivity in Multitask Learning.
  15. Possibility of Adaptation with A Fewer Number of Tasks.
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $P$ be any distribution supported on $\mathcal{X}\times\mathcal{Y}$ such that $h^{*}=\mathop{\mathrm{arg\,min}}\limits_{h\in\mathcal{H}}\text{er}_{P}(h)$. Then, $P$ satisfies the Bernstein class condition (Definition def:Bernstein-class-condition) if and only if

Theorems & Definitions (36)

  • Definition 2.1: Bernstein class condition
  • Definition 2.2: Transfer exponent
  • Lemma 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4: Upper bound of Algo. \ref{['algo:adaptive-algo']}
  • Theorem 5.1
  • Theorem 5.2: Adaptivity is Impossible
  • Lemma 5.3: hanneke2022no
  • ...and 26 more