Adaptive Sample Aggregation In Transfer Learning

Abstract: Transfer Learning aims to optimally aggregate samples from a target distribution, with related samples from a so-called source distribution to improve target risk. Multiple procedures have been proposed over the last two decades to address this problem, each driven by one of a multitude of possible divergence measures between source and target distributions. A first question asked in this work is whether there exist unified algorithmic approaches that automatically adapt to many of these divergence measures simultaneously. We show that this is indeed the case for a large family of divergences proposed across classification and regression tasks, as they all happen to upper-bound the same measure of continuity between source and target risks, which we refer to as a weak modulus of transfer. This more unified view allows us, first, to identify algorithmic approaches that are simultaneously adaptive to these various divergence measures via a reduction to particular confidence sets. Second, it allows for a more refined understanding of the statistical limits of transfer under such divergences, and in particular, reveals regimes with faster rates than might be expected under coarser lenses. We then turn to situations that are not well captured by the weak modulus and corresponding divergences: these are situations where the aggregate of source and target data can improve target performance significantly beyond what's possible with either source or target data alone. We show that common such situations -- as may arise, e.g., under certain causal models with spurious correlations -- are well described by a so-called strong modulus of transfer which supersedes the weak modulus. We finally show that the strong modulus also admits adaptive procedures, which achieve near optimal rates in terms of the unknown strong modulus, and therefore apply in more general settings.

• The paper introduces a unified framework using weak and strong moduli to quantify the relationship between source and target domains.
• It demonstrates how the weak modulus derives tight upper bounds on target error rates while establishing critical lower bounds.
• The study validates its theoretical findings with empirical examples and confidence set constructions for practical transfer learning applications.

A More Unified Theory of Transfer Learning

In their paper, "A More Unified Theory of Transfer Learning," Steve Hanneke and Samory Kpotufe tackle the critical domain adaptation problem, specifically focusing on scenarios where learners leverage data from a source domain $P$ to optimize performance on a target domain $Q$. This area has seen renewed interest spurred by real-world applications often marked by insufficient high-quality target data.

Transfer Learning: Addressing Discrepancies

Domain adaptation's core challenge lies in quantifying the relatedness between $P$ and $Q$. Past metrics such as total-variation refinements, Wasserstein distances, and covariance structure comparisons have struggled to provide a unified perspective. Hanneke and Kpotufe propose a unifying measure called the moduli of transfer, which are demonstrated to implicitly bound many existing relatedness notions, offering as tight or tighter transfer rates than current metrics.

Moduli of Transfer: Weak and Strong

Weak Modulus of Transfer

The weak modulus, denoted as $(\epsilon)$, captures the source-target relationship by measuring how well low-risk predictors under $P$ perform under $Q$: $$\delta(\epsilon) \doteq \sup_{h \in H: E_P(h) \leq \epsilon} E_Q(h)$$ This metric forms the basis of most analyses in transfer learning literature by implicitly bounding quantities tied to many existing notions of relatedness.

Strong Modulus of Transfer

The strong modulus, $(\epsilon_1, \epsilon_2)$, refines this concept by better accounting for available target data: $$\delta(\epsilon_1, \epsilon_2) \doteq \sup_{h \in H_Q(\epsilon_1): E_P(h;H_Q(\epsilon_1)) \leq \epsilon_2} E_Q(h)$$ This accounts for situations where distinct predictors under $P$ yield different risks under $Q$.

Practical and Theoretical Implications

1. Constructive Upper Bounds on Transfer Rates:

The paper establishes how the weak modulus can derive upper bounds on $E_Q(\hat{h})$, indicating: $E_Q(\hat{h}) \lesssim \min{\epsilon_Q, \delta(\epsilon_P))$ This critical finding suggests that combining source and target data sample sizes yields adaptable transfer rates.

2. Lower Bounds Establish Transfer Rate Limits:

Through rigorous lower-bound proofs, the authors confirm that these bounds, particularly for the weak modulus, are sensitive to structure assumptions, implying that no better transfer rates are generally achievable without additional information.

3. Empirical Justifications and Confidence Sets:

The paper explores practical examples, demonstrating how classical methods in classification and regression can instantiate confidence sets leading to the derived bounds. For instance, empirical risk minimizers over weak confidence sets achieve target classifiers with low error rates.

Looking Ahead: Broadening Transfer Learning's Horizons

This unified theory's inherent adaptability makes it a versatile framework for future transfer learning research. The significant compatibility with various existing metrics repositions these moduli as key analytical tools, moving toward a more generalized theory governing transfer learning.

Conclusion

The insights from Hanneke and Kpotufe's "A More Unified Theory of Transfer Learning" provide a consolidated view of transfer learning that bridges multiple existing relatedness measures. By introducing the weak and strong moduli of transfer, the authors offer a robust framework that unites classical and modern transfer learning concepts, providing a platform for adaptable and theoretically grounded algorithmic designs. Future advancements in AI and domain adaptation would likely build upon this unified perspective, exploring new efficient adaptations and practical implementations.