Features are fate: a theory of transfer learning in high-dimensional regression (2410.08194v1)

Abstract: With the emergence of large-scale pre-trained neural networks, methods to adapt such "foundation" models to data-limited downstream tasks have become a necessity. Fine-tuning, preference optimization, and transfer learning have all been successfully employed for these purposes when the target task closely resembles the source task, but a precise theoretical understanding of "task similarity" is still lacking. While conventional wisdom suggests that simple measures of similarity between source and target distributions, such as $\phi$-divergences or integral probability metrics, can directly predict the success of transfer, we prove the surprising fact that, in general, this is not the case. We adopt, instead, a feature-centric viewpoint on transfer learning and establish a number of theoretical results that demonstrate that when the target task is well represented by the feature space of the pre-trained model, transfer learning outperforms training from scratch. We study deep linear networks as a minimal model of transfer learning in which we can analytically characterize the transferability phase diagram as a function of the target dataset size and the feature space overlap. For this model, we establish rigorously that when the feature space overlap between the source and target tasks is sufficiently strong, both linear transfer and fine-tuning improve performance, especially in the low data limit. These results build on an emerging understanding of feature learning dynamics in deep linear networks, and we demonstrate numerically that the rigorous results we derive for the linear case also apply to nonlinear networks.

• The paper introduces a theory where feature space overlap, not traditional data similarity, predicts transfer learning performance.
• The methodology uses deep linear networks to derive a transferability phase diagram, defining effective transfer based on target data size and feature alignment.
• The results provide practical guidelines for designing algorithms that prioritize transferable features, with simulations extending findings to nonlinear models.

Overview of "Features are fate: a theory of transfer learning in high-dimensional regression"

The paper, "Features are fate: a theory of transfer learning in high-dimensional regression," provides a rigorous theoretical analysis of transfer learning, emphasizing the critical role of feature space over classical dataset similarity measurements. In the context of adapting large-scale pre-trained neural networks to data-limited downstream tasks, the authors challenge conventional wisdom that task similarity—often measured by distributional metrics like $\phi$-divergences or integral probability metrics—directly correlates with transfer learning success.

Key Contributions

1. Feature-Centric Viewpoint: The authors argue that the feature space learned during pretraining is more predictive of transfer learning performance than traditional dataset similarity metrics. They show that dataset discrepancies measured by popular metrics can be misleading regarding transferability.
2. Deep Linear Networks: By focusing on deep linear networks as a minimal model, the authors analytically derive conditions under which transfer learning outperforms training from scratch. They develop a "transferability phase diagram" based on target dataset size and feature space overlap, showing conditions for positive and negative transfer.
3. Role of Feature Overlap: They demonstrate that when source and target tasks share a feature space, linear transfer and fine-tuning significantly enhance performance, especially in low data regimes.
4. Phase Diagram of Transfer Efficiency: A novel aspect of the work is the establishment of a phase diagram characterizing different regimes of transfer learning efficiency based on model parameters and task similarities.
5. Numerical Validation: Through numerical simulations, the authors extend their theoretical insights from linear to nonlinear networks, demonstrating the qualitative applicability of their findings.

Implications

The findings have both theoretical and practical implications for the development and application of transfer learning techniques in machine learning:

• Practical Transfer Strategies:

The insights on when transfer learning is beneficial have direct applications in choosing practical fine-tuning and transfer strategies for pre-trained models.

• Refined Metrics for Transferability:

The notion that feature representation is more critical than the traditional dataset similarity suggests a need for developing new metrics that better capture task relatedness in terms of feature overlap.

• Algorithm Design:

The paper’s results can inform the design of algorithms that prioritize learning transferable features, potentially increasing efficiency in practical applications involving foundation models.

Future Directions

The paper suggests several areas for future research, including:

• Expanding Beyond Linear Models:

While the authors provide a robust analysis in the context of deep linear networks and preliminary insights into nonlinear models, extending these results more broadly to complex architectures remains an open area for investigation.

• Development of Feature-Based Metrics:

Developing algorithm-independent metrics that capture the notion of feature space overlap could enhance the predictability and applicability of transfer learning across diverse domains.

• Exploration of Non-convex Settings:

Investigating the role of feature overlap in more sophisticated, non-convex settings where models are not linearly separable could provide deeper insights into transfer learning dynamics.

In summary, this paper shifts the understanding of transfer learning efficacy from traditional data similarity measures to a nuanced consideration of feature space alignment, offering a compelling framework for future exploration in high-dimensional regression and beyond.