Provable Meta-Learning of Linear Representations (2002.11684v5)

Abstract: Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning -- a key tool for performing meta-learning -- learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the problem of multi-task linear regression -- in which multiple linear regression models share a common, low-dimensional linear representation. Here, we provide provably fast, sample-efficient algorithms to address the dual challenges of (1) learning a common set of features from multiple, related tasks, and (2) transferring this knowledge to new, unseen tasks. Both are central to the general problem of meta-learning. Finally, we complement these results by providing information-theoretic lower bounds on the sample complexity of learning these linear features.

• The paper demonstrates meta-learning methods with provable guarantees for efficiently learning shared linear representations across tasks.
• It introduces two algorithmic approaches—local minimizers of empirical risk and a method-of-moments estimator—to achieve statistically efficient feature recovery.
• The study establishes sample complexity bounds and transfer conditions that enhance multi-task learning performance with limited data.

Provable Meta-Learning of Linear Representations

The paper "Provable Meta-Learning of Linear Representations" addresses the theoretical and computational aspects of meta-learning, specifically in the context of multi-task linear regression. Meta-learning, or learning-to-learn, is a paradigm aimed at designing algorithms capable of leveraging past experience to efficiently adapt to new tasks with minimal data. This research focuses on the theoretical underpinnings of meta-learning algorithms, particularly concerning linear representation learning, an area that has been historically underexplored despite significant practical interest.

Key Contributions

1. Learning Linear Representations: The authors address multi-task linear regression, where multiple related tasks share a common low-dimensional feature space. They propose algorithms with provable guarantees for learning these features efficiently and sample-effectively. The central challenge lies in learning a unified feature representation from diverse tasks and then transferring this knowledge to new, unseen tasks.
2. Algorithmic Approaches:

Two main approaches are discussed for feature learning: - Local Minimizers of Empirical Risk: The authors prove that all local minima of a regularized empirical risk function are close to the optimal solution, suggesting that first-order methods like gradient descent can effectively recover the feature representation. - Method-of-Moments Estimator: This approach provides a statistically efficient method for feature recovery, aggregating information across tasks even when individual task parameters are difficult to estimate due to data scarcity.

1. Sample Complexity Bounds: The paper establishes information-theoretic lower bounds on the sample complexity required for feature learning, demonstrating the near-optimality of their proposed method-of-moments estimator. This estimator efficiently leverages data from multiple tasks to achieve a significantly reduced error in subspace recovery.
2. Transfer Learning: The research explores how learned representations can improve learning on new tasks. The analysis reveals a bias-variance trade-off inherent in the transfer of learned features, providing conditions under which positive transfer is achievable. Notably, when the learned representation is considerably smaller than the original feature dimension, substantial improvements in generalization performance can be achieved on new tasks with limited data.

Implications and Future Directions

The paper's insights into feature learning and transfer offer valuable contributions to the theoretical landscape of meta-learning. The demonstrated benefits of efficient representation learning have practical implications for areas such as few-shot image classification, deep reinforcement learning, and natural language processing, where tasks often share underlying structures that can be exploited.

Future research could extend these findings to non-linear models and deep learning contexts, exploring how such theoretical guarantees can drive the development of scalable algorithms for broader applications. Additionally, understanding the implications of task diversity and adaptation in more realistic settings, with complex data distributions and non-identical tasks, remains an essential area for further investigation.

In summary, this paper provides a solid theoretical foundation for the efficiency and effectiveness of meta-learning in linear representation settings, with promising directions for expanding these concepts to more complex domains in artificial intelligence.