TL;DR#
The study investigates the efficiency of Empirical Risk Minimization (ERM) in feature learning, focusing on regression problems. Traditional approaches suggest that performance depends heavily on the size of the model class. However, this research explores scenarios where a model jointly learns appropriate feature maps from the data along with a linear predictor. A key challenge here is that the feature selection burden falls on the model and data, potentially requiring more samples for successful learning. The non-convex nature of this combined learning problem poses additional challenges for analysis.
The paper offers asymptotic and non-asymptotic analysis quantifying the excess risk. Remarkably, it finds that when the feature map set isn’t excessively large, and a unique optimal feature map exists, the asymptotic quantiles of the excess risk of ERM closely match (within a factor of two) those of an oracle procedure. This oracle procedure has prior knowledge of the optimal feature map. Further, a non-asymptotic analysis shows how the complexity of the feature map set impacts ERM’s performance, linking it to the size of the sub-optimal feature maps’ sublevel sets. These results are applied to the best subset selection process in sparse linear regression.
Key Takeaways#
Why does it matter?#
This paper is crucial because it challenges conventional wisdom in feature learning, demonstrating that the model class size’s impact on ERM performance diminishes significantly with sufficient data. This opens new avenues for designing more efficient and effective feature learning models, especially for large-scale applications where the traditional complexity trade-offs become less relevant. The refined theoretical analysis offers crucial insights for researchers working on efficient machine learning algorithms, especially concerning high-dimensional data and feature selection.
Visual Insights#
Full paper#
