On the Efficiency of ERM in Feature Learning
This work addresses theoretical guarantees for feature learning in machine learning, offering insights into model selection and sparse regression, but it is incremental as it builds on existing ERM frameworks.
The paper analyzes the efficiency of empirical risk minimization (ERM) in feature learning for regression problems, showing that under certain conditions, ERM's excess risk quantiles match those of an oracle procedure up to a factor of two, and provides non-asymptotic bounds linking complexity to performance.
Given a collection of feature maps indexed by a set $\mathcal{T}$, we study the performance of empirical risk minimization (ERM) on regression problems with square loss over the union of the linear classes induced by these feature maps. This setup aims at capturing the simplest instance of feature learning, where the model is expected to jointly learn from the data an appropriate feature map and a linear predictor. We start by studying the asymptotic quantiles of the excess risk of sequences of empirical risk minimizers. Remarkably, we show that when the set $\mathcal{T}$ is not too large and when there is a unique optimal feature map, these quantiles coincide, up to a factor of two, with those of the excess risk of the oracle procedure, which knows a priori this optimal feature map and deterministically outputs an empirical risk minimizer from the associated optimal linear class. We complement this asymptotic result with a non-asymptotic analysis that quantifies the decaying effect of the global complexity of the set $\mathcal{T}$ on the excess risk of ERM, and relates it to the size of the sublevel sets of the suboptimality of the feature maps. As an application of our results, we obtain new guarantees on the performance of the best subset selection procedure in sparse linear regression under general assumptions.