Sparse Algorithms Are Not Stable: A No-Free-Lunch Theorem - Huan Xu; Caramanis, C.; Mannor, S. (IEEE transactions on pattern analysis and machine intelligence, 2012)
We consider two desired properties of learning algorithms: sparsity and algorithmic stability. Both properties are believed to lead to good generalization ability. We show that these two properties are fundamentally at odds with each other: A sparse algorithm cannot be stable and vice versa. Thus, one has to trade off sparsity and stability in designing a learning algorithm. In particular, our general result implies that ℓ 1 -regularized regression (Lasso) cannot be stable, while ℓ 2 -regularized regression is known to have strong stability properties and is therefore not sparse.