Sparse Algorithms Are Not Stable: A No-Free-Lunch Theorem

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.