Lasso, or $\ell^1$ regularized least squares, has been explored extensively for its remarkable sparsity properties. It is shown in this paper that the solution to Lasso, in addition to its sparsity, has robustness properties: it is the solution to a robust optimization problem. This has two important consequences. First, robustness provides a connection of the regularizer to a physical property, namely, protection from noise. This allows a principled selection of the regularizer, and in particular, generalizations of Lasso that also yield convex optimization problems are obtained by considering different uncertainty sets. Secondly, robustness can itself be used as an avenue to exploring different properties of the solution. In particular, it is shown that robustness of the solution explains why the solution is sparse. The analysis as well as the specific results obtained differ from standard sparsity results, providing different geometric intuition. Furthermore, it is shown that the robust optimization formulation is related to kernel density estimation, and based on this approach, a proof that Lasso is consistent is given using robustness directly. Finally, a theorem saying that sparsity and algorithmic stability contradict each other, and hence Lasso is not stable, is presented.

本文证明了 Lasso 的鲁棒性质，并将其与物理属性，即对噪声的保护，联系起来。通过考虑不同的不确定性集合，可以得出Lasso的一般化形式并获得凸优化问题。同时，通过鲁棒性质可以解释为何Lasso解是稀疏的，并且与标准稀疏结果不同。最后，证明了稀疏性和算法稳定性是相互矛盾的，因此Lasso是不稳定的。

稳健回归和Lasso