### Abstract

Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these “sacrifices” do not always require more computational efforts. To shed light on such a “free-lunch” phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can directly apply proximal algorithms (e.g. proximal gradient descent, proximal Newton, and proximal quasi-Newton algorithms) without worrying about the nonsmoothness of the loss function. Theoretically, we prove that the proximal algorithms enjoy fast local convergence with high probability. Our numerical experiments also show that when further combined with the pathwise optimization scheme, the proximal algorithms significantly outperform other competing algorithms.

Original language | English (US) |
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State | Published - Jan 1 2019 |

Event | 35th Conference on Uncertainty in Artificial Intelligence, UAI 2019 - Tel Aviv, Israel Duration: Jul 22 2019 → Jul 25 2019 |

### Conference

Conference | 35th Conference on Uncertainty in Artificial Intelligence, UAI 2019 |
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Country | Israel |

City | Tel Aviv |

Period | 7/22/19 → 7/25/19 |

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### Cite this

*On fast convergence of proximal algorithms for SQRT-lasso optimization: Don’t worry about its nonsmooth loss function*. Paper presented at 35th Conference on Uncertainty in Artificial Intelligence, UAI 2019, Tel Aviv, Israel.