Convergence and statistical properties of the Autotune Lasso algorithm

Characterize the limit points of the Autotune Lasso iterative algorithm that alternates coordinate descent updates for Lasso coefficients with noise variance updates based on partial residual ranking and sequential F-tests; derive rigorous statistical properties of the resulting estimator and identify conditions under which the iterative procedure fails.

Background

The paper introduces Autotune, an automatic tuning parameter selection strategy for Lasso that alternates between estimating regression coefficients via coordinate descent at a data-driven penalty level and estimating the noise variance using a procedure based on partial residuals and sequential F-tests. This yields a fast, accurate tuning method and a noise variance estimator useful for high-dimensional inference.

Although extensive empirical results are provided, the theoretical behavior of the iterative procedure is not analyzed. In particular, the paper explicitly notes that the convergence analysis, characterization of limit points, and statistical properties (including conditions for failure) remain to be addressed. These are crucial to understanding when and why the algorithm succeeds or may fail.