Exact post-selection inference for the generalized lasso path

Abstract

We study tools for inference conditioned on model selection events that are defined by the generalized lasso regularization path. The generalized lasso estimate is given by the solution of a penalized least squares regression problem, where the penalty is the $\ell_{1}$ norm of a matrix $D$ times the coefficient vector. The generalized lasso path collects these estimates as the penalty parameter $\lambda$ varies (from $\infty$ down to 0). Leveraging a (sequential) characterization of this path from Tibshirani and Taylor [37], and recent advances in post-selection inference from Lee at al. [22], Tibshirani et al. [38], we develop exact hypothesis tests and confidence intervals for linear contrasts of the underlying mean vector, conditioned on any model selection event along the generalized lasso path (assuming Gaussian errors in the observations).

Our construction of inference tools holds for any penalty matrix $D$. By inspecting specific choices of $D$, we obtain post-selection tests and confidence intervals for specific cases of generalized lasso estimates, such as the fused lasso, trend filtering, and the graph fused lasso. In the fused lasso case, the underlying coordinates of the mean are assigned a linear ordering, and our framework allows us to test selectively chosen breakpoints or changepoints in these mean coordinates. This is an interesting and well-studied problem with broad applications; our framework applied to the trend filtering and graph fused lasso cases serves several applications as well. Aside from the development of selective inference tools, we describe several practical aspects of our methods such as (valid, i.e., fully-accounted-for) post-processing of generalized lasso estimates before performing inference in order to improve power, and problem-specific visualization aids that may be given to the data analyst for he/she to choose linear contrasts to be tested. Many examples, from both simulated and real data sources, are presented to examine the empirical properties of our inference methods.