An MCMC-free approach to post-selective inference

We develop a Monte Carlo-free approach to inference post output from randomized algorithms with a convex loss and a convex penalty. The pivotal statistic based on a truncated law, called the selective pivot, usually lacks closed form expressions. Inference in these settings relies upon standard Monte Carlo sampling techniques at a reference parameter followed by an exponential tilting at the reference. Tilting can however be unstable for parameters that are far off from the reference parameter. We offer in this paper an alternative approach to construction of intervals and point estimates by proposing an approximation to the intractable selective pivot. Such an approximation solves a convex optimization problem in |E| dimensions, where |E| is the size of the active set observed from selection. We empirically show that the confidence intervals obtained by inverting the approximate pivot have valid coverage.