  We consider the problem of selective inference after solving a (randomized)
convex statistical learning program in the form of a penalized or constrained
loss function. Our first main result is a change-of-measure formula that
describes many conditional sampling problems of interest in selective
inference. Our approach is model-agnostic in the sense that users may provide
their own statistical model for inference, we simply provide the modification
of each distribution in the model after the selection.
  Our second main result describes the geometric structure in the Jacobian
appearing in the change of measure, drawing connections to curvature measures
appearing in Weyl-Steiner volume-of-tubes formulae. This Jacobian is necessary
for problems in which the convex penalty is not polyhedral, with the
prototypical example being group LASSO or the nuclear norm. We derive explicit
formulae for the Jacobian of the group LASSO.
  To illustrate the generality of our method, we consider many examples
throughout, varying both the penalty or constraint in the statistical learning
problem as well as the loss function, also considering selective inference
after solving multiple statistical learning programs. Penalties considered
include LASSO, forward stepwise, stagewise algorithms, marginal screening and
generalized LASSO. Loss functions considered include squared-error, logistic,
and log-det for covariance matrix estimation.
  Having described the appropriate distribution we wish to sample from through
our first two results, we outline a framework for sampling using a projected
Langevin sampler in the (commonly occuring) case that the distribution is
log-concave.
