  In sparse regression modeling via regularization such as the lasso, it is
important to select appropriate values of tuning parameters including
regularization parameters. The choice of tuning parameters can be viewed as a
model selection and evaluation problem. Mallows' $C_p$ type criteria may be
used as a tuning parameter selection tool in lasso-type regularization methods,
for which the concept of degrees of freedom plays a key role. In the present
paper, we propose an efficient algorithm that computes the degrees of freedom
by extending the generalized path seeking algorithm. Our procedure allows us to
construct model selection criteria for evaluating models estimated by
regularization with a wide variety of convex and non-convex penalties. Monte
Carlo simulations demonstrate that our methodology performs well in various
situations. A real data example is also given to illustrate our procedure.
