The minimum description length (MDL) principle in supervised learning is studied. One of the most important theories for the MDL principle is Barron and Cover's theory (BC theory), which gives a mathematical justification of the MDL principle. The original BC theory, however, can be applied to supervised learning only approximately and limitedly. Though Barron et al. recently succeeded in removing a similar approximation in case of unsupervised learning, their idea cannot be essentially applied to supervised learning in general. To overcome this issue, an extension of BC theory to supervised learning is proposed. The derived risk bound has several advantages inherited from the original BC theory. First, the risk bound holds for finite sample size. Second, it requires remarkably few assumptions. Third, the risk bound has a form of redundancy of the two-stage code for the MDL procedure. Hence, the proposed extension gives a mathematical justification of the MDL principle to supervised learning like the original BC theory. As an important example of application, new risk and (probabilistic) regret bounds of lasso with random design are derived. The derived risk bound holds for any finite sample size $n$ and feature number $p$ even if $n\ll p$ without boundedness of features in contrast to the past work. Behavior of the regret bound is investigated by numerical simulations. We believe that this is the first extension of BC theory to general supervised learning with random design without approximation.