Improved MDL Estimators Using Fiber Bundle of Local Exponential Families for Non-exponential Families

Minimum Description Length (MDL) estimators, using two-part codes for universal coding, are analyzed. For general parametric families under certain regularity conditions, we introduce a two-part code whose regret is close to the minimax regret, where regret of a code with respect to a target family M is the difference between the code length of the code and the ideal code length achieved by an element in M. This is a generalization of the result for exponential families by Gr\"unwald. Our code is constructed by using an augmented structure of M with a bundle of local exponential families for data description, which is not needed for exponential families. This result gives a tight upper bound on risk and loss of the MDL estimators based on the theory introduced by Barron and Cover in 1991. Further, we show that we can apply the result to mixture families, which are a typical example of non-exponential families.

Approximate Spectral Decomposition of Fisher Information Matrix for Simple ReLU Networks

We investigate the Fisher information matrix (FIM) of one hidden layer networks with the ReLU activation function and obtain an approximate spectral decomposition of FIM under certain conditions. From this decomposition, we can approximate the main eigenvalues and eigenvectors. We confirmed by numerical simulation that the obtained decomposition is approximately correct when the number of hidden nodes is about 10000.

Minimum Description Length Principle in Supervised Learning with Application to Lasso

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.