Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory

In regular statistical models, it is well known that the cross validation leaving one out is asymptotically equivalent to Akaike information criterion. However, a lot of learning machines are singular statistical models, resulting that the asymptotic behavior of the cross validation has been left unknown. In previous papers, we established singular learning theory and proposed a widely applicable information criterion whose expectation value is asymptotically equal to the average Bayes generalization loss. In this paper, we theoretically compare the Bayes cross validation loss and the widely applicable information criterion and prove two theorems. Firstly, the Bayes cross validation loss is asymptotically equivalent to the widely applicable information criterion. Therefore, model selection and hyperparameter optimization using these two values are asymptotically equivalent to each other. Secondly, the sum of the Bayes generalization error and the Bayes cross validation error is asymptotically equal to $2\lambda/n$, where $\lambda$ is the log canonical threshold and $n$ is the number of training samples. This fact shows that the relation between the cross validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine.