Continuous optimization of hyper-parameters
Many machine learning algorithms can be formulated as the minimization of a training criterion which involves a hyper-parameter. This hyper-parameter is usually chosen by trial and error with a model selection criterion. In this paper we present a methodology to optimize several hyper-parameters, based on the computation of the gradient of a model selection criterion with respect to the hyper-parameters. In the case of a quadratic training criterion, the gradient of the selection criterion with respect to the hyper-parameters is efficiently computed by back-propagating through a Cholesky decomposition. In the more general case, we show that the implicit function theorem can be used to derive a formula for the hyper-parameter gradient involving second derivatives of the training criterion.