The Loss Rank Principle for Model Selection

We introduce a new principle for model selection in regression and classification. Many regression models are controlled by some smoothness or flexibility or complexity parameter c, e.g. the number of neighbors to be averaged over in k nearest neighbor (kNN) regression or the polynomial degree in regression with polynomials. Let f̂ c D be the (best) regressor of complexity c on data D. A more flexible regressor can fit more data D′ well than a more rigid one. If something (here small loss) is easy to achieve it’s typically worth less. We define the loss rank of f̂ c D as the number of other (fictitious) data D ′ that are fitted better by f̂ c D′ than D is fitted by f̂ c D. We suggest selecting the model complexity c that has minimal loss rank (LoRP). Unlike most penalized maximum likelihood variants (AIC,BIC,MDL), LoRP only depends on the regression functions and the loss function. It works without a stochastic noise model, and is directly applicable to any non-parametric regressor, like kNN. In this paper we formalize, discuss, and motivate LoRP, study it for specific regression problems, in particular linear ones, and compare it to other model selection schemes.