# An Experimental and Theoretical Comparison of Model Selection Methods

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## Abstract

We investigate the problem of *model selection* in the setting of supervised learning of boolean functions from independent random examples. More precisely, we compare methods for finding a balance between the complexity of the hypothesis chosen and its observed error on a random training sample of limited size, when the goal is that of minimizing the resulting generalization error. We undertake a detailed comparison of three well-known model selection methods — a variation of Vapnik's *Guaranteed Risk Minimization* (GRM), an instance of Rissanen's *Minimum Description Length Principle* (MDL), and (hold-out) cross validation (CV). We introduce a general class of model selection methods (called *penalty-based* methods) that includes both GRM and MDL, and provide general methods for analyzing such rules. We provide both controlled experimental evidence and formal theorems to support the following conclusions:

•Even on simple model selection problems, the behavior of the methods examined can be both complex and incomparable. Furthermore, no amount of “tuning” of the rules investigated (such as introducing constant multipliers on the complexity penalty terms, or a distribution-specific “effective dimension”) can eliminate this incomparability.

•It is possible to give rather general bounds on the generalization error, as a function of sample size, for penalty-based methods. The quality of such bounds depends in a precise way on the extent to which the method considered automatically limits the complexity of the hypothesis selected.

•For *any* model selection problem, the additional error of cross validation compared to *any* other method can be bounded above by the sum of two terms. The first term is large only if the learning curve of the underlying function classes experiences a phase transition” between (1-γ)*m* and *m* examples (where gamma is the fraction saved for testing in CV). The second and competing term can be made arbitrarily small by increasing γ.

•The class of penalty-based methods is fundamentally handicapped in the sense that there exist two types of model selection problems for which every penalty-based method must incur large generalization error on at least one, while CV enjoys small generalization error on both.

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