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We exploit a formal correspondence between thermodynamics and inference,where the number of samples can be thought of as the inverse temperature, todefine a "learning capacity'' which is a measure of the effectivedimensionality of a model. We show that the learning capacity is a tinyfraction of the number of parameters for many deep networks trained on typicaldatasets, depends upon the number of samples used for training, and isnumerically consistent with notions of capacity obtained from the PAC-Bayesianframework. The test error as a function of the learning capacity does notexhibit double descent. We show that the learning capacity of a model saturatesat very small and very large sample sizes; this provides guidelines, as towhether one should procure more data or whether one should search for newarchitectures, to improve performance. We show how the learning capacity can beused to understand the effective dimensionality, even for non-parametric modelssuch as random forests and $k$-nearest neighbor classifiers.

Learning Capacity: A Measure of the Effective Dimensionality of a Model