A typical approach to the problem of selecting between models of differing complexity is to choose the model with the minimum Akaike Information Criterion (AIC) score. This paper examines a common scenario in which there is more than one candidate model with the same number of free parameters which violates the conditions under which AIC was derived. The main result of this paper is a novel upper bound that quantifies the poor performance of the AIC criterion when applied in this setting. Crucially, the upper-bound does not depend on the sample size and will not disappear even asymptotically. Additionally, an AIC-like criterion for sparse feature selection in regression models is derived, and simulation results in the case of denoising a signal by wavelet thresholding demonstrate the new AIC approach is competitive with SureShrink thresholding.
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