Eluder Dimension and Generalized Rank

We study the relationship between the eluder dimension for a function class and a generalized notion of rank, defined for any monotone “activation” σ : R → R, which corresponds to the minimal dimension required to represent the class as a generalized linear model. When σ has derivatives bounded away from 0, it is known that σ-rank gives rise to an upper bound on eluder dimension for any function class; we show however that eluder dimension can be exponentially smaller than σ-rank. We also show that the condition on the derivative is necessary; namely, when σ is the relu activation, we show that eluder dimension can be exponentially larger than σ-rank.