Functional ANOVA models are considered in the context of generalized regression, which includes logistic regression, probit regression and Poisson regression as special cases. The multivariate predictor function is modeled as a specified sum of a constant term, main effects and interaction terms. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen approximating spaces. We allow general linear spaces and their tensor products as building blocks for the approximating spaces. It is shown that the $L_2$ rates of convergence of the maximum likelihood estimates and their ANOVA components are determined by the approximation power and dimension of the approximating spaces. When the approximating spaces are appropriately chosen, the optimal rates of convergence can be achieved.