The ACE (alternating conditional expectations) algorithm developed by Breiman and Friedman is an iterative method for finding optimal transformations of variables in multiple regression. Recently, several authors have extended ACE to discriminant analysis, time series and principal components. The central idea of ACE and of each of these extensions is that an optimal transformation $\phi^\ast$ minimizes a squared error-related functional over a Hilbert space, subject to nonlinear functional constraints. An estimate $\hat{\phi}^{(N)}$ is obtained by minimizing an estimate of the functional, subject to estimates of the constraints, over a smoothness restricted class of transformations. Using the method of sieves, conditions are established for consistency of $\hat{\phi}^{(N)}$ in the $\mathbf{L}^2$ sense.