We are interested in maximizing a general (but continuous) function where observations are noisy and may be expensive. We derive a knowledge gradient policy, which chooses measurements which maximize the expected value of information, while using a locally parametric belief model which uses linear approximations around regions of the function, known as clouds. The method, called DC-RBF (Dirichlet Clouds with Radial Basis Functions) is well suited to recursive estimation, and uses a compact representation of the function which avoids storing the entire history. Our technique allows for correlated beliefs within adjacent subsets of the alternatives and does not pose any a priori assumption on the global shape of the underlying function. Experimental work suggests that the method adapts to a range of arbitrary, continuous functions, and appears to reliably find the optimal solution.