Sup-norm approximation bounds for networks through probabilistic methods

We consider the problem of approximating a smooth target function and its derivatives by networks involving superpositions and translations of a fixed activation function. The approximation is with respect to the sup-norm and the rate is shown to be of order O(n^-1/2); that is, the rate is independent of the dimension d. The results apply to neural and wavelet networks and extend the work of Barren(see Proc. 7th Yale Workshop on Adaptive and Learning Systems, May, 1992, and ibid., vol.39, p.930, 1993). The approach involves probabilistic methods based on central limit theorems for empirical processes indexed by classes of functions