Radial basis functions (RBFs) are the natural generalization of coarse
coding to continuous-valued features. Rather than each feature being
either 0 or 1, it can be anything in the interval 
, reflecting
various degrees to which the feature is present. A typical RBF
feature,
i, has a gaussian (bell-shaped) response, 
, dependent only on
the distance between the state, s, and the feature's prototypical or
center state, 
, and relative to the feature's width, 
:
The norm or distance metric of course can be chosen in whatever way seems most appropriate to the states and task at hand. Figure 8.7 shows a 1-dimensional example with a euclidean distance metric.
Figure 8.7: One-dimensional radial basis functions.
An RBF network is simply a linear function approximator using RBFs for its features. Learning is defined by equations, (8.3) and (8.8) in exactly the same way it is for other linear function approximators. The primary advantage of RBFs over binary features is that they produce approximate functions that vary smoothly and are differentiable. In addition, some learning methods for RBF networks change the centers and widths of the features as well. Such nonlinear methods may be able to fit the target function much more precisely. The downside to RBF networks, and to nonlinear RBF networks especially, is greater computational complexity and, often, more manual tuning before learning is robust and efficient.