Learning Nonlinearly Parametrized Decision Regions

In this paper we present a deterministic analysis of an online scheme for learning very general classes of nonlinearly parametrized decision regions. The only input required is a sequence ((xk ; yk )) k2Z + of data samples, where yk = 1 if xk belongs to the decision region of interest, and yk = Gamma1 otherwise. Averaging results and Lyapunov theory are used to prove the stability of the scheme. In the course of this proof, conditions on both the parametrization and the sequence of input examples arise which are sufficient to guarantee convergence of the algorithm. A number of examples are presented, including the problem of learning an intersection of half spaces using only data samples.