In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact Krylov is used, search direction is not anymore in span of Krylov subspace. I am in process of analyzing convergence of one such version of CG where search directions are not anymore in span of Krylov subspace of initial residual. I wonder if any such method has been already discussed in literature. In particular, what will be condition of any instance of search direction (more than just being a descent direction) for CG to successfully converge. How does the convergence rate depends on initial search direction?
I recommend Simoncini and Szyld, Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput., 25 (2003) pp. 454-477. It looks like it discusses exactly what you want.