  A number of modern learning tasks involve estimation from heterogeneous
information sources. This includes classification with labeled and unlabeled
data as well as other problems with analogous structure such as competitive
(game theoretic) problems. The associated estimation problems can be typically
reduced to solving a set of fixed point equations (consistency conditions). We
introduce a general method for combining a preferred information source with
another in this setting by evolving continuous paths of fixed points at
intermediate allocations. We explicitly identify critical points along the
unique paths to either increase the stability of estimation or to ensure a
significant departure from the initial source. The homotopy continuation
approach is guaranteed to terminate at the second source, and involves no
combinatorial effort. We illustrate the power of these ideas both in
classification tasks with labeled and unlabeled data, as well as in the context
of a competitive (min-max) formulation of DNA sequence motif discovery.
