Stability Region based Methods for Learning and Discovery

Many problems that arise in machine learning and data mining domains deal with nonlinearity and quite often demand users to obtain global optimal solutions rather than local optimal ones. Several algorithms had been proposed in the optimization literature and inherited by the machine learning community. Popularly known as the initialization problem, the ideal set of parameters required will significantly depend on the initial values given by the user. In this paper, we propose stability region based methods for systematically exploring the subspace of the parameters to obtain the neighborhood local optimal solutions. The proposed algorithm takes advantage of TRUST-TECH (TRansformation Under STability-reTaining Equilibria CHaracterization) to compute neighborhood local optimal solutions on the nonlinear surface in a systematic manner using stability regions. Our method explores the dynamic and geometric characteristics of stability boundaries of a nonlinear dynamical system corresponding to the nonlinear function of interest. Basically, our method coalesces the advantages of the traditional local optimizers with that of the dynamic and geometric characteristics of the stability regions of the corresponding nonlinear dynamical system of the log-likelihood function. Two phases namely, the local phase and the stability region phase, are repeated alternatively in the parameter space to achieve improvements in the quality of the solutions. The local phase obtains the local maximum of the nonlinear function and the stability region phase helps to escape out of the local maximum by moving towards the neighboring stability regions. The stability region based algorithms are applied to three important machine learning problems in: (1) Unsupervised learning model-based clustering, (2) Pattern discovery motif finding problem and (3) Supervised learning training artificial neural networks. Our algorithms were tested on both synthetic and real datasets and the advantages of using this stability region based framework are clearly manifested. This framework not only reduces the sensitivity to initialization, but also allows the flexibility for the practitioners to use various global and local methods that work well for a particular problem of interest.