Stability analysis of the decomposition method for solving support vector machines

In situations where processing memory is limited, the Support Vector Machine quadratic program can be decomposed into smaller sub-problems and solved sequentially. The convergence of this method has been proven previously through (he use of a counting method. In this initial investigation, we approach the convergence analysis by treating the decomposed sub-problems as subsystems of a general system. The gradients of the sub-problems and the inequality constraints are explicitly modelled as system variables. The change in these variables during optimization form a dynamic system modelled by vector differential equations. We show that the change in the objective function can be written as the energy in the system. This makes it a natural Lyapunov function, which has an asymptotically stable point at the origin. The asymptotic stability of the whole system then follows under certain assumptions.