The AdaBoost flow

We introduce a dynamical system which we call the AdaBoost flow. The flow is defined by a system of ODEs with control. We show that three algorithms of the AdaBoost family (i) the AdaBoost algorithm of Schapire and Freund (ii) the arc-gv algorithm of Breiman (iii) the confidence rated prediction of Schapire and Singer can be can be embedded in the AdaBoost flow. The nontrivial part of the AdaBoost flow equations coincides with the equations of dynamics of nonperiodic Toda system written in terms of spectral variables. We provide a novel invariant geometrical description of the AdaBoost algorithm as a gradient flow on a foliation of the simplex of probability measures defined by level sets of a potential function. We propose a new approach for constructing boosting algorithms as a continuous time gradient flow on measures defined by various metrics and potential functions. Finally, we explain similarity of the AdaBoost flow with the Perelman's construction for the Ricci flow.