The AdaBoost Flow

TL;DR

This paper introduces the AdaBoost flow, a dynamical system modeled by ODEs, unifying several AdaBoost algorithms and relating them to Toda systems, gradient flows, and Ricci flow, offering a new geometric perspective.

Contribution

It presents a novel dynamical system framework for AdaBoost algorithms, connecting them to integrable systems and geometric flows, and proposes a continuous-time approach for constructing boosting algorithms.

Findings

AdaBoost algorithms can be embedded in a dynamical system called AdaBoost flow.

The AdaBoost flow equations relate to the Toda system dynamics.

A geometric interpretation of AdaBoost as a gradient flow is provided.

Abstract

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 defined by level sets of the 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 algorithm with the Perelman's construction for the Ricci flow.