  We present an effective framework for improving the breakdown point of robust
regression algorithms. Robust regression has attracted widespread attention due
to the ubiquity of outliers, which significantly affect the estimation results.
However, many existing robust least-squares regression algorithms suffer from a
low breakdown point, as they become stuck around local optima when facing
severe attacks. By expanding on the previous work, we propose a novel framework
that enhances the breakdown point of these algorithms by inserting a prior
distribution in each iteration step, and adjusting the prior distribution
according to historical information. We apply this framework to a specific
algorithm and derive the consistent robust regression algorithm with iterative
local search (CORALS). The relationship between CORALS and momentum gradient
descent is described, and a detailed proof of the theoretical convergence of
CORALS is presented. Finally, we demonstrate that the breakdown point of CORALS
is indeed higher than that of the algorithm from which it is derived. We apply
the proposed framework to other robust algorithms, and show that the improved
algorithms achieve better results than the original algorithms, indicating the
effectiveness of the proposed framework.
