We have been writing for a while about the convergence of Newton steps applied to a logistic regression (See: What does a generalized linear model do?, How robust is logistic regression? and Newton-Raphson can compute an average). This is all based on our principle of working examples for understanding. This eventually progressed to some writing on the nature of problem solving (a nice complement to our earlier writing on calculation). In the course of research we were directed to a very powerful technique called the MM algorithm (see: “The MM Algorithm” Kenneth Lang, 2007; “A Tutorial on MM Algorithms”, David R. Hunter, Kenneth Lange, Amer. Statistician 58:30–37, 2004; and “Monotonicity of Quadratic-Approximation Algorithms”, Dankmar Bohning, Bruce G. Lindsay, Ann. Inst. Statist. Math, Vol. 40, No. 4, pp 641-664, 1988). The MM algorithm introduces an essential idea: majorized functions (not to be confused with the majorized order on R^d). Majorization it is an interesting way to modify Newton methods to be reliable contractions (and therefore converge in a manner similar to EM algorithms).
Here we will work an example of the MM method. We will not work it in its most general form, but in a form that quickly reveals much of the beauty of the method. We also introduce a “collared Newton step” which guarantees convergence without resorting to line-search (essentially resolving the issues in solving a logistic regression by Newton style methods). Continue reading