I recently reviewed a thesis project wherein the student used recursion to minimize a function. The student used number of iterations as a stopping criterion. When I revised the manuscript, it gave me a flashback to 1991.
Back then I was inmersed into Numerical Dynamics and Chaos Theory conferences. I wrote a paper wherein one little component consisted in stopping a routine if it reaches 10,000 iterations. The piece landed in the hands of a picky reviewer. It was rejected on the grounds that number of iterations was not a good stopping criterion.
The reviewer was absolutely right. I missed the point back then, though.
Why use 1,000, 10,000, or 1,000,000 iterations? The point is that no matter how many iterations one uses there will always be someone out there asking: Why use n number of iterations? Why not more?
Indeed, number of iterations as a stopping criterion is not recommended, especially if the goal is optimization. Not to mention that this is a subjective approach which requires initializing an extra parameter. The parameter might be sensitive to other variables or initial conditions.
Instead of number of iterations, a better approach consists in using a comparative function F. F can be defined as the absolute error or absolute relative error between a sequence of results.
If F is less than a threshold value, we stop the iterative procedure; otherwise, we continue with it.
Such functions are given below:
F = | (X(n) – X(n+1)) | * 100 < threshold value
F = | (X(n) – X(n+1))/X(n+1) | * 1000 < threshold value
and so forth.
For example if F is less than, let say, 10^-6, we stop the recursions.
In this case, F is based on an objective, statistical criterion: the relative error between two consecutive results. Note that:
1. the analyst states the threshold value that best sweets his/her precision needs.
2. F is independent of any initial condition or parameter.
Of course, for this to work the recursion should be toward what we call in Chaos Theory, an attractive fixed point. Methods for evaluating if this is the case do exist in Numerical Dynamics (e.g., the Fixed Point Theorem).
Fortunately, the student’s problem was not of this kind and he was not dealing with a dynamical system at all.