“Go with the winners” algorithms

We can view certain randomized optimization algorithms as rules for randomly moving a particle around in a state space; each state might correspond to a distinct solution to the optimization problem, or more generally, the state space might express some other structure underlying the optimization algorithm. In this setting, a general paradigm for designing heuristics is to run several simulations of the algorithm simultaneously, and every so often classify the particles as “doing well” or “doing badly”, and move each particle that is “doing badly” to the position of one that is “doing well”. In this paper, we give a rigorous analysis of such a “go with the winners” scheme in the concrete setting of searching for a deep leaf in a tree. There are two relevant parameters of the tree: its depth d, and another parameter κ which is a measure of the imbalance of the tree. We prove that the running time of the “go with the winners” scheme (to achieve 99% probability of success) is bounded by a polynomial in d and κ. By contrast, the simple restart scheme: run several independent simulations and pick the deepest leaf encountered takes time exponential in κ and d in the worst-case. We also show that any algorithm that guarantees a constant probability of success must have worst case running time at least κd