A Robbins--Monro Sequence That Can Exploit Prior Information For Faster
  Convergence
Siwei Liu Ke Ma Stephan M. Goetz
We propose a new method to improve the convergence speed of the Robbins-Monro
algorithm by introducing prior information about the target point into the
Robbins-Monro iteration. We achieve the incorporation of prior information
without the need of a -- potentially wrong -- regression model, which would
also entail additional constraints. We show that this prior-information
Robbins-Monro sequence is convergent for a wide range of prior distributions,
even wrong ones, such as Gaussian, weighted sum of Gaussians, e.g., in a kernel
density estimate, as well as bounded arbitrary distribution functions greater
than zero. We furthermore analyse the sequence numerically to understand its
performance and the influence of parameters. The results demonstrate that the
prior-information Robbins-Monro sequence converges faster than the standard
one, especially during the first steps, which are particularly important for
applications where the number of function measurements is limited, and when the
noise of observing the underlying function is large. We finally propose a rule
to select the parameters of the sequence.