  A surrogate function is often employed to reduce the number of objective
function evaluations for optimization. However, the effect of using a surrogate
model in evolutionary approaches has not been theoretically investigated. This
paper theoretically analyzes the information-geometric optimization framework
using a surrogate function. The value of the expected objective function under
the candidate sampling distribution is used as the measure of progress of the
algorithm. We assume that the surrogate function is maintained so that the
population version of the Kendall's rank correlation coefficient between the
surrogate function and the objective function under the candidate sampling
distribution is greater than or equal to a predefined threshold. We prove that
information-geometric optimization using such a surrogate function leads to a
monotonic decrease in the expected objective function value if the threshold is
sufficiently close to one. The acceptable threshold value is analyzed for the
case of the information-geometric optimization instantiated with Gaussian
distributions, i.e., the rank-$\mu$ update CMA-ES, on a convex quadratic
objective function. As an alternative to the Kendall's rank correlation
coefficient, we investigate the use of the Pearson correlation coefficient
between the weights assigned to candidate solutions based on the objective
function and the surrogate function.
