Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: Global minimization via Monte Carlo

TLDR: In this article, the authors studied the asymptotic behavior of the systems where the objective function values can only be sampled via Monte Carlo, where the discrete algorithm is a combination of stochastic approximation and simulated annealing.

Abstract: The asymptotic behavior of the systems $X_{n + 1} = X_n + a_n b( {X_n ,\xi _n } ) + a_n \sigma ( X_n )\psi_n $ and $dy = \bar b( y )dt + \sqrt {a( t )} \sigma ( y )dw$ is studied, where $\{ {\psi _n } \}$ is i.i.d. Gaussian, $\{ \xi _n \}$ is a (correlated) bounded sequence of random variables and $a_n \approx A_0/\log (A_1 + n )$. Without $\{ \xi _n \}$, such algorithms are versions of the “simulated annealing” method for global optimization. When the objective function values can only be sampled via Monte Carlo, the discrete algorithm is a combination of stochastic approximation and simulated annealing. Our forms are appropriate. The $\{ \psi _n \}$ are the “annealing” variables, and $\{ \xi _n \}$ is the sampling noise. For large $A_0 $, a full asymptotic analysis is presented, via the theory of large deviations: Mean escape time (after arbitrary time n) from neighborhoods of stable sets of the algorithm, mean transition times (after arbitrary time n) from a neighborhood of one stable set to another, a...