Multidimensional stochastic approximation schemes are presented, and conditions are given for these schemes to converge a.s. (almost surely) to the solutions of $k$ stochastic equations in $k$ unknowns and to the point where a regression function in $k$ variables achieves its maximum.
Julius R. Blum. "Multidimensional Stochastic Approximation Methods." Ann. Math. Statist. 25 (4) 737 - 744, December, 1954. https://doi.org/10.1214/aoms/1177728659