Data-driven iterative learning control can achieve high performance for systems performing repeating tasks without the need for modeling. The aim of this paper is to develop a fast data-driven method for iterative learning control that is suitable for massive MIMO systems through the use of efficient unbiased gradient estimates. A stochastic conjugate gradient descent algorithm is developed that uses dedicated experiments to determine the conjugate search direction and optimal step size at each iteration. The approach is illustrated on a multivariable example, and it is shown that the method is superior to both the earlier stochastic gradient descent and deterministic conjugate gradient descent methods.