  Greedy algorithms have been successfully analyzed and applied in training
neural networks for solving variational problems, ensuring guaranteed
convergence orders. However, their practical applicability is limited due to
the subproblems, which often require an exhaustive search over a discrete
dictionary and incur significant computational costs. This limitation becomes
critical, especially in high-dimensional problems. In this paper, we propose a
more practical approach of randomly discretizing the dictionary at each
iteration of the greedy algorithm. We quantify the required size of the
randomized discrete dictionary and prove that, with high probability, the
proposed algorithm realizes a weak greedy algorithm, achieving optimal
convergence orders. Through numerous numerical experiments, we demonstrate the
advantage of using randomized discrete dictionaries over a deterministic one by
showing orders of magnitude reductions in the size of the discrete dictionary,
particularly in higher dimensions.
