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.

翻译：贪心算法在训练神经网络以求解变分问题方面已得到成功分析与应用，确保了收敛阶的严格保证。然而，其实际应用受到子问题的限制，这些子问题通常需要对离散字典进行穷举搜索，并产生显著的计算成本。这一局限在高维问题中尤为关键。本文提出一种更实用的方法：在贪心算法的每次迭代中随机离散化字典。我们量化了随机化离散字典所需的规模，并证明所提算法以高概率实现弱贪心算法，达到最优收敛阶。通过大量数值实验，我们展示了使用随机化离散字典相较于确定性字典的优势，证明了离散字典规模可降低数个数量级，尤其在高维情况下。