关于广义保序递归划分算法正确性的研究
On the Correctness of the Generalized Isotonic Recursive Partitioning Algorithm
AI总结：
本文指出广义保序递归划分算法可能产生非保序解，通过证明递归二元划分可找到正确解，并提出修改中间解选择及将划分从三元简化为二元的方法，从而保证算法正确性。
摘要：
本文对广义保序递归划分（GIRP）算法进行了深入分析，该算法用于在可分离凸损失下拟合保序模型，最初由 Luss 和 Rosset [J. Comput. Graph. Statist., 23 (2014), pp. 192--201] 针对可微损失提出，并由 Painsky 和 Rosset [IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), pp. 308-321] 扩展到不可微损失情形。GIRP 算法具有一个吸引人的特性：在算法的每一步中，中间解都满足保序性约束。本文首先通过一个例子表明，文献中描述的 GIRP 算法可能无法产生保序模型，这说明必须仔细处理保序回归问题解的存在性与唯一性问题。随后本文证明，在可能存在的多个解中，确实存在一个可以通过对观测数据集进行递归二元划分而找到的解。对 GIRP 算法进行一处小的修改就足以获得正确解，并保持所有中间解均为保序的这一理想性质。所提出的修改包括对中间解的恰当选择，以及将划分步骤从三元简化为二元。
英文摘要：
This paper presents an in-depth analysis of the generalized isotonic recursive partitioning (GIRP) algorithm for fitting isotonic models under separable convex losses, proposed by Luss and Rosset [J. Comput. Graph. Statist., 23 (2014), pp. 192--201] for differentiable losses and extended by Painsky and Rosset [IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), pp. 308-321] for nondifferentiable losses. The GIRP algorithm poseses an attractive feature that in each step of the algorithm, the intermediate solution satisfies the isotonicity constraint. The paper begins with an example showing that the GIRP algorithm as described in the literature may fail to produce an isotonic model, suggesting that the existence and uniqueness of the solution to the isotonic regression problem must be carefully addressed. It proceeds with showing that, among possibly many solutions, there indeed exists a solution that can be found by recursive binary partitioning of the set of observed data. A small modification of the GIRP algorithm suffices to obtain a correct solution and preserve the desired property that all the intermediate solutions are isotonic. This proposed modification includes a proper choice of intermediate solutions and a simplification of the partitioning step from ternary to binary.
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