Multiple-gradient Descent Algorithm for Pareto-Front Identification
Multiple-gradient Descent Algorithm for Pareto-Front Identification
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Pareto前沿识别的多梯度下降算法
DOI：
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发表时间：
2014
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通讯作者：
J. Désidéri
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作者：
J. Désidéri
本文综合并扩展了多篇出版物，其中提出并测试了多目标可微优化处理的多梯度下降算法（MGDA）。该方法最初在[3]中介绍，并在[8]中进行了测试和重新表述。与进化策略相比，其识别帕累托前沿的功效[18]已在[22]中得到证明。最近，提出了一种变体 MGDA-II，其中下降方向是通过基于带有特殊归一化的 Gram-Schmidt 正交化过程 (GSP) 的直接过程 [6] 来计算的。该算法在域分区模拟的环境中进行了测试，作为一种同时匹配不同界面组件的技术[4]。实验揭示了缩放的重要性，并提出了稍微修改的归一化程序（“MGDA-IIb”）。此后提出了两种新颖的变体。第一个是 MGDA-III，实现了两项增强。首先，每当测试表明当前对搜索方向的估计也足够时，GSP 的执行就不完整。尚未考虑梯度；这种改进简化了当梯度大致指向同一方向时搜索方向的识别，并使多个目标函数共同的方向导数更大。其次，GSP 中考虑不同梯度的顺序是以一种独特的方式定义的，该方式旨在支持不完整的 GSP。在第二个变体 MGDA-IV 中，当 Hessians 已知时，缩放问题就得到解决。还提出了一种变体，其中 Hessian 矩阵通过 Broyden-Fletcher-Goldfarb-Shanno (BFGS) 公式进行估计。最后，提出了一种在下降步骤中优化调整步长的解决方案。
This article compounds and extends several publications in which a Multiple-Gradient Descent Algorithm (MGDA), has been proposed and tested for the treatment of multi-objective differentiable optimization. Originally introduced in [3], the method has been tested and reformulated in [8]. Its efficacy to identify the Pareto front [18] has been demonstrated in [22], in comparison with an evolutionary strategy. Recently, a variant, MGDA-II, has been proposed in which the descent direction is calculated by a direct procedure [6] based on a Gram-Schmidt orthogonalization process (GSP) with special normalization. This algorithm was tested in the context of a simulation by domain partitioning, as a technique to match the different interface components concurrently [4]. The experimentation revealed the importance of scaling, and a slightly modified normalization procedure was proposed (“MGDA-IIb”). Two novel variants have been proposed since. The first, MGDA-III, realizes two enhancements. Firstly, the GSP is conducted incompletely whenever a test reveals that the current estimate of the direction of search is adequate also w.r.t. the gradients not yet taken into account; this improvement simplifies the identification of the search direction when the gradients point roughly in the same direction, and makes the directional derivative common to several objective-functions larger. Secondly, the order in which the different gradients are considered in the GSP is defined in a unique way devised to favor an incomplete GSP. In the second variant, MGDA-IV, the question of scaling is addressed when the Hessians are known. A variant is also proposed in which the Hessians are estimated by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula. Lastly, a solution is proposed to adjust the step-size optimally in the descent step.