Many modern machine learning applications, such as multi-task learning, require finding optimal model parameters to trade-off multiple objective functions that may conflict with each other. The notion of the Pareto set allows us to focus on the set of (often infinite number of) models that cannot be strictly improved. But it does not provide an actionable procedure for picking one or a few special models to return to practical users. In this paper, we consider \emph{optimization in Pareto set (OPT-in-Pareto)}, the problem of finding Pareto models that optimize an extra reference criterion function within the Pareto set. This function can either encode a specific preference from the users, or represent a generic diversity measure for obtaining a set of diversified Pareto models that are representative of the whole Pareto set. Unfortunately, despite being a highly useful framework, efficient algorithms for OPT-in-Pareto have been largely missing, especially for large-scale, non-convex, and non-linear objectives in deep learning. A naive approach is to apply Riemannian manifold gradient descent on the Pareto set, which yields a high computational cost due to the need for eigen-calculation of Hessian matrices. We propose a first-order algorithm that approximately solves OPT-in-Pareto using only gradient information, with both high practical efficiency and theoretically guaranteed convergence property. Empirically, we demonstrate that our method works efficiently for a variety of challenging multi-task-related problems.

翻译：许多现代机器学习应用程序,例如多任务学习,需要找到最佳模型参数来交换相互冲突的多重目标功能。 Pareto集的概念使我们能够集中关注一组无法严格改进的(往往是无限数量的)模型。但是它并没有提供一个可操作的程序来选择一种或几种特殊模型,以便返回实际用户。在本文中,我们认为在Pareto集(OPT-in-Pareto)中找到一个最佳模型来交换可能相互冲突的多重目标功能。找到在Pareto集中优化额外参照标准功能的Pareto模型的问题。这个功能可以将用户的具体偏好编码成一套(往往是无限数量的)模型,或者代表一种通用的多样性衡量标准,以获得一套代表整个Pareto集的多样化的多种模型。不幸的是,尽管这是一个非常有用的框架,但巴雷托LOma-in-Pareto集(OPT-in-in-in-inal-inal-legal-lical-al-lical-liversional-al-le)的算算法, 也显示一个高成本-rolegal-ral-ral-ral-deal-ral-ral-deal-de-deal-deal-deal-deal-deal-deal-deal-deal-lexal-deal-deal-legal-lex-lex-lex-lex-lex-le)的计算方法。一个高一个高要求。