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Many modern machine learning applications, such as multi-task learning,require finding optimal model parameters to trade-off multiple objectivefunctions that may conflict with each other. The notion of the Pareto setallows us to focus on the set of (often infinite number of) models that cannotbe strictly improved. But it does not provide an actionable procedure forpicking one or a few special models to return to practical users. In thispaper, we consider \emph{optimization in Pareto set (OPT-in-Pareto)}, theproblem of finding Pareto models that optimize an extra reference criterionfunction within the Pareto set. This function can either encode a specificpreference from the users, or represent a generic diversity measure forobtaining a set of diversified Pareto models that are representative of thewhole Pareto set. Unfortunately, despite being a highly useful framework,efficient algorithms for OPT-in-Pareto have been largely missing, especiallyfor large-scale, non-convex, and non-linear objectives in deep learning. Anaive approach is to apply Riemannian manifold gradient descent on the Paretoset, which yields a high computational cost due to the need foreigen-calculation of Hessian matrices. We propose a first-order algorithm thatapproximately solves OPT-in-Pareto using only gradient information, with bothhigh practical efficiency and theoretically guaranteed convergence property.Empirically, we demonstrate that our method works efficiently for a variety ofchallenging multi-task-related problems.

Pareto Navigation Gradient Descent: a First-Order Algorithm for Optimization in Pareto Set