Regret Ratio Minimization in Multi-Objective Submodular Function Maximization

Abstract

Submodular function maximization has numerous applications in machine learning and artificial intelligence. Many real applications require multiple submodular objective func-tions to be maximized, and which function is regarded as important by a user is not known in advance. In such cases, it is desirable to have a small family of representative solutions that would satisfy any user’s preference. A traditional approach for solving such a problem is to enumerate the Pareto optimal solutions. However, owing to the massive number of Pareto optimal solutions (possibly exponentially many), it is difficult for a user to select a solution. In this paper, we propose two efficient methods for finding a small family of representative solutions, based on the notion of regret ratio. The first method outputs a family of fixed size with a nontrivial regret ratio. The second method enables us to choose the size of the output family, and in the biobjective case, it has a provable trade-off between the size and the regret ratio. Using real and synthetic data, we empirically demonstrate that our methods achieve a small regret ratio.

Multi-Objective Submodular Maximization by Regret Ratio Minimization with Theoretical Guarantee

“…Note that αapproximation implies that the generated solution X * satisfies f (X * ) ≥ α • max X∈C f (X), and thus the regret ratio 1−f (X * )/ max X∈C f (X) ≤ 1−α. When d = 2, the upper bound becomes 1−α+O(1/k), which is the same as that by the only existing algorithm POLYTOPE (Soma and Yoshida 2017). When d ≥ 3, this is the first theoretical guarantee.…”

“…Definition 1 (Regret Ratio Minimization (Soma and Yoshida 2017)). Given submodular functions f 1 , f 2 , .…”

“…that is to find a solution maximizing the minimum of all submodular objectives. Recently, Soma and Yoshida (2017) considered the formulation of minimizing regret ratio:…”

“…( 2), and often considered in real-world robust submodular maximization applications, e.g., robust influence maximization (Chen et al 2016). Soma and Yoshida (2017) proposed a geometric algorithm POLYTOPE for regret ratio minimization. Given an α-approximation algorithm for single-objective submodular maximization, POLYTOPE first gets an initial set of d solutions by applying the α-approximation algorithm to solve each objective function, and then iteratively generates new solutions until finding k solutions in total.…”

Happiness Maximizing Sets under Group Fairness Constraints (Technical Report)

“…There have been many studies on regret minimizing set (RMS) [5,35,36,41,55], happiness maximizing set (HMS) [39,56], and different variants of them [2,6,8,12,14,15,29,30,34,38,43,44,50,52,53,59] (see [54] for an extensive survey). The RMS problem was first proposed by Nanongkai et al [35] to alleviate the deficiencies of top-𝑘 and skyline queries.…”