A Test Score-Based Approach to Stochastic Submodular Optimization

Abstract

We study the canonical problem of maximizing a stochastic submodular function subject to a cardinality constraint, where the goal is to select a subset from a ground set of items with uncertain individual performances to maximize their expected group value. Although near-optimal algorithms have been proposed for this problem, practical concerns regarding scalability, compatibility with distributed implementation, and expensive oracle queries persist in large-scale applications. Motivated by online platforms that rely on individual item scores for content recommendation and team selection, we study a special class of algorithms that select items based solely on individual performance measures known as test scores. The central contribution of this work is a novel and systematic framework for designing test score–based algorithms for a broad class of naturally occurring utility functions. We introduce a new scoring mechanism that we refer to as replication test scores and prove that as long as the objective function satisfies a diminishing-returns condition, one can leverage these scores to compute solutions that are within a constant factor of the optimum. We then extend these scoring mechanisms to the more general stochastic submodular welfare-maximization problem, where the goal is to partition items into groups to maximize the sum of the expected group values. For this more difficult problem, we show that replication test scores can be used to develop an algorithm that approximates the optimal solution up to a logarithmic factor. The techniques presented in this work bridge the gap between the rigorous theoretical work on submodular optimization and simple, scalable heuristics that are useful in certain domains. In particular, our results establish that in many applications involving the selection and assignment of items, one can design algorithms that are intuitive and practically relevant with only a small loss in performance compared with the state-of-the-art approaches. This paper was accepted by Chung Piaw Teo, optimization.

Test Score Algorithms for Budgeted Stochastic Utility Maximization

“…We also give a more refined analysis for the regime of regular item costs by parametrizing variations in item costs. As a byproduct of our analysis, we recover the result of Sekar et al (2020) for the cardinality constrained case. We further derive approximation guarantees for the sub-classes of value functions with bounded curvature, for the notion of curvature originally defined by Conforti and Cornuéjols (1984) initially for set functions.…”

“…Kleinberg and Raghu (2015) used the test scores for the problem of constructing a team whose best r players have high performance (we will discuss this further in Section 2.2). Sekar et al (2020) showed that the greedy algorithm based on the replication test score achieves a constant approximation for the cardinality constrained submodular maximization if the objective satisfies the extended diminishing returns property. They also showed that using the mean or quantile value as a score metric does not guarantee a constant approximation for a function with the property (the CES function, defined in Section 2.2).…”

“…where X S is the vector with elements X i , i ∈ S. By the linearity of expectation, if g has the diminishing returns property, u in (1) is submodular. This model of stochastic submodular utility functions was studied in Asadpour and Nazerzadeh (2016) and Sekar et al (2020). Again, stochastic utility functions as in (1) can formulate situations where the performance of individual items can be tested and observed.…”

“…The important idea is that each individual item is evaluated based on a score metric which depends only on the distribution of a single item. Sekar et al (2020) applied the concept of using test scores to more general objective functions.…”

“…We consider functions g that satisfy the extended diminishing returns property, defined in Section 2.2, thus covering the class of objective functions considered in Sekar et al (2020), but under more general budget constraints. The extended diminishing returns property holds for different instances of value functions that arise in various applications.…”