Adaptive Regularized Submodular Maximization

TL;DR

This paper addresses the challenge of maximizing the difference between an adaptive submodular revenue function and a modular cost function under uncertainty, proposing policies with provable approximation guarantees.

Contribution

It introduces new adaptive policies with theoretical performance bounds for maximizing revenue minus cost in uncertain, adaptive settings.

Findings

Effective policy $ ilde{ ext{pi}}^l$ achieves near-optimal revenue-cost trade-off with polynomial queries.

Randomized policy $ ilde{ ext{pi}}^r$ guarantees a constant-factor approximation.

Applicable to adaptive submodular functions with negative and positive values.

Abstract

In this paper, we study the problem of maximizing the difference between an adaptive submodular (revenue) function and an non-negative modular (cost) function under the adaptive setting. The input of our problem is a set of $n$ items, where each item has a particular state drawn from some known prior distribution $p$. The revenue function $g$ is defined over items and states, and the cost function $c$ is defined over items, i.e., each item has a fixed cost. The state of each item is unknown initially, one must select an item in order to observe its realized state. A policy $\pi$ specifies which item to pick next based on the observations made so far. Denote by $g_{avg}(\pi)$ the expected revenue of $\pi$ and let $c_{avg}(\pi)$ denote the expected cost of $\pi$. Our objective is to identify the best policy $\pi^o\in \arg\max_{\pi}g_{avg}(\pi)-c_{avg}(\pi)$ under a $k$-cardinality constraint. Since our objective function can take on both negative and positive values, the existing results of submodular maximization may not be applicable. To overcome this challenge, we develop a series of effective solutions with performance grantees. Let $\pi^o$ denote the optimal policy. For the case when $g$ is adaptive monotone and adaptive submodular, we develop an effective policy $\pi^l$ such that $g_{avg}(\pi^l) - c_{avg}(\pi^l) \geq (1-\frac{1}{e}-\epsilon)g_{avg}(\pi^o) - c_{avg}(\pi^o)$, using only $O(n\epsilon^{-2}\log \epsilon^{-1})$ value oracle queries. For the case when $g$ is adaptive submodular, we present a randomized policy $\pi^r$ such that $g_{avg}(\pi^r) - c_{avg}(\pi^r) \geq \frac{1}{e}g_{avg}(\pi^o) - c_{avg}(\pi^o)$.