Partial-Monotone Adaptive Submodular Maximization

TL;DR

This paper introduces a unified framework for adaptive submodular maximization that interpolates between monotone and non-monotone cases using an adaptive monotonicity ratio, providing new approximation guarantees.

Contribution

It generalizes existing results by defining the adaptive monotonicity ratio and deriving approximation ratios for partial-monotone adaptive submodular functions under various constraints.

Findings

Random greedy policy achieves an approximation ratio of m(1-1/e)+(1-m)(1/e).

Sampling-based policy achieves an approximation ratio of (m+1)/10 under knapsack constraints.

Results imply near-optimal solutions are possible for functions close to monotone.

Abstract

Many sequential decision making problems, including pool-based active learning and adaptive viral marketing, can be formulated as an adaptive submodular maximization problem. Most of existing studies on adaptive submodular optimization focus on either monotone case or non-monotone case. Specifically, if the utility function is monotone and adaptive submodular, \cite{golovin2011adaptive} developed a greedy policy that achieves a $(1-1/e)$ approximation ratio subject to a cardinality constraint. If the utility function is non-monotone and adaptive submodular, \cite{tang2021beyond} showed that a random greedy policy achieves a $1/e$ approximation ratio subject to a cardinality constraint. In this work, we aim to generalize the above mentioned results by studying the partial-monotone adaptive submodular maximization problem. To this end, we introduce the notation of adaptive monotonicity ratio $m\in[0,1]$ to measure the degree of monotonicity of a function. Our main result is to show that a random greedy policy achieves an approximation ratio of $m(1-1/e)+(1-m)(1/e)$ if the utility function is $m$-adaptive monotone and adaptive submodular. Notably this result recovers the aforementioned $(1-1/e)$ and $1/e$ approximation ratios when $m = 0$ and $m = 1$, respectively. We further extend our results to consider a knapsack constraint. We show that a sampling-based policy achieves an approximation ratio of $(m+1)/10$ if the utility function is $m$-adaptive monotone and adaptive submodular. One important implication of our results is that even for a non-monotone utility function, we still can achieve an approximation ratio close to $(1-1/e)$ if this function is ``close'' to a monotone function. This leads to improved performance bounds for many machine learning applications whose utility functions are almost adaptive monotone.