Problem
Research questions and friction points this paper is trying to address.
Submodular Markov Decision Processes
Submodular Orienteering
Monotone Submodular Function
Strong and Compact Policies for Submodular Markov Decision Processes via LP-Based Submodular Orienteering
🤖 AI Summary
本文提出了一种基于线性规划的新算法，用于解决具有次模奖励函数的马尔可夫决策过程问题，提高了近似解的质量。
📝 Abstract
Finding policies for Markov Decision Processes (MDPs) is a central problem in areas such as Reinforcement Learning and Operations Research. Here, we have to repeatedly choose an action that should be performed by an agent. Depending on the action and the current state of the agent, the agent collects a reward and randomly transitions into a new state. The goal is to maximize the reward in expectation over a finite time horizon of length $H$. We consider a recently introduced variant that generalizes the traditionally additive reward function in the model to a monotone submodular one, which allows for capturing a range of interesting applications.
Without the stochastic component, this problem is equivalent to the Submodular Orienteering problem, where the goal is to find an $s$-$t$ walk in a directed graph maximizing a monotone submodular function under a length constraint. We present a novel LP-based algorithm for Submodular Orienteering using ideas from the Sherali-Adams hierarchy and Round-or-Cut. Our guarantees are comparable to the known quasi-polynomial time logarithmic approximation for Submodular Orienteering, but also extend to the setting of Submodular Markov Decision Processes. In the polynomial time regime, we present an $O(n^{\varepsilon})$-approximation (and $O(H^{\varepsilon})$ for Submodular MDPs) for every $\varepsilon >0$, where $n$ is the number of vertices, which was unknown even for Submodular Orienteering. Prior to our work, the best known approximation guarantee for Submodular MDPs had an approximation ratio linear in $H$.
Beyond these algorithmic results, our methods reveal a trade-off between the approximation guarantee and the number of previously visited vertices on which an agent conditions its decision.
Innovation
Methods, ideas, or system contributions that make the work stand out.
LP-based Algorithm
Submodular Markov Decision Processes
Sherali-Adams Hierarchy
Round-or-Cut
Polynomial Time Approximation
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