LTL目的のためのベルマン方程式の解の一意性について
On the Uniqueness of Solution for the Bellman Equation of LTL Objectives
LTL目的の代理報酬で用いられる2つの割引率を持つベルマン方程式が一意解を持つ条件を明らかにし、棄却BSCC内の状態の解を0とする条件が十分であることを証明した。
著者: Zetong Xuan, Alper Kamil Bozkurt, Miroslav Pajic, Yu Wang
分類: cs.AI, cs.RO
原文アブストラクト
Surrogate rewards for linear temporal logic (LTL) objectives are commonly utilized in planning problems for LTL objectives. In a widely-adopted surrogate reward approach, two discount factors are used to ensure that the expected return approximates the satisfaction probability of the LTL objective. The expected return then can be estimated by methods using the Bellman updates such as reinforcement learning. However, the uniqueness of the solution to the Bellman equation with two discount factors has not been explicitly discussed. We demonstrate with an example that when one of the discount factors is set to one, as allowed in many previous works, the Bellman equation may have multiple solutions, leading to inaccurate evaluation of the expected return. We then propose a condition for the Bellman equation to have the expected return as the unique solution, requiring the solutions for states inside a rejecting bottom strongly connected component (BSCC) to be 0. We prove this condition is sufficient by showing that the solutions for the states with discounting can be separated from those for the states without discounting under this condition
