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Reward Bound for Behavioral Guarantee of Model-based Planning Agents

Zhiyu An, Xianzhong Ding, Wan Du

TL;DR

This work addresses guaranteeing that a model-based planning agent reaches a designated goal within a finite horizon in a deterministic, reward-preserving MDP. It derives two core conditions: (i) the goal must be reachable within $J$ steps, i.e., $s_g\in R(s_0,J)$, and (ii) there must exist a goal-containing trajectory with discounted reward exceeding all alternatives. Under these conditions, and with full forward-reachable-set exploration, the guarantee is established, and the framework is extended to handle multiple goals with a strict preference order. The results offer a principled, reward-based criterion to enforce behavioral guarantees and goal prioritization in planning agents.

Abstract

Recent years have seen an emerging interest in the trustworthiness of machine learning-based agents in the wild, especially in robotics, to provide safety assurance for the industry. Obtaining behavioral guarantees for these agents remains an important problem. In this work, we focus on guaranteeing a model-based planning agent reaches a goal state within a specific future time step. We show that there exists a lower bound for the reward at the goal state, such that if the said reward is below that bound, it is impossible to obtain such a guarantee. By extension, we show how to enforce preferences over multiple goals.

Reward Bound for Behavioral Guarantee of Model-based Planning Agents

TL;DR

This work addresses guaranteeing that a model-based planning agent reaches a designated goal within a finite horizon in a deterministic, reward-preserving MDP. It derives two core conditions: (i) the goal must be reachable within steps, i.e., , and (ii) there must exist a goal-containing trajectory with discounted reward exceeding all alternatives. Under these conditions, and with full forward-reachable-set exploration, the guarantee is established, and the framework is extended to handle multiple goals with a strict preference order. The results offer a principled, reward-based criterion to enforce behavioral guarantees and goal prioritization in planning agents.

Abstract

Recent years have seen an emerging interest in the trustworthiness of machine learning-based agents in the wild, especially in robotics, to provide safety assurance for the industry. Obtaining behavioral guarantees for these agents remains an important problem. In this work, we focus on guaranteeing a model-based planning agent reaches a goal state within a specific future time step. We show that there exists a lower bound for the reward at the goal state, such that if the said reward is below that bound, it is impossible to obtain such a guarantee. By extension, we show how to enforce preferences over multiple goals.
Paper Structure (9 sections, 2 theorems, 9 equations)

This paper contains 9 sections, 2 theorems, 9 equations.

Table of Contents

  1. Introduction & Preliminaries
  2. Main Result
  3. Single Goal State
  4. Multiple Goal States with Preference Order
  5. Conclusion
  6. Appendix
  7. Proof of Lemma \ref{['lemma 1']}
  8. Proof of Section \ref{['Sec: single goal state']}
  9. Proof of Lemma \ref{['lemma 2']}

Key Result

Lemma 1

Given that the underlying MDP is deterministic and the model is accurate, if Eq. eq: condtion 2 is false, then the probability of the agent reaching $s_g$ within $J$ steps is strictly less than $1$. The proof of Lemma lemma 1 is postponed to Appendix app, Proof of lemma 1. If the agent is determinis

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2