insight - Algorithms and Data Structures - # Stochastic Shortest Path Problem with Failure Probability

Core Concepts

This paper proposes a framework to design optimal policies for stochastic shortest path problems that consider the probability of task failure, by expanding the search range beyond policies that solely minimize failure probability.

Abstract

The paper addresses limitations of the standard stochastic shortest path (SSP) problem, which cannot handle cases where a catastrophic event may occur during an episode for any policy. To address this, the authors introduce the concept of "dead-ends" to express catastrophic events.

The key contributions are:

- Formulation of a constrained SSP problem that considers the task failure probability as a constraint, in addition to minimizing the expected total cost.
- Approximation of the original problem by treating it as a combination of a Bayesian adaptive Markov decision process (BAMDP) and a two-person zero-sum game.
- Derivation of the optimal policy, which is shown to be a mixed policy that stochastically selects from a set of deterministic semi-Markov policies.
- Demonstration of the effectiveness of the proposed methods through a motion planning problem with obstacle avoidance for a mobile robot.

The authors show that by appropriately setting the parameters (c, ε, γ), the optimal policy for the approximation problem can be made to closely approximate the optimal policy for the original problem.

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by Ritsusamuel ... at **arxiv.org** 09-26-2024

Deeper Inquiries

The proposed framework for the Stochastic Shortest Path Problem (SSP) with failure probability can be extended to handle more complex environments by incorporating additional layers of uncertainty and constraints. For instance, in environments with dynamic obstacles or varying task requirements, the framework can integrate real-time data inputs to adjust the policies dynamically. This can be achieved by employing adaptive Markov Decision Processes (MDPs) that utilize reinforcement learning techniques to continuously update the policy based on observed outcomes and changing conditions.
Moreover, the introduction of multi-objective optimization can allow the framework to balance multiple competing goals, such as minimizing cost while maximizing safety or efficiency. This can be particularly useful in scenarios like autonomous vehicle navigation, where the vehicle must consider not only the shortest path but also the safety of passengers and pedestrians.
Additionally, the framework can be adapted to include hierarchical decision-making structures, where high-level policies dictate overall strategies while low-level policies manage immediate actions. This hierarchical approach can facilitate the management of complex tasks that require coordination among multiple agents or systems, such as in robotic swarms or collaborative multi-robot systems.

The two-person zero-sum game approach, while effective in deriving optimal policies under uncertainty, has several potential limitations. One significant drawback is the assumption of adversarial behavior between the two players, which may not accurately reflect the dynamics of many real-world decision-making scenarios. In many applications, agents may not be strictly competing against each other; instead, they may have cooperative or mixed motives, which a zero-sum framework does not accommodate.
Furthermore, the complexity of solving zero-sum games can lead to computational challenges, especially in high-dimensional state spaces. The need to compute mixed strategies and equilibria can result in increased computational overhead, making the approach less scalable for larger or more complex environments.
Additionally, the reliance on the assumption that one player's gain is exactly equal to the other's loss may oversimplify the interactions in multi-agent systems. This can lead to suboptimal strategies that do not account for collaborative opportunities or synergies that could be exploited for better overall outcomes.

Yes, the insights from this work can be applied to a variety of sequential decision-making problems under uncertainty beyond the stochastic shortest path setting. The framework's emphasis on managing failure probabilities and incorporating constraints can be particularly relevant in fields such as healthcare, finance, and supply chain management.
For instance, in healthcare, the framework can be utilized to optimize treatment plans for patients, where the goal is to minimize the risk of adverse outcomes while considering the costs associated with different treatment paths. Similarly, in finance, the principles can be applied to portfolio management, where investors seek to maximize returns while minimizing the risk of significant losses.
In supply chain management, the framework can help in designing robust logistics strategies that account for uncertainties in demand, supply disruptions, and transportation risks. By adapting the proposed methods to these contexts, decision-makers can develop policies that are not only optimal in terms of cost but also resilient to the uncertainties inherent in their respective environments.
Overall, the versatility of the proposed framework allows it to be tailored to various domains, enhancing its applicability to a wide range of sequential decision-making challenges under uncertainty.

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