Probably Approximately Correct MDP Learning and Control With Temporal Logic Constraints (1404.7073v2)

Abstract: We consider synthesis of control policies that maximize the probability of satisfying given temporal logic specifications in unknown, stochastic environments. We model the interaction between the system and its environment as a Markov decision process (MDP) with initially unknown transition probabilities. The solution we develop builds on the so-called model-based probably approximately correct Markov decision process (PAC-MDP) methodology. The algorithm attains an $\varepsilon$-approximately optimal policy with probability $1-\delta$ using samples (i.e. observations), time and space that grow polynomially with the size of the MDP, the size of the automaton expressing the temporal logic specification, $\frac{1}{\varepsilon}$, $\frac{1}{\delta}$ and a finite time horizon. In this approach, the system maintains a model of the initially unknown MDP, and constructs a product MDP based on its learned model and the specification automaton that expresses the temporal logic constraints. During execution, the policy is iteratively updated using observation of the transitions taken by the system. The iteration terminates in finitely many steps. With high probability, the resulting policy is such that, for any state, the difference between the probability of satisfying the specification under this policy and the optimal one is within a predefined bound.

• The paper extends PAC-MDP learning to synthesize control policies for unknown MDPs that satisfy temporal logic constraints.
• It proposes an iterative algorithm that achieves an approximately optimal policy with high probability using finite samples, requiring polynomial resources.
• The research has practical implications for robotics and autonomous systems that need to operate reliably in uncertain environments.

Analysis of Probably Approximately Correct MDP Learning and Control With Temporal Logic Constraints

The paper by Jie Fu and Ufuk Topcu tackles the challenge of synthesizing control policies that adhere to temporal logic specifications in unknown, stochastic environments, modeled as Markov Decision Processes (MDPs) with initially unknown transition probabilities. Building upon the PAC-MDP methodology, the authors present an algorithm that achieves an $\varepsilon$-approximately optimal policy with high probability by leveraging finite samples. This iterative approach is notable for its polynomial growth in computational resources required relative to the size of the MDP and the temporal logic automaton.

Key Contributions

The primary contribution of the paper is the extension of PAC-MDP algorithms to incorporate temporal logic constraints in the synthesis process of control policies. The model-based learning technique balances exploration and exploitation by continuously updating the policy based on observations. When all states become known, the learned MDP approximates the true MDP with high fidelity, ensuring the policy's near-optimality by finite termination of the policy iterations. This efficiency in convergence to an optimal policy is underscored by the fact that only polynomial time, space, and sample complexities are required.

Numerical Results and Claims

Within the context of unknown MDPs, the paper guarantees that the resulting policy's probability of satisfying temporal logic specifications does not deviate from the optimal policy beyond a predefined bound. This strong numerical claim emphasizes the effectiveness of the PAC-MDP method when facing incomplete knowledge of the environment. Moreover, the algorithm adapts to novel observations during live execution, facilitating reliable control synthesis under uncertainty.

Theoretical and Practical Implications

The theoretical implications of this research extend to settings where MDPs represent systems with incomplete knowledge. The proposed strategy allows the agent to efficiently explore its environment while advancing towards optimal policy solutions. Practically, this has significant implications for robotics and autonomous systems where an agent must adapt to uncertain terrains or dynamics.

Future Directions

Future developments may explore extending the PAC-MDP approach to two-player stochastic games, where policy synthesis requires integration with different strategy classes. Additionally, adopting model-free methods could reduce space complexity, offering alternatives for systems with dynamic objectives. Other objectives, such as cost minimization, within temporal logic constraints are identified as promising research trajectories, indicating potential integration with advanced control optimization techniques.

In conclusion, the paper provides a robust framework for deploying PAC-MDP methodology in control synthesis tasks involving temporal logic constraints. By achieving probabilistic guarantees on policy performance with scalable computational requirements, this paper makes a substantial contribution to the intersection of reinforcement learning and formal methods in systems engineering.

This analysis encapsulates the research within the field of reinforcement learning for unknown systems, highlighting the methodological advancements and anticipating future exploration in similar domains.