Technical Summary: Reinforcement Learning for Reachability: Guaranteeing Asymptotic Optimality

1. Problem Statement

The paper addresses the challenge of learning optimal policies in unknown Markov Decision Processes (MDPs) under reachability specifications. While Reinforcement Learning (RL) for reward-based objectives has strong theoretical guarantees (asymptotic convergence and PAC guarantees), learning under high-level temporal logic specifications (such as Linear Temporal Logic, LTL) remains theoretically underexplored.

Specifically, the authors highlight two gaps in existing literature:

1. PAC Limitations: General LTL specifications are not learnable with PAC guarantees without knowing internal MDP parameters (e.g., minimum non-zero transition probability $p_{min}$, mixing time). These parameters are typically unknown in standard RL settings.
2. Asymptotic Limitations: Prior work achieving asymptotic convergence for LTL (e.g., Le et al., 2024) relies on converting specifications to limit-average reward objectives. This approach depends on external parameters (discount factors) disconnected from the MDP's intrinsic structure, offering no insight into when optimality emerges or the dynamics of convergence.

The core problem is to design a learning algorithm for reachability in unknown MDPs that achieves asymptotic convergence (converging to the optimal policy with probability 1) while providing observable guarantees on when the optimal policy emerges, without relying on reward conversion or external parameters.

2. Methodology

The authors propose a staged direct learning algorithm that operates without converting the reachability objective into a reward function. Instead, it relies on iteratively refining estimates of internal MDP parameters.

Core Algorithmic Framework

The algorithm proceeds in discrete stages $k = 1, 2, 3, \dots$. In each stage, three key parameters are set to decrease exponentially:

• Guess for minimum transition probability: $p_k = 1/2^k$
• Confidence error: $\delta_k = 1/2^k$
• Approximation factor: $\epsilon_k = 1/2^k$

The procedure for each stage involves:

1. Simulation and Model Building: The algorithm simulates the MDP $N_k$ times to build a partial model, maintaining counters for state-action transitions.
2. Transition Estimation: It calculates lower estimates of transition probabilities using Hoeffding bounds.
3. Handling End-Components (ECs): To ensure convergence of value iteration in the presence of ECs (cycles where the agent might get stuck), the algorithm detects ECs and collapses them into super-states. This creates a "collapsed MDP" where ECs are treated as single states with specific boundary conditions (value 0 or 1 depending on whether the EC contains a goal state).
4. Bounded Value Iteration (BVI): The algorithm runs BVI on the collapsed partial model to compute lower ($L$) and upper ($U$) bounds on the reachability probability.
5. Policy Extraction: A memoryless deterministic policy is extracted based on the upper bounds in the partial model.

Theoretical Mechanism for Convergence

The convergence relies on three theoretical pillars:

1. PAC Guarantee (Theorem 3.1): Once the stage index $k$ is large enough such that the guessed $p_k \le p_{min}$ (the true minimum non-zero transition probability), the algorithm guarantees that the extracted policy is $\epsilon_k$-optimal with probability $1 - \delta_k$.
2. Exact Optimality (Theorem 3.2): The authors define $\epsilon_{diff}$ as the minimum value separation between the optimal policy and the best sub-optimal memoryless deterministic policy. Since there are finitely many such policies, $\epsilon_{diff} > 0$. As $k$ increases, $\epsilon_k$ shrinks. Eventually, $\epsilon_k < \epsilon_{diff}$. At this point (stage $K_{opt}$), any $\epsilon_k$-optimal policy must be exactly optimal.
3. Almost-Sure Convergence (Theorem 3.3): By choosing $\delta_k = 1/2^k$, the sum of failure probabilities across all stages is finite. By the Borel–Cantelli Lemma, with probability 1, only finitely many stages will fail. Consequently, beyond a finite stage $K$, the algorithm produces only optimal policies.

3. Key Contributions

• Direct Learning Approach: The paper presents the first direct learning approach for reachability specifications that avoids converting LTL to reward-based objectives, thereby preserving the intrinsic structure of the MDP.
• Stronger than Asymptotic Convergence: Unlike standard asymptotic guarantees where policies merely approach optimality, this work proves that with probability 1, there exists a finite stage $K_{opt}$ after which only optimal policies are observed.
• Characterization of Convergence Dynamics: The stage $K_{opt}$ is explicitly characterized by intrinsic MDP quantities, specifically the value separation $\epsilon_{diff}$. This provides a theoretical bound on when optimality emerges.
• Bounds on Value Separation: The paper derives a lower bound for $\epsilon_{diff}$ based on the transition complexity of the MDP (Theorem 4.1), showing that the gap between optimal and sub-optimal policies depends on the denominators of the transition probabilities.
• Empirical Validation: The algorithm is implemented and tested on 9 standard MDP benchmarks. Results show that optimal policies emerge rapidly (median stage $k=2$), often well before the value bounds ($L$ and $U$) fully converge.

4. Results

• Theoretical: The algorithm is proven to converge asymptotically with probability 1. Furthermore, it guarantees that beyond a specific finite stage, the output is strictly the optimal policy, not just an approximation.
• Empirical:
  • Experiments on benchmarks (e.g., Dining Philosophers, CSMA, Zeroconf) demonstrate that the "optimal-policy-only" regime emerges quickly.
  • A notable observation is that policy accuracy stabilizes significantly earlier than value bounds. For instance, in the Dining Philosophers benchmark, policy accuracy converged at stage $k=2$, while value bounds only began converging around $k=16$. This suggests the algorithm finds the correct policy even when the value estimates are not yet tight.
  • The algorithm successfully handles benchmarks with varying reachability probabilities (from 0.00 to 1.00).

5. Significance and Claims

The authors claim that this work provides a concrete foundation for practical RL systems learning from rich temporal specifications.

• Fundamental Primitive: Reachability is identified as the core computational primitive for all $\omega$-regular and LTL specifications. Therefore, a principled algorithm for reachability convergence immediately induces one for the full class of LTL objectives.
• Observable Guarantees: By characterizing convergence in terms of intrinsic MDP parameters rather than external reward conversion, the approach offers deeper insights into algorithm design and enables implementations with observable convergence guarantees.
• Modesty: The authors acknowledge that their current approach is model-based. They explicitly state that extending this framework to model-free methods (learning policies directly without explicit model construction) represents a critical path for future practical deployment and scalability. They do not claim immediate societal impact but position the work as an advancement in trustworthy machine learning theory.