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Omega-Regular Objectives in Model-Free Reinforcement Learning
Ernst Moritz Hahn, Mateo Perez, Sven Schewe, Fabio Somenzi, Ashutosh Trivedi, Dominik Wojtczak
TL;DR
The paper tackles model-free RL for maximizing satisfaction of ω-regular objectives in MDPs, a problem not adequately handled by prior reward constructions. It compiles specifications into limit-deterministic Büchi automata and reduces satisfaction to reachability, yielding optimal-strategy learning from MDP observations. The construction also provides a practical path toward increasingly better intermediate strategies in the almost-sure case.
Problem
The paper addresses computing a strategy that maximizes satisfaction probability for an ω-regular objective in an MDP with unknown transitions.
Method
The method compiles objectives into limit-deterministic Büchi automata, augments the product MDP with a sink, and reduces almost-sure satisfaction to reachability solvable by model-free RL.
Results
The constructed augmented MDP yields an optimal strategy for the product and the original MDP, with every ζ ∈ ]0, 1[ valid for probability-1 objectives.
Takeaways & Limitations
The approach supports off-the-shelf temporal-difference algorithms for learning ω-regular objectives and can provide increasingly better intermediate strategies in the almost-sure case.
Takeaways & Limitations
Prior Rabin-reward approaches can fail even when a probability-1 strategy exists, including objectives with Rabin index 2 and some objectives expressible by deterministic Büchi automata.
Abstract
from arXiv · showhide
We provide the first solution for model-free reinforcement learning of ω-regular objectives for Markov decision processes (MDPs). We present a constructive reduction from the almost-sure satisfaction of ω-regular objectives to an almost- sure reachability problem and extend this technique to learning how to control an unknown model so that the chance of satisfying the objective is maximized. A key feature of our technique is the compilation of ω-regular properties into limit- deterministic Buechi automata instead of the traditional Rabin automata; this choice sidesteps difficulties that have marred previous proposals. Our approach allows us to apply model-free, off-the-shelf reinforcement learning algorithms to compute optimal strategies from the observations of the MDP. We present an experimental evaluation of our technique on benchmark learning problems.
1 Introduction
The paper addresses how to encode ω-regular objectives as scalar rewards so model-free RL can maximize their satisfaction probability in MDPs. It replaces problematic Rabin-based approaches with limit-deterministic Büchi automata and a reachability-based construction.
- The paper defines ω-regular rewards as scalar transition rewards enabling off-the-shelf model-free RL to maximize satisfaction probability for ω-regular objectives.
- ω-regular objectives capture long-term safety and progress behavior, where infinitely recurring accepting transitions matter more than reward frequency.
- Previous model-free approaches based on deterministic Rabin automata may fail to find optimal strategies and underestimate objective-satisfaction probabilities.
- The proposed method uses limit-deterministic Büchi automata to reduce almost-sure ω-regular satisfaction constructively to almost-sure reachability.
- The paper focuses on model-free learning for infinitary behaviors in finite MDPs and evaluates the construction experimentally.
2 Preliminaries
The preliminaries formalize MDPs, strategies, ω-regular automata, satisfaction probabilities, products, end components, and limit-deterministic Büchi automata. These definitions establish the product-based representation used for controlling ω-regular objectives.
- Markov Decision Processes: An MDP is a finite state-action system with probabilistic transitions, atomic propositions, and state labeling.
- Markov Decision Processes: Strategies map finite runs to action distributions, with pure, mixed, stationary, and positional strategies distinguished by their action-selection behavior.
- ω-Regular Objectives: An ω-regular objective is represented by an automaton whose accepting runs define the language, and the paper optimizes the probability that an MDP run satisfies it.
- Automata: Büchi acceptance requires some designated transition to occur infinitely often, whereas Rabin acceptance combines infinitely-often and finitely-often conditions across pairs.
- Limit-Deterministic Büchi Automata: Every ω-regular language has deterministic Rabin and nondeterministic Büchi representations, while limit-deterministic Büchi automata retain full expressive power with restricted nondeterminism.
- Products: The product of an MDP and deterministic automaton combines MDP states with automaton states and transfers the automaton acceptance condition to product transitions.
- Products and End Components: End components can be visited permanently, and accepting end components are those whose eventual runs satisfy the product acceptance condition.
3 Problem Statement and Motivation
The paper formulates model-free RL for maximizing satisfaction of ω-regular objectives in unknown MDPs and examines why Rabin-based reward reductions can fail. It motivates limit-deterministic Büchi automata as a route to reachability-based RL.
- Problem: The task is to compute a strategy maximizing the probability that an unknown MDP satisfies an ω-regular objective.
- Problem: Model-free RL requires rewards derived from MDP observations and an automaton monitoring objective satisfaction.
- Limitations of Rabin reductions: Earlier approaches use Rabin automata and multiple Rabin reward functions, but the paper identifies cases where their optimal strategies fail to satisfy the objective with probability 1.
- Counterexamples: In the two-pair example, a strategy satisfies the objective with probability 1, whereas optimal average strategies for every Rabin reward do not.
- Counterexamples: Even a deterministic Büchi objective can separate optimal expected-average strategies from strategies that satisfy the ω-regular objective.
- Proposed direction: Limit-deterministic Büchi automata avoid preselecting a Rabin pair and enable reducing almost-sure ω-regular satisfaction to reachability solvable by off-the-shelf RL.
4 Learning from Omega-Regular Rewards
The paper augments the product of an MDP and a limit-deterministic Büchi automaton with a probabilistic reachability target, then proves that optimizing this reachability objective recovers optimal ω-regular strategies for sufficiently large ζ. In the almost-sure case, every ζ in (0,1) suffices.
- Correctness: The Büchi-product probability provides a lower bound on the original ω-regular satisfaction probability, and an optimal product strategy attains the original optimum.This establishes the bridge between optimization on the automaton product and the original MDP objective.
- Construction: Mζ adds a target state t and diverts each accepting transition to t with probability 1−ζ while scaling other destinations by ζ.The target has a self-loop, and removing t maps strategies on Mζ back to strategies on the product MDP.
- Construction: Accepting and rejecting BSCCs in the product determine whether the reachability probability of t is 1 or 0 under a positional strategy.Rejecting BSCCs cannot reach t, while an accepting BSCC makes t the only reachable BSCC in the augmented MDP.
- Correctness: For sufficiently large ζ, every strategy maximizing reachability in Mζ is optimal on the product and induces an optimal strategy for the original MDP.The threshold is chosen using the finite gap between the best product strategy and the best non-optimal positional strategy, together with a bound on transient accepting transitions.
- Almost-sure case: Every ζ in (0,1) works when the objective can be satisfied with probability 1.This follows from the absence of reachable rejecting BSCCs under an almost-surely satisfying strategy.
5 Experimental Results
The implementation evaluates the reduction with model-free reinforcement learning on benchmark MDPs and confirms learned satisfaction probabilities using a model checker. In the deferred model, increasing ζ suppresses misleading transient accepting transitions and selects the truly satisfying strategy.
- Implementation: MUNGOJERRIE builds Mζ from PRISM MDPs and HOA automata, exposes the construction to reinforcement-learning algorithms, and supports probabilistic model checking.The algorithm computes the maximum satisfaction probability after each state-action pair.
- Benchmarks: The benchmark evaluation compares RL-computed maximum satisfaction probabilities with model-checker results across models including twoPairs, riskReward, grid5x5, trafficNtk, windy, and frozen examples.The model checker has full access to the MDP and is used to confirm the RL results.
- Model deferred: Figure 3 studies two strategies in Model deferred: action a reaches a nonaccepting end component after transient accepting transitions, whereas action b reaches an accepting end component.Only the initial action differs, so the two pure strategies are directly comparable.
- Model deferred: 20,000 episodes of length 80 and five experiments per ζ value are used to estimate strategy a’s satisfaction probability with Q-learning.The plotted curve pa is the average across the five experiments.
- Model deferred: At ζ=0.9999, strategy a’s probability reaches its true value of 0 while strategy b remains at 1, so the selected strategy yields pϕ=1.For ζ near 0, Q-learning treats both strategies as approximately equal and the mixed strategy gives pϕ=0.5; increasing ζ separates them.
- Interpretation: The experiment illustrates that high ζ may be needed numerically to distinguish transient from recurrent accepting transitions, while intermediate strategies can improve toward the target strategy.The reported change does not contradict the theorem because pb=1 remains greater than pa for every ζ.