Approximate State Abstraction

Summary: A relaxed theoretical framework in reinforcement learning that allows grouped states to have near-identical rather than exactly identical action values. This enables practical state abstraction with bounded performance guarantees.

Overview

Approximate state abstraction extends the classical State Abstraction theory by allowing some tolerance in the grouping of states. Instead of requiring states in the same abstract group to have identical optimal action values, approximate state abstraction permits small differences while maintaining theoretical performance bounds.

The key insight is captured by the abstraction error metric: ε(g_c) = max_{s,s': g(s)=g(s')} max_a |Q*(s,a) - Q*(s',a)|, where g(s) represents the abstract state mapping. This measures the maximum difference in optimal Q-values between any two states that are grouped together by the abstraction function g.

The relaxation from exact to approximate abstraction is crucial for practical applications, as real-world state spaces rarely admit perfect abstractions that preserve all decision-relevant information while significantly reducing complexity.

Key Details

  • Performance bound: The approximation error translates directly to policy performance loss via V^π*(s) - V^π_c*(s) ≤ 2ε(g_c)/(1-γ)², where γ is the discount factor
  • Abstraction error: Quantifies the maximum Q-value difference between states assigned to the same abstract group
  • Decision-relevance principle: States should only be grouped if they have similar optimal actions, making the abstraction decision-preserving
  • The framework enables automatic concept selection by formulating abstraction quality as an optimization problem
  • Mixed-integer linear programming can solve the concept selection problem with O(n_d² + K) variables where n_d is the number of distinct abstract states
  • Probabilistic variants handle uncertainty in state grouping through soft constraints

Relationships

  • State Abstraction — Approximate version relaxes the exact equality requirement of classical state abstraction
  • Decision-Relevant Concepts — Provides the theoretical foundation for selecting concepts that minimize abstraction error
  • Concept-Based Models — Enables interpretable RL by ensuring concept representations preserve decision structure
  • Abstraction Error — Core metric quantifying the quality of the approximate abstraction
  • Q-Learning — Performance bounds connect abstraction quality to optimal Q-function preservation
  • Markov Decision Processes — Provides the underlying framework where approximate abstractions operate

Sources