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Near Optimal Behavior via Approximate State Abstraction
David Abel, D. Ellis Hershkowitz, Michael L. Littman
TL;DR
Planning and RL face state-space growth that makes exact abstraction difficult or ineffective when states are not identical. This paper develops four approximate abstraction types, proves bounded behavioral error, and empirically shows reduced state-space size with minor or bounded loss in behavior quality.
Problem
State-space growth makes planning and RL difficult, while exact abstractions can require intractable knowledge and fail when no situations are identical.
Method
The paper defines four approximate abstraction functions based on optimal Q-values, model similarity, and Boltzmann or multinomial distributions over optimal Q-values.
Results
Approximate abstractions induce abstract MDPs whose optimal policies have bounded error in the ground MDP and reduce state-space size with minor behavioral loss across environments.
Takeaways & Limitations
Relaxing state aggregation from equality to similarity supports greater compression and adjustable abstraction aggressiveness while retaining bounded behavioral quality.
Takeaways & Limitations
The paper gives sufficient conditions for bounded error but leaves open which conditions are strictly necessary.
Abstract
from arXiv · showhide
The combinatorial explosion that plagues planning and reinforcement learning (RL) algorithms can be moderated using state abstraction. Prohibitively large task representations can be condensed such that essential information is preserved, and consequently, solutions are tractably computable. However, exact abstractions, which treat only fully-identical situations as equivalent, fail to present opportunities for abstraction in environments where no two situations are exactly alike. In this work, we investigate approximate state abstractions, which treat nearly-identical situations as equivalent. We present theoretical guarantees of the quality of behaviors derived from four types of approximate abstractions. Additionally, we empirically demonstrate that approximate abstractions lead to reduction in task complexity and bounded loss of optimality of behavior in a variety of environments.
1 Introduction
State abstraction reduces the effective complexity of planning and reinforcement learning, but exact equality-based abstractions can fail when situations are merely similar. The paper studies approximate abstractions that relax equality to similarity while bounding behavioral error and supporting adjustable compression.
- Motivation: Exact abstractions preserve optimality but may require computationally intractable knowledge and offer little compression when no states are identical.They aggregate only fully identical situations or states with equal values of specified quantities.
- Motivation: Planning and RL remain difficult because state-space size grows super-polynomially with the number of domain variables.Polynomial complexity in state-space size can therefore remain ineffective for sufficiently complex tasks.
- Core thesis: Approximate abstraction relaxes state aggregation from equality to similarity, producing polynomially bounded behavioral error.The relaxed criterion treats sufficiently similar states as identical.
- Benefits: Approximate abstractions can increase compression and tune abstraction aggressiveness by changing what counts as sufficiently similar.The paper also empirically examines the relationship between compression and incurred error across MDPs.
- Core approach: The paper analyzes four approximate abstraction functions based on optimal Q-values, rewards and transitions, Boltzmann distributions, and multinomial distributions.These are denoted eφQ∗,ε, eφmodel,ε, eφbolt,ε, and eφmult,ε, respectively.
2 MDPs and Sequential Decision Making
An MDP represents sequential decision making with states, actions, transition probabilities, rewards, and discounting. Its solution is a policy, with optimal behavior defined by maximizing expected discounted reward.
- MDP definition: An MDP is represented by the five-tuple ⟨S, A, T, R, γ⟩.S is the finite state space, A the available actions, T the transition function, R the reward function, and γ the discount factor.
- MDP definition: The discount factor γ ∈[0, 1] determines how strongly the agent prefers future rewards over immediate rewards.Reward functions are normalized to the range [0, 1].
- Optimal behavior: A policy π maps states to actions, and the objective is to maximize expected discounted reward from every state.The maximizing policy is denoted π∗.
- Value functions: Vπ(s) denotes the expected discounted reward from state s under policy π, while Qπ(s, a) evaluates taking action a before following π.Qπ is defined using the Bellman equation.
- Value functions: With maximum reward RMax = 1, the maximum Q-value is QMax = RMax/(1−γ).The notation distinguishes optimal value and Q-functions as V∗ and Q∗.
3 Related Work
Related work develops state aggregation, similarity metrics, and planning or RL algorithms with value guarantees. This paper differs by bounding the suboptimality of applying an abstract MDP’s optimal policy in the original MDP across four approximate abstraction types.
- MDP abstraction: Prior MDP work used bisimulation to cluster states with ε-close transition models and reward functions, with Interval Value Iteration computing bounds for abstract MDPs.The resulting family is called Bounded MDPs.
- MDP abstraction: Bisimulation-metric research bounded value differences between ground and abstract states, whereas this paper bounds suboptimality when applying the abstract optimal policy in the ground MDP.The paper covers four abstraction types, including one closely paralleling bisimulation.
- RL aggregation: Hutter’s results address similar Q-value aggregation and include bounds on ground–abstract value differences and applying an optimal abstract policy in the ground environment.The paper notes that its proof strategies differ while the relevant result is the same.
- Planning: Planning research applied abstraction to Monte Carlo Tree Search, expectimax search, and rapidly computed abstractions with bounded value.These approaches also analyze the value of abstract actions or plans in ground settings.
- Reinforcement learning: Other RL work studied Bayesian aggregation, Thompson Clustering, and choosing between candidate abstractions by trading approximation error against estimation error.The latter yields a loss bound for the selected policy.
- Exact abstraction: Li et al. defined five exact state-aggregation types, and this paper generalizes φQ∗ and φmodel to approximate criteria.The generalizations coincide with the exact versions when ε = 0, when the bounds indicate no value loss.
4 Abstraction Notation
The paper formalizes abstraction as a mapping from a ground MDP to an abstract MDP whose states aggregate ground states. Abstract rewards and transitions are weighted combinations of the corresponding ground quantities.
- MDP abstraction: An abstraction maps a ground MDP MG to an abstract MDP MA through a state aggregation scheme.The two MDPs share the action set and discount factor while having distinct state spaces and dynamics.
- State aggregation: The aggregation function φ maps each ground state to an abstract state.Applying φ constructs the abstract state space SA.
- State aggregation: The function G retrieves the ground states associated with either a ground state or an abstract state under φ.It identifies the states aggregated together.
- Abstract dynamics: Each abstract state’s reward and transition dynamics are weighted combinations of those for its constituent ground states.The abstract reward and transition functions are defined over abstract states, actions, and next abstract states.
- Abstract dynamics: The weighting scheme assigns each ground state a weight and must induce a probability distribution within every abstract state.This restriction determines how ground-state rewards and transitions are aggregated.
5 Approximate State Abstraction
The paper formalizes approximate state abstraction for MDPs and shows that four aggregation families produce abstract policies with polynomially bounded suboptimality in the ground MDP.
- Framework: The analysis evaluates abstract policies by defining their execution in the ground MDP and comparing them with ground optimal behavior.The derived ground policy follows the optimal abstract policy after mapping ground states to abstract states.
- Framework: Approximate abstractions aggregate states whose relevant quantities are similar within ε, rather than requiring exact equality.For function-based aggregation, all grouped states have values within ε for every action.
- Main result: Four abstraction families—approximate optimal-Q, model, Boltzmann-Q, and multinomial-Q similarity—yield abstract optimal policies with suboptimality polynomially bounded in ε.The bounds depend on an abstraction-family-specific η_f and, for distribution-based abstractions, bounded differences in normalizing terms.
- Scope of guarantees: When ε=0, the bounds are exactly zero; increasing ε interpolates between exact abstraction and the maximum-value bound, with different degrees of bounded loss.The stated bound becomes uninformative when 2εη_f ≥ R_max, while rewards are assumed to lie in [0,1].
- Optimal-Q abstraction: The approximate optimal-Q abstraction groups states whose optimal Q-values differ by at most ε, and its proof bounds abstract and ground Q-values before bounding policy suboptimality.The proof proceeds through Claims 1 and 2 to establish Lemma 1 for the resulting ground policy.
- Model abstraction: Model similarity aggregates states with rewards and transitions within ε, and the resulting guarantee follows by treating this abstraction as a special approximate-Q abstraction.The paper explicitly derives Lemma 2 by applying Lemma 1 after bounding the induced Q-value difference.
- Distributional abstractions: Boltzmann-Q abstraction can aggregate states with similar Q-value ratios despite different magnitudes, while multinomial-Q abstraction uses a simpler distribution over optimal Q-values.The Boltzmann result is reduced to the approximate-Q case with a bound parameter tied to ε.
6 Example Domains
The paper evaluates approximate abstraction across five diverse MDP domains, using visual and quantitative examples to examine compression and behavior preservation. Upworld provides an exact-abstraction case with arbitrary state reduction at no loss of optimal value.
- Domains and setup: Experiments cover NChain, Upworld, Taxi, Minefield, and Random, selected to span simple, hierarchical, stochastic, goal-based, and near-optimal-policy-rich domains.The implementations visualize and evaluate both ground and abstract MDPs, using γ = 0.95 throughout.
- Upworld: Upworld is an N × M grid where moving up is always optimal because only top-row transitions provide positive reward.The experiments use N = 10 and M = 4.
- Upworld: Exact Q*-based abstraction reduces Upworld from N × M ground states to N abstract states without changing the optimal policy value.All states in the same row have identical Q values, enabling aggregation across each row.
- Taxi: Taxi is a goal-based Grid World task with movement, passenger pickup, and passenger drop-off actions, evaluated on a 626-instance problem.The goal is achieved after all passengers reach their destinations.
- Minefield and Random: Minefield combines Grid World dynamics with slip, mine states, and reward variation, while Random uses 100 states and three actions with stochastic transitions to two randomly selected states.Minefield uses N = 10, M = 4, ε = 0.5, κ = 5, and x = 0.01.
7 Empirical Results
Across the tested domains, increasing approximation generally reduces abstract state-space size while preserving behavior within bounded error. The value–compression trade-off depends on domain structure: Taxi shows a threshold, whereas Random degrades smoothly because many near-optimal policies remain available.
- Overall pattern: Approximate abstraction reduces the number of states while retaining bounded behavioral error across the empirical domains.At ε = 0, Random has no compression because its states have no identical optimal Q-values.
- NChain and Minefield: In NChain, increasing ε reduces the planned-over state count while fully maintaining optimal behavior; Minefield retains behavior very nearly.
- Taxi: Taxi reduces state count while fully maintaining value for ε between .02 and .025, but further compression after .025 costs value.
- Taxi: Taxi’s state-space size and behavior remain relatively stable up to ε = .02, then both decline sharply because aggregation can remove information needed to achieve the goal and obtain reward.
- Random: Random shows a smooth reduction in abstract states as ε increases, with a corresponding decrease in derived-policy value because many near-optimal policies remain available.
8 Conclusion
Approximate abstraction reduces state-space size with bounded behavioral loss and offers tunable compression, but several theoretical and practical questions remain open.
- Approximate abstractions can reduce state-space size with minor loss in behavior quality.
- Their relaxed criteria enable greater compression than exact abstraction while retaining bounded error.
- Abstraction aggressiveness can be tuned while incurring bounded error.
- Future work includes learning approximate abstraction functions online in planning or reinforcement learning settings.
- The paper gives sufficient conditions for bounded error, but whether those conditions are necessary remains unresolved.