Source-linked AI summary
Scalable methods for computing state similarity in deterministic Markov Decision Processes
Pablo Samuel Castro
TL;DR
Bisimulation metrics offer behavioral state similarity with theoretical guarantees, but expensive computation and full state enumeration limit them in large or continuous MDPs. The paper develops an on-policy variant plus sampling and differentiable approximation methods, with results supporting use in large and continuous state spaces while identifying stochastic systems as an open extension.
Problem
Bisimulation metrics are expensive and require full state enumeration, making them impractical for large state spaces and incompatible with continuous state spaces.
Method
The paper introduces an on-policy bisimulation metric, a convergent sampling-based computation method, and a differentiable neural-network loss for metric approximation.
Results
The sampling method converges to the true metric, while the differentiable method learns a good approximant in continuous state spaces.
Takeaways & Limitations
Bisimulation metrics can be used effectively for deep reinforcement learning and for state representation, replay-buffer compression, and exploration.
Takeaways & Limitations
The presented results are for deterministic MDPs, and extending them to stochastic systems remains an important research avenue.
Abstract
from arXiv · showhide
We present new algorithms for computing and approximating bisimulation metrics in Markov Decision Processes (MDPs). Bisimulation metrics are an elegant formalism that capture behavioral equivalence between states and provide strong theoretical guarantees on differences in optimal behaviour. Unfortunately, their computation is expensive and requires a tabular representation of the states, which has thus far rendered them impractical for large problems. In this paper we present a new version of the metric that is tied to a behavior policy in an MDP, along with an analysis of its theoretical properties. We then present two new algorithms for approximating bisimulation metrics in large, deterministic MDPs. The first does so via sampling and is guaranteed to converge to the true metric. The second is a differentiable loss which allows us to learn an approximation even for continuous state MDPs, which prior to this work had not been possible.
Introduction
Bisimulation metrics capture behavioral similarity between MDP states, but their computational and tabular-state requirements limit use in large or continuous problems. This paper addresses those limitations with an on-policy metric and scalable sampling- and neural-network-based methods.
- MDP foundations: Finite MDPs model sequential decision problems with states, actions, transitions, rewards, and discounting.States should encode enough environmental information for consistent behavior.
- Motivation: Duplicating an MDP can double the state count even when corresponding states remain behaviorally indistinguishable.The grid example illustrates why state labels alone are insufficient for planning.
- Bisimulation metrics: Bisimulation metrics assign distances to states and provide theoretical guarantees that nearby states have similar optimal value functions.They support reasoning about similarity, state aggregation, abstraction, policy transfer, representation discovery, and exploration.
- Limitations: Existing bisimulation metrics are expensive, require full state enumeration, and can be overly pessimistic because they measure worst-case differences.These requirements make them impractical for large state spaces and incompatible with continuous state spaces.
- Contributions: The paper introduces an on-policy bisimulation variant that focuses on behavior of interest rather than worst-case scenarios.The paper analyzes the variant's theoretical properties.
- Contributions: The paper presents a sampling-based algorithm with guaranteed convergence and a differentiable neural-network loss for approximating bisimulation metrics in large and continuous-state MDPs.The neural-network approach is presented as the first mechanism for approximating bisimulation metrics with neural networks.
On-policy bisimulation
Standard bisimulation metrics can be pessimistic because they compare states under every action, even when behavior follows a particular policy. On-policy bisimulation instead compares policy-induced dynamics and retains value-function guarantees for that policy.
- Motivation: Standard bisimulation can assign large distances to states with equal optimal values because it includes worst-case actions.In Figure 2, V*(s) = V*(t) = 10K while d∼(s, t) = 10K.
- Motivation: Exact action matching can compare states improperly when identically labeled actions produce different behaviors from different states.Lax bisimulation addresses differing optimal action labels, but remains susceptible to the pessimism of standard metrics.
- On-policy bisimulation: On-policy bisimulation excludes actions that a behavior policy never chooses and removes the requirement to match action labels.It considers dynamics induced by π rather than dynamics induced separately by each action.
- Definition: A π-bisimulation relation compares policy-induced rewards and transitions, defining π-bisimilar states through the largest such relation.The associated metric is a pseudometric whose zero-distance equivalence relation is the largest π-bisimulation relation.
- Guarantee: π-bisimulation metrics bound differences in the value function induced by π.Theorem 3 states that |Vπ(s) − Vπ(t)| ≤ dπ∼.
Bisimulation metrics for deterministic MDPs
For deterministic MDPs, Wasserstein transition comparisons reduce to distances between unique successor states, enabling simpler metric updates. Sampling these updates yields an almost-surely convergent computation, while large and continuous spaces still motivate scalable approximations.
- Deterministic MDPs: A deterministic MDP has a unique successor state N(s, a) for every state-action pair.Its transition distribution assigns probability 1 to that successor.
- Deterministic MDPs: In deterministic MDPs, the Wasserstein distance between same-action transitions equals the metric distance between their unique successor states.Thus W1(d)(P(s, a), P(t, a)) = d(N(s, a), N(t, a)).
- Metric operators: For deterministic policies, the metric operators reduce to reward differences plus discounted distances between policy-selected successor states.The original metric still maximizes this quantity over actions, whereas the on-policy metric evaluates the policy’s selected actions.
- Scope: Even with deterministic transitions, exact metric computation remains intractable for very large or continuous state spaces.The paper therefore extends the approach toward scalable algorithms and learned approximants.
- Sampling algorithm: The sampling algorithm updates only sampled state pairs using the larger of their current estimate and the sampled bisimulation backup.It samples transition pairs from a distribution assigning positive probability to every valid transition pair.
- Sampling algorithm: lim n→∞ dn = d∼ almost surely when updates begin from the everywhere-zero metric.The convergence argument uses repeated sampling, monotonicity, and contraction.
Learning an approximation
The paper learns bisimulation distances with neural networks by training on sampled transition pairs, extending approximation beyond tabular state spaces. The approach supports large and continuous-state MDPs, although the learned function is not guaranteed to be a proper pseudometric.
- Neural approximation: A neural network ψθ receives concatenated state representations and approximates the bisimulation distance between the represented states.The representation φ maps each state to R^k, while ψθ maps two representations to a scalar estimate.
- Training procedure: The learning procedure uses online and target network parameters to define a differentiable target objective from transition samples.Target parameters are updated every C iterations, and analogous mini-batch losses support GPU training.
- Training procedure: The matrix formulation masks action mismatches when approximating d∼ and removes diagonal entries from the learned distance matrix.ψ is applied elementwise, and the Hadamard product combines the relevant batchwise terms.
- Neural approximation: Figure 3 depicts a neural network learning ψ as an approximant to d∼ or dπ∼.The figure concerns the learned function’s role as an approximant to either metric.
- Evaluation setting: Figure 4 represents GridWorld states by (x, y) coordinates and visualizes noisy sampled states with black densities.The representation is used when adding noise to create a continuous variant.
- Limitation: The learned approximant is not guaranteed to satisfy all properties of a proper pseudometric.The paper also uses a gradually increasing parameter to grow the effective horizon of the backup and maximization.
Empirical evaluation
The authors evaluate their bisimulation approximants on an exactly solvable GridWorld, a noisy continuous variant, and Atari agents. Results indicate robust learning under continuous noise and behaviorally meaningful distances in Atari frames.
- GridWorld: The evaluation begins with a 31-state GridWorld whose exact bisimulation metric enables quantitative assessment of the learned approximant.The same setup also supports a continuous-state variant created by adding Gaussian noise to state representations.
- GridWorld: The GridWorld experiments measure both absolute and normalized metric error across 10 independent runs with 95% confidence intervals.Normalized error captures relative distances, which the authors describe as the practically relevant quantity.
- GridWorld: The learned metric performs similarly on the discrete GridWorld and its continuous noisy variant, with noise not hurting performance and possibly helping.The authors hypothesize that the noise may act as regularization, but state that this requires further investigation.
- Atari 2600: For Atari evaluation, the authors use deterministic agents trained on three Atari 2600 games and learn a policy-tied approximant from replay-buffer transitions.Sticky-action agents are excluded because sticky actions introduce stochasticity into action outcomes.
- Atari 2600: On Space Invaders, learned distances capture meaningful events beyond pixel differences: game starts are relatively close, enemy destruction creates peaks, and frames preceding death are typically farthest.Similar results were observed for Asterix and Pong.
Conclusion
The paper introduces scalable methods for bisimulation metrics in large deterministic MDPs, including an asymptotically exact sampling method and a differentiable approximation method. It also establishes theoretical properties for on-policy metrics and demonstrates applicability to continuous and deep-RL settings.
- Empirical scope: The learned approximant captures behavioral similarities in continuous state spaces, while the Atari results are described as promising despite nonoptimal architecture and hyperparameters.The authors suggest applications including auxiliary representation learning, replay-buffer compression, and more efficient exploration.
- Limitations and future work: Extending the methods to stochastic systems remains future work because Wasserstein computation without a generative model is challenging.The paper suggests estimating transition dynamics from replay-buffer transitions as one possible approach.
- On-policy metric: The on-policy metric has a fixed-point formulation and bounds differences in policy values.The paper proves that |Vπ(s) − Vπ(t)| ≤ dπ for any two states.
- Theoretical guarantees: The deterministic-state transition Wasserstein term reduces to the distance between successor states.This identity enables the sampling-based updates used by the algorithm.
- Theoretical guarantees: The sampling-based algorithm converges almost surely to the true bisimulation metric.The proof uses maximizing-action sampling, monotonicity, and contraction arguments.
- Differentiable approximation: The differentiable loss trains neural-network approximants from batches of state, action, reward, and next-state samples.Batch matrix operations support efficient training on specialized hardware, including GPUs.
Bisimulation distances between all states in the GridWorld
Figure 7 shows bisimulation distances from every state in the GridWorld MDP. The distances capture behavioral similarity beyond physical proximity.
- GridWorld distances: The hallway state is farthest from the neighboring bottom row in the top room under the bisimulation metric.This illustrates that physical adjacency does not determine behavioral similarity.
Configuration file for GridWorld
The supplied passage identifies the configuration file as specifying the hyperparameters used in subsection 7.1.
- Configuration: The configuration file specifies the hyperparameters used in subsection 7.1.
Aggregating states
The continuous-MDP experiment aggregates sampled states using the learned bisimulation metric approximant. The resulting clusters capture behavioral symmetries in the environment.
- Sampling and clustering: The experiment samples 100 independent states per underlying cell and incrementally grows clusters whose states remain within a distance threshold.Pairwise learned distances determine whether sampled states can be grouped together.
- Behavioral aggregation: The learned distance captures symmetries by grouping near-goal states and other behaviorally similar regions.The text identifies orange, dark-brown, and dark-blue clusters as examples of these groupings.
Distance plots for all games
The experiments visualize distances from a source state to other frames alongside value-function differences, using four panels for each game. Hyperparameters were selected through three-run sweeps for overall performance across games.
- Distance plots for all games: Each game is shown with source, closest-so-far, and current frames, plus a plot comparing distances with value-function differences.The distance plot uses black for ψπ θ ([φ(s), φ(t)]) and blue for |V̂π(φ(s)) − V̂π(φ(t))|.
- Distance plots for all games: Hyperparameters were chosen from three independent runs for each setting, selecting the best overall performance across all games.The sweep varied hidden layers over [1, 2, 3] and hidden units per layer over [16, 32, 64, 128].
- Distance plots for all games: Evaluation visualizations are provided for Space Invaders, Pong, and Asterix at frame 500.The Pong caption notes that peaks occur when the agent scores, while the Asterix caption notes that plateaus occur when the agent is not moving.
Training curves for dπ
The paper provides training curves for the policy-tied bisimulation metric dπ learned on trained reinforcement-learning agents.
- Training curves for dπ: Training curves are provided for the bisimulation metric dπ learned on trained reinforcement-learning agents.The supplied passage identifies these as training curves for dπ learned on trained agents.
Lax bisimulation metrics
Lax bisimulation metrics compare rewards and transition distributions across action choices, with a fixed-point operator whose least fixed point characterizes lax behavioral equivalence and bounds optimal-value differences.
- Definitions and theoretical results: A lax bisimulation relation requires matching rewards and aggregated transition probabilities across suitably paired actions.For related states, every action must have another action with equal reward and equal probability for every equivalence class.
- Definitions and theoretical results: The metric δ(d) combines absolute reward differences with discounted Wasserstein distance between transition distributions.The definition is δ(d)((s, a), (t, b)) = |R(s, a) − R(t, b)| + γW(d)(P(s, a), P(t, b)).
- Definitions and theoretical results: The operator F applies a Hausdorff metric to action-state sets constructed from the state-action pairs Xs.The supplied definitions introduce Xs = {(s, a)|a ∈ A} and the Hausdorff metric H(d) used by F.
- Definitions and theoretical results: The least fixed point dlax is zero exactly for lax-bisimilar states and bounds optimal-value differences: |V∗(s) − V∗(t)| ≤ dlax(s, t) ≤ d∼(s, t).Theorem 5 establishes monotonicity and the least fixed point; Theorem 6 gives the value-difference bounds.
- Empirical illustrations: Training curves are also reported for Space Invaders, Asterix, and Pong.The supplied figure captions identify these as training curves for the three games.