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Invariant Causal Prediction for Block MDPs

Amy Zhang, Clare Lyle, Shagun Sodhani, Angelos Filos, Marta Kwiatkowska, Joelle Pineau, Yarin Gal, Doina Precup

arXiv:2003.06016v2cs.LGcs.AIstat.ML

TL;DR

The paper asks how reinforcement learning can generalize across environments with shared latent dynamics but changing observations. It uses invariant causal prediction to learn model-irrelevance state abstractions, proving and empirically supporting generalization across linear and nonlinear settings. The approach is bounded by assumptions such as identifiable observation blocks and realizable value learning.

  • Problem

    Reinforcement learning needs abstractions that generalize beyond training environments when observations vary despite shared latent dynamics, while standard methods can overfit environment-specific features.

  • Method

    The paper applies invariant causal prediction to learn state abstractions that retain causal, reward-relevant structure and discard spurious observation features across environments.

  • Results

    The method learns optimal or model-irrelevance state abstractions with high probability in certain linear settings and improves generalization over single-task and multitask baselines in nonlinear settings.

  • Takeaways & Limitations

    Causal feature selection can provide a state-abstraction framework with error bounds and identifiability results for generalization across environments sharing causal structure.

  • Takeaways & Limitations

    The analysis assumes each observation identifies its latent state; relaxing this can make value learning ill-posed or partially observable.

Abstract

from arXiv · show

Generalization across environments is critical to the successful application of reinforcement learning algorithms to real-world challenges. In this paper, we consider the problem of learning abstractions that generalize in block MDPs, families of environments with a shared latent state space and dynamics structure over that latent space, but varying observations. We leverage tools from causal inference to propose a method of invariant prediction to learn model-irrelevance state abstractions (MISA) that generalize to novel observations in the multi-environment setting. We prove that for certain classes of environments, this approach outputs with high probability a state abstraction corresponding to the causal feature set with respect to the return. We further provide more general bounds on model error and generalization error in the multi-environment setting, in the process showing a connection between causal variable selection and the state abstraction framework for MDPs. We give empirical evidence that our methods work in both linear and nonlinear settings, attaining improved generalization over single- and multi-task baselines.

1. Introduction

The paper addresses reinforcement learning generalization across environments that share latent dynamics but vary in observations. It uses invariant causal prediction to learn state abstractions that discard spurious features and generalize with fewer training environments.

  • Standard RL often overfits environment-specific observations, such as sensor placement or lighting, despite unchanged physical dynamics.
  • Multitask generalization bounds can require many training tasks, which is especially costly because reinforcement learning is sample inefficient.
  • The paper studies multienvironment RL, where environments share latent causal structure while their observations vary and their reward-relevant structure remains shared.
  • Its central contribution is an invariant causal prediction method for learning state abstractions that generalize across environments.
  • In linear settings, the method can learn an optimal generalizing state abstraction with high probability using fewer training environments than standard PAC bounds require.
  • In nonlinear settings, analogous invariant prediction methods improve generalization over single-task and multitask baselines.

2. Background

The background connects state abstraction and bisimulation with invariant causal prediction. These perspectives aim to retain reward-relevant structure while eliminating spurious information and identifying representations stable across environments.

  • State abstractions compress observations by distinguishing decision-relevant information from irrelevant information for planning and decision making.
  • Bisimulation defines shared structure by requiring equivalent states to have the same expected reward and equivalent distributions over next bisimilar states.
  • The cross-environment extension treats different experimental conditions as parts of a super-MDP, allowing bisimilar states to be identified across MDPs.
  • Invariant Causal Prediction searches for the minimal causal feature set by exploiting the invariance of causal models across interventions or environments.
  • IRM learns representations whose optimal linear classifier is invariant across environments, aiming to remove spurious correlations and maintain low risk in structurally similar new environments.
  • Because the IRM constraint is generally difficult to optimize directly, prior work replaces it with a penalized objective and a penalty schedule.

3. Problem Setup

The problem setup considers block MDP families with shared latent states and dynamics but environment-specific emissions, including correlated spurious variables. The goal is to learn a state abstraction that generalizes to unseen emissions under causal and realizability assumptions.

  • The learner uses training environments to learn a representation that enables a policy to generalize across the entire environment family.
  • A block MDP has finite latent states and actions, observable emissions, latent transitions, and rewards defined over the latent process.
  • The block assumption requires each observation to uniquely identify its generating latent state, partitioning observations into disjoint state-specific blocks.
  • Across environments, latent states, actions, transitions, and rewards are shared while emission functions vary; the objective is to generalize across these emissions.
  • The setup focuses on structured overlap among environment-specific emissions because disjoint emission ranges can make the problem arbitrarily difficult.
  • The relaxed block MDP permits spurious variables to have environment-specific transition dynamics provided those correlations do not affect expected reward.
  • Relaxations: The analysis assumes realizability: without the block assumption, identical observations could correspond to different latent states and values, making value learning partially observable.
  • The causal formulation assumes current state variables affect only the next state and current reward, and environments intervene on individual observation variables.

4. Connecting State Abstractions to Causal Feature Sets

Invariant causal prediction connects causal feature selection to model-irrelevance state abstractions: selecting reward ancestors yields an abstraction valid across environments, while temporal dependence can require ancestors beyond direct reward parents. Under suitable intervention conditions, multi-environment ICP can identify the minimal causal variables.

  • ICP removes irrelevant observation variables by identifying causal predictors, paralleling state abstraction’s removal of irrelevant environmental information.
  • In Figure 2, x2 directly causes reward, but x1 is also required because x2’s next-timestep distribution depends on x1.
  • A causal-variable state abstraction is a model-irrelevance abstraction for every environment in the block MDP family.
  • The abstraction must include reward ancestors, not only direct reward parents, because return prediction depends on multi-timestep interactions.
  • Under identifiable training-environment interventions, multi-environment ICP can recover the minimal causal variable set, whereas a single environment may not suffice.

5. Block MDP Generalization Bounds

The paper develops generalization and model-error bounds for invariant state abstractions in block MDPs. These results relate value discrepancy to abstraction errors and improve the dependence of generalization bounds from the number of environments to aggregate samples.

  • Theorem 2 bounds test-environment model error for a coarser bisimulation using abstraction error and the Wasserstein distance between abstract-state distributions.
  • The learned invariant MDP can be analyzed through discrepancies in its dynamics and reward models relative to the underlying MDP.
  • Theorem 3 bounds the value difference for any L-Lipschitz policy using the learned invariant MDP’s model discrepancies.
  • The generalization bound depends on supremum dynamics and reward errors, which reduce to regression problems over the available data.
  • Using aggregate samples n = Σ_e n_e yields bounds scaling with sample count rather than the number of training environments, improving over prior multitask bounds.

6. Methods

The paper proposes two invariant-representation methods for block MDPs: linear ICP-based variable selection and a nonlinear gradient-based representation-learning approach. Both seek state abstractions that preserve relevant dynamics and reward information while discarding environment-specific information.

  • The methods use invariant causal prediction to learn state abstractions across stochastic observations and interventions.
  • Linear MISA: Linear MISA applies ICP iteratively to replay-buffer transitions tagged by environment to identify causal ancestors of reward.
  • Linear MISA: Under suitable identifiability conditions, linear ICP returns the exact causal variable set with a specified confidence level.
  • Linear MISA: Linear MISA is exponential in the number of variables and restricted to linear relationships between observations, reward, and next state.
  • Nonlinear MISA: Nonlinear MISA learns a dynamics-preserving invariant encoder with shared dynamics and reward models, task-specific dynamics, and a decoder across multiple environments.
  • Nonlinear MISA: An adversarial task-classifier loss encourages the shared representation to exclude task-specific information.

7. Results

The experiments evaluate MISA across linear and nonlinear dynamics, rich observations, imitation learning, and reinforcement learning. Across settings, MISA generalizes to unseen environments, camera angles, and spurious-correlations shifts better than comparison methods.

  • 7.1.2. RICH OBSERVATION SETTING: MISA zero-shot generalizes to unseen environments in both low-dimensional and high-dimensional observation settings.The evaluation covers linear and nonlinear versions of MISA with corresponding Block MDP dynamics.
  • 7.1.1. LINEAR SETTING: MISA recovers {x1, x2} and achieves zero generalization error by eliminating the spurious x3 variable.Any nonzero least-squares weight on x3 produces arbitrarily large error under novel hard interventions.
  • 7.1.2. RICH OBSERVATION SETTING: MISA is more effective than IRM with two training environments at disentangling invariant, dynamics-relevant features from noncausal features.The comparison reports that IRM’s loss starts higher, decreases slowly, and is brittle to tune.
  • 7.2. Imitation Learning: MISA’s test error grows significantly more slowly than single-task and multitask baselines when camera angles change across Cheetah Run environments.The evaluation uses imitation learning from an expert policy and tests action prediction on new camera angles.
  • 7.3. Reinforcement Learning: MISA drastically improves the generalization gap over aggregate SAC training and IRM in cartpole swingup with spurious correlated dimensions.The setup trains on two environments and adds Gaussian noise to the causal state, encouraging reliance on a noiseless spurious dimension.

8. Related Work

The related work reviews PAC-style generalization bounds, multitask reinforcement learning, shared representations, transfer methods, state abstractions, and causal prediction. It distinguishes this paper’s multienvironment setting by its explicit use of shared environment structure.

  • Generalization Bounds: PAC and PAC-Bayes bounds relate test error to training error and model or hypothesis-class complexity, with extensions to reinforcement learning and multitask settings.The reviewed bounds include PAC-MDP and multitask PAC-Bayes formulations.
  • Multitask Reinforcement Learning: Prior multitask reinforcement-learning methods share policies or representations but generally assume only a common state and action space.These approaches do not explicitly model how tasks relate through environment structure.

9. Discussion

The paper concludes that causal inference can produce invariant state representations that generalize across environments sharing causal structure. It establishes a framework spanning linear and deep reinforcement learning while identifying broader generalization challenges.

  • Conclusions: Causal inference methods can learn invariant causal representations that generalize across environments with shared causal structure.The paper provides a framework for defining relevant environment families and extracting causal state representations.
  • Conclusions: The framework covers low-dimensional linear value-function approximation and deep reinforcement learning, with error bounds and identifiability results.These results connect representation learning for generalization with formal guarantees in both settings.
  • Future Scope: The authors characterize the work as a first step toward useful representations that generalize across a broader class of environments.Potential applications include third-person imitation learning and sim2real transfer.

B. Proofs

The proofs establish when causal ancestors of reward form valid and identifiable model-irrelevance state abstractions, and bound errors arising from abstraction and approximate dynamics. They also show why sufficient interventions and block-MDP assumptions are necessary.

  • Interventions must avoid causal reward ancestors; otherwise environments can have different latent dynamics and violate the block-MDP assumption.
  • The reward ancestors φS form a model-irrelevance abstraction for every environment under the stated causal assumptions.
  • The causal feature set is identifiable when training environments provide sufficient interventions for each reward ancestor.
  • Insufficient interventions can leave an abstraction valid on training environments but invalid globally, as shown by a variable constant during training and perturbed at test time.
  • Theorems provide bounds on value difference and model error for Lipschitz policies, approximate transition models, and coarser bisimulations.
  • A union of environment-specific abstractions is valid when shared observations receive the same abstract-state mapping across environments.
  • A joint abstraction incurs a lower-bounded error when different latent states from different environments induce the same observation.

C. Implementation Details

The implementation uses convolutional encoders and decoders for model learning, multilayer perceptrons for dynamics and rewards, and an augmented SAC architecture with shared representations and regularization.

  • Model learning uses an encoder based on a convolutional trunk with added layers, ReLU activations, 3 × 3 kernels, and 32 channels.
  • The decoder uses one fully connected layer followed by four deconvolutional layers with ReLU activations except at the pixel-output layer.
  • Dynamics and reward models are multilayer perceptrons with two hidden layers of 200 neurons and ReLU activations.
  • Reinforcement learning modifies SAC with a shared actor-critic encoder, general and task-specific forward models, and a 50-dimensional hidden representation.
  • The experiments add L1 regularization on the latent representation, a spurious-correlation dimension, an environment identifier, and Gaussian observation noise.
  • SAC performs soft policy evaluation from replay-buffer transitions and policy improvement by minimizing KL divergence to a Q-induced Boltzmann distribution.

D. Additional Results

Additional results show that baseline SAC fails to generalize with only two environments even without noise on the ground-truth states; Figure 8 reports this comparison across 10 seeds.

  • With only two environments and no ground-truth state noise, baseline SAC fails in the Cartpole Swingup experiment.
  • Figure 8 measures the SAC generalization gap on Cartpole Swingup from DMC using two training environments and 10 seeds.
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