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Rich-Observation Reinforcement Learning with Continuous Latent Dynamics

Yuda Song, Lili Wu, Dylan J. Foster, Akshay Krishnamurthy

TL;DR

RichCLD addresses reinforcement learning with rich, high-dimensional observations and continuous latent dynamics by coupling representation learning with exploration. The core method, BCRL.C, learns a Bellman-consistent representation in the latent space, ensuring Lipschitz-structured backups, while CRIEE interleaves representation learning with exploration via discretization and optimistic planning. The authors establish statistical tractability under approximate coverability, prove a separation from weaker latent-Lipschitz notions, and demonstrate practical performance on visual maze and locomotion benchmarks with competitive latent representations and downstream rewards. This work advances sample-efficient learning in realistic settings where perception is rich and the latent dynamics are nonlinear and continuous, enabling more scalable RL for vision-based control tasks.

Abstract

Sample-efficiency and reliability remain major bottlenecks toward wide adoption of reinforcement learning algorithms in continuous settings with high-dimensional perceptual inputs. Toward addressing these challenges, we introduce a new theoretical framework, RichCLD (Rich-Observation RL with Continuous Latent Dynamics), in which the agent performs control based on high-dimensional observations, but the environment is governed by low-dimensional latent states and Lipschitz continuous dynamics. Our main contribution is a new algorithm for this setting that is provably statistically and computationally efficient. The core of our algorithm is a new representation learning objective; we show that prior representation learning schemes tailored to discrete dynamics do not naturally extend to the continuous setting. Our new objective is amenable to practical implementation, and empirically, we find that it compares favorably to prior schemes in a standard evaluation protocol. We further provide several insights into the statistical complexity of the RichCLD framework, in particular proving that certain notions of Lipschitzness that admit sample-efficient learning in the absence of rich observations are insufficient in the rich-observation setting.

Rich-Observation Reinforcement Learning with Continuous Latent Dynamics

TL;DR

RichCLD addresses reinforcement learning with rich, high-dimensional observations and continuous latent dynamics by coupling representation learning with exploration. The core method, BCRL.C, learns a Bellman-consistent representation in the latent space, ensuring Lipschitz-structured backups, while CRIEE interleaves representation learning with exploration via discretization and optimistic planning. The authors establish statistical tractability under approximate coverability, prove a separation from weaker latent-Lipschitz notions, and demonstrate practical performance on visual maze and locomotion benchmarks with competitive latent representations and downstream rewards. This work advances sample-efficient learning in realistic settings where perception is rich and the latent dynamics are nonlinear and continuous, enabling more scalable RL for vision-based control tasks.

Abstract

Sample-efficiency and reliability remain major bottlenecks toward wide adoption of reinforcement learning algorithms in continuous settings with high-dimensional perceptual inputs. Toward addressing these challenges, we introduce a new theoretical framework, RichCLD (Rich-Observation RL with Continuous Latent Dynamics), in which the agent performs control based on high-dimensional observations, but the environment is governed by low-dimensional latent states and Lipschitz continuous dynamics. Our main contribution is a new algorithm for this setting that is provably statistically and computationally efficient. The core of our algorithm is a new representation learning objective; we show that prior representation learning schemes tailored to discrete dynamics do not naturally extend to the continuous setting. Our new objective is amenable to practical implementation, and empirically, we find that it compares favorably to prior schemes in a standard evaluation protocol. We further provide several insights into the statistical complexity of the RichCLD framework, in particular proving that certain notions of Lipschitzness that admit sample-efficient learning in the absence of rich observations are insufficient in the rich-observation setting.
Paper Structure (83 sections, 37 theorems, 216 equations, 4 figures, 2 tables, 5 algorithms)

This paper contains 83 sections, 37 theorems, 216 equations, 4 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. The RichCLD framework
  3. Contributions
  4. Paper organization
  5. Problem Setting
  6. Markov decision processes
  7. The RichCLD model
  8. Function approximation and learning objective
  9. Additional notation
  10. Statistical Complexity for the RichCLD Framework
  11. Upper bound for the RichCLD framework
  12. Lower bound under weaker continuity
  13. Efficient Algorithms for the RichCLD Framework
  14. Representation Learning with Continuous Latent Dynamics: BCRL.C
  15. Algorithm and guarantee
  16. ...and 68 more sections

Key Result

Theorem 1

Suppose ass:lipschitzassum:decoder_realizability hold. For any $\delta \in (0,1)$ and $\varepsilon\in(0,1)$, with probability at least $1-\delta$, GOLF.DBR(alg:golf) outputs a policy $\widehat{\pi}$ satisfying $J(\pi^\star) - J(\widehat{\pi}) \leq \varepsilon$ with sample complexity

Figures (4)

  • Figure 1: Results for BCRL.C on maze environments: $K$-means clusterings ($K=16$) visualize the learned latent space. Baseline does not enforce Lipschitzness of the $g$ functions in \ref{['eq:rep_learn']}. BCRL.C learns to respect walls and boundaries; Baseline does not.
  • Figure 2: Results on visual D4RL: learning curves of TD3-BC using pre-trained (frozen) decoders from each representation learning method. Random denotes a randomly initialized decoder.
  • Figure 3: Visualization of the coverage of the offline dataset in the latent space for the Spiral Maze environment. Each dot denotes one sample and each line denotes the transition between the states. Here we subsample 20000 samples out of the 500k samples collected in the maze environment.
  • Figure 4: Left: 3-frames Cheetah Run; Right: 3-frames Walker Walk.

Theorems & Definitions (45)

  • Example 1
  • Example 2
  • Theorem 1: PAC upper bound for RichCLD framework; informal
  • Theorem 2: Lipschitz $Q^\star/V^\star$ lower bound; informal
  • Remark 1
  • Theorem 3: Guarantee of BCRL.C
  • Proposition 1: Informal
  • Theorem 4: PAC guarantee for CRIEE; informal
  • Definition B.1
  • Proposition 2
  • ...and 35 more