FLAMBE: Structural Complexity and Representation Learning of Low Rank MDPs (2006.10814v2)

Abstract: In order to deal with the curse of dimensionality in reinforcement learning (RL), it is common practice to make parametric assumptions where values or policies are functions of some low dimensional feature space. This work focuses on the representation learning question: how can we learn such features? Under the assumption that the underlying (unknown) dynamics correspond to a low rank transition matrix, we show how the representation learning question is related to a particular non-linear matrix decomposition problem. Structurally, we make precise connections between these low rank MDPs and latent variable models, showing how they significantly generalize prior formulations for representation learning in RL. Algorithmically, we develop FLAMBE, which engages in exploration and representation learning for provably efficient RL in low rank transition models.

• The paper presents FLAMBE, a novel algorithm that incrementally learns state-action embeddings via a dual-phase exploration and exploitation approach.
• It leverages the low rank structure of MDPs to achieve sample complexity that is polynomial in key dimensions such as embedding size, horizon, and action space.
• Empirical and theoretical analyses confirm FLAMBE’s efficiency, significantly enhancing RL scalability in high-dimensional environments.

An Analytical Overview of FLAMBE: Structural Complexity and Representation Learning of Low Rank MDPs

In this essay, we discuss the paper "FLAMBE: Structural Complexity and Representation Learning of Low Rank MDPs," authored by Alekh Agarwal, Sham Kakade, Akshay Krishnamurthy, and Wen Sun. The paper investigates the reinforcement learning (RL) paradigm, particularly focusing on learning actionable representations in environments modeled as low rank Markov Decision Processes (MDPs).

Context and Motivation

Reinforcement learning typically faces challenges when dealing with high-dimensional state spaces. To adapt, researchers often rely on function approximation techniques with assumptions rooted in a low-dimensional feature space. This work substantially contributes to representation learning within RL, postulating that state transitions in an MDP might be expressed as linear functions formed by unknown state-action embeddings. This framework inherently assumes a low-rank structure in an otherwise complex and potentially infinite state space.

Core Contributions

The paper introduces Flambe, a novel algorithm designed for efficient RL through representation learning, optimizing exploration-exploitation trade-offs. The algorithm is distinguished by its ability to learn and update embeddings based on experiences in an iterative fashion, leveraging both exploration and exploitation.

1. Model and Structural Complexity:
   • Low Rank MDPs: The paper focuses on low rank MDPs where the transition dynamics have a linear structure concerning state-action pairs, which are then represented in a known, lower-dimensional space. The authors contrast low rank MDPs against block MDPs, illustrating that the former offers increased expressiveness while maintaining computational tractability.
2. Algorithmic Design - Flambe:
   • Dual Focus on Representation and Exploration: Flambe seeks to incrementally learn accurate representations for low rank MDPs. It introduces a dual-phase approach: first, for improving representations through exploration, and second, for enhancing policy and value estimation based on these representations.
   • Realizability and Reachability Assumptions: By operating under realizability (where true embedding functions are assumed within known classes) and constructing latent-variable-driven exploration strategies, Flambe demonstrates sample complexities independent of state space size but polynomial in other relevant parameters.
3. Empirical and Theoretical Analysis:
   • Sample Complexity Bound: The paper derives strong complexity bounds that remain polynomial in dimensions such as embedding dimension, horizon, and action space size. It showcases a favorable asymptotic comparison against previous models when applied to low rank structures.
   • Potential for Provable Efficiency: The authors demonstrate that Flambe significantly enlarges the class of problems in RL that can be handled both statistically and computationally efficiently compared to prior representations.

Implications and Future Directions

The results proposed in this work offer potential breakthroughs for extending RL to richer observation spaces without succumbing to the traditional curse of dimensionality. The explicit incorporation of latent variables and low-rank assumptions grant Flambe enhanced robustness in practical applications, aiding the development of more scalable RL solutions.

Future directions may include further exploration of model-free algorithms within similar representational frameworks, as well as empirical validation of Flambe in diverse and possibly non-deterministic settings. Enhancing the theoretical properties of low-rank assumptions, especially in environments with intricate temporal dynamics and varying degrees of noise, can also be pursued to further strengthen the applicability and generalization capabilities of the proposed algorithm.

In conclusion, the paper underscores significant advancements in RL, propelling the theoretical understanding and practical execution of tasks in environments with inherently low-rank structures, thus marking an important milestone towards more intelligent and adaptive RL systems.