Flow Models for Unbounded and Geometry-Aware Distributional Reinforcement Learning (2505.04310v1)

Abstract: We introduce a new architecture for Distributional Reinforcement Learning (DistRL) that models return distributions using normalizing flows. This approach enables flexible, unbounded support for return distributions, in contrast to categorical approaches like C51 that rely on fixed or bounded representations. It also offers richer modeling capacity to capture multi-modality, skewness, and tail behavior than quantile based approaches. Our method is significantly more parameter-efficient than categorical approaches. Standard metrics used to train existing models like KL divergence or Wasserstein distance either are scale insensitive or have biased sample gradients, especially when return supports do not overlap. To address this, we propose a novel surrogate for the Cram`er distance, that is geometry-aware and computable directly from the return distribution's PDF, avoiding the costly CDF computation. We test our model on the ATARI-5 sub-benchmark and show that our approach outperforms PDF based models while remaining competitive with quantile based methods.

Overview of Flow Models for Unbounded and Geometry-Aware Distributional Reinforcement Learning

The paper presents a novel architecture for Distributional Reinforcement Learning (DistRL) that employs normalizing flows to model return distributions, addressing limitations in prior methodologies such as categorical and quantile-based approaches. Traditional RL techniques focus on expected returns, which can obscure key information regarding the return distribution's spread and potential risk. Distributional RL, by contrast, enhances decision-making by capturing the entire return distribution. Existing models like Categorical DQN (C51) and quantile methods like QR-DQN encounter limitations due to their fixed and bounded representations. Notably, these existing methods struggle with environments characterized by unbounded rewards, such as those seen in certain robotics and financial applications, where asymmetric and multimodal return distributions are prevalent.

DistRL via Normalizing Flows

The proposed method—Normalizing Flow Distributional Reinforcement Learning (NFDRL)—eschews fixed support structures in favor of a dynamic architecture using normalizing flows. These flows adaptively model the return distribution with continuous, unbounded densities, allowing for greater expressivity and parameter efficiency. Central to this approach is the architecture's ability to adjust support and resolution based on the data it encounters, enabling intricate modeling of multi-modality and distributional tail behaviors. The choice of normalizing flows also introduces the capability of exact density computation through invertible and differentiable transformations.

Instead of using the KL divergence, which is insensitive to scale and problematic when there is minimal overlap between predicted and target distributions, the paper proposes leveraging a novel surrogate based on the Cramér distance for optimization. This innovation is particularly noteworthy as it aligns distributions with stronger theoretical guarantees compared to biased gradients inherent in Wasserstein distance approximations, ensuring more stable and interpretable learning outcomes.

Key Methodological Insights

1. Expressivity and Parameter Efficiency:
   • NFDRL's use of normalizing flows offers high modeling expressivity with fewer parameters than categorical methods, effectively capturing return distributions beyond mean estimations. This capability is vital for environments with open-ended reward structures.
2. Handling Unbounded Returns:
   • Unlike C51, which imposes a capped return range, NFDRL supports continuous and unbounded return distributions. This is crucial for applications such as lifelong learning or transfer learning, where reward distributions are inherently unbounded and skewed.
3. Surrogate Cramér Distance:
   • By using a surrogate for the Cramér distance, NFDRL captures the geometry of return distributions more effectively than standard metrics like KL divergence or quantile regression losses. This choice enhances the stability and accuracy of DistRL learning through unbiased sample gradients.

Empirical Validation and Implications

The effectiveness of NFDRL is demonstrated across a range of controlled experiments and the ATARI-5 benchmark, highlighting its superiority in handling environments with complex reward structures. Results indicate that NFDRL not only surpasses PDF-based models in performance but remains competitive with quantile-based approaches. This positions NFDRL as an impactful advancement in DistRL, particularly for tasks characterized by diverse and challenging distributional dynamics.

Future Directions

This research opens several promising avenues for further exploration:

• Architectural Enhancements: Exploring alternative flow-based enhancements, including diffusion models, could offer additional benefits in stability and expressiveness.
• Risk-Sensitive Learning: The ability to accurately model full return distributions provides a foundation for incorporating distortion risk measures, allowing for tailored risk-aware policy development.

In summary, NFDRL sets a new standard for distributional modeling in reinforcement learning by leveraging the inherent strengths of normalizing flows. It overcomes prior limitations in representation support and scale alignment, thereby providing a robust framework suitable for an array of challenging real-world applications. As such, it represents a significant stride toward more holistic and adaptable DistRL systems.