[Paper Review] Hyperbolic Deep Reinforcement Learning

Edoardo Cetin, Benjamin Chamberlain+2 more|arXiv (Cornell University)|Oct 4, 2022

Reinforcement Learning in Robotics4 citations

TL;DR

This paper introduces a novel deep reinforcement learning framework that models latent representations in hyperbolic space to better capture hierarchical structures in sequential decision-making. By designing spectrally-regularized hyperbolic mappings (S-RYM), the method stabilizes training and enables end-to-end learning, achieving near-universal performance and generalization gains across Procgen and Atari 100K benchmarks compared to Euclidean baselines.

ABSTRACT

We propose a new class of deep reinforcement learning (RL) algorithms that model latent representations in hyperbolic space. Sequential decision-making requires reasoning about the possible future consequences of current behavior. Consequently, capturing the relationship between key evolving features for a given task is conducive to recovering effective policies. To this end, hyperbolic geometry provides deep RL models with a natural basis to precisely encode this inherently hierarchical information. However, applying existing methodologies from the hyperbolic deep learning literature leads to fatal optimization instabilities due to the non-stationarity and variance characterizing RL gradient estimators. Hence, we design a new general method that counteracts such optimization challenges and enables stable end-to-end learning with deep hyperbolic representations. We empirically validate our framework by applying it to popular on-policy and off-policy RL algorithms on the Procgen and Atari 100K benchmarks, attaining near universal performance and generalization benefits. Given its natural fit, we hope future RL research will consider hyperbolic representations as a standard tool.

Motivation & Objective

• To address the limitation of current deep RL models in learning generalizable, hierarchical features due to lack of proper inductive bias.
• To overcome optimization instabilities in hyperbolic RL caused by non-stationary and high-variance RL gradients.
• To enable stable, end-to-end training of deep RL agents with hyperbolic latent representations.
• To validate that hyperbolic geometry improves policy generalization and performance across diverse RL benchmarks.
• To establish hyperbolic representations as a standard inductive bias for future RL research.

Proposed method

• Proposes spectrally-regularized hyperbolic mappings (S-RYM), combining scaling and spectral normalization to stabilize hyperbolic representations.
• Applies S-RYM to both on-policy (PPO) and off-policy (DQN, SAC) RL algorithms for seamless integration with existing frameworks.
• Uses hyperbolic geometry to model hierarchical relationships in state representations, leveraging exponential volume growth for efficient tree-structure encoding.
• Employs δ-hyperbolicity as a metric to quantify the hierarchical structure of learned representations and correlate it with performance.
• Integrates the hyperbolic framework with existing RL components without architectural overhaul, ensuring compatibility with standard training pipelines.
• Validates the method on Procgen and Atari 100K benchmarks using standard evaluation protocols and ablation studies.

Experimental results

Research questions

• RQ1 Can hyperbolic geometry serve as an effective inductive bias for deep reinforcement learning by better encoding hierarchical state relationships?
• RQ2 How can hyperbolic representations be stably trained end-to-end in deep RL, given the high variance and non-stationarity of RL gradients?
• RQ3 Does incorporating hyperbolic geometry lead to improved generalization and performance across diverse RL environments?
• RQ4 How does the hierarchical structure of learned representations, measured via δ-hyperbolicity, correlate with RL performance?
• RQ5 Is the hyperbolic representation framework compatible with other state-of-the-art RL techniques such as PPG and data augmentation?

Key findings

• The proposed hyperbolic RL framework achieves near-universal performance improvements across both Procgen and Atari 100K benchmarks compared to Euclidean baselines.
• Performance gains correlate strongly with increased δ-hyperbolicity of the latent representation space, validating the importance of hierarchical inductive bias.
• Hyperbolic representations reduce overfitting to training data, particularly improving test performance on unseen environments such as bigfish and dodgeball.
• The method maintains consistent benefits even when combined with advanced techniques like Phasic Policy Gradient (PPG), demonstrating orthogonal compatibility and generalization gains.
• Data augmentation via random shifts harms performance on some environments, but the hyperbolic framework still outperforms its Euclidean counterpart, indicating robustness and general advantage.
• The framework enables stable training without additional compute costs, suggesting potential for efficient, compact representation learning in RL.