GRL: Generalized Reinforcement Learning

Actions as Operators on State Space

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🎯 What is GRL?

Generalized Reinforcement Learning (GRL) redefines the concept of "action" in reinforcement learning. Instead of treating actions as discrete indices or fixed-dimensional vectors, GRL models actions as parametric operators that transform the state space.

flowchart TB
    subgraph TRL["🔵 Traditional RL"]
        direction LR
        S1["<b>State</b><br/>s"] --> P1["<b>Policy</b><br/>π"]
        P1 --> A1["<b>Action Symbol</b><br/>a ∈ A"]
        A1 --> NS1["<b>Next State</b><br/>s'"]
    end
    
    TRL --> GRL
    
    subgraph GRL["✨ Generalized RL"]
        direction LR
        S2["<b>State</b><br/>s"] --> P2["<b>Policy</b><br/>π"]
        P2 --> AP["<b>Operator Params</b><br/>θ"]
        AP --> OP["<b>Operator</b><br/>Ô<sub>θ</sub>"]
        OP --> ST["<b>State Transform</b><br/>s' = Ô<sub>θ</sub>(s)"]
    end
    
    style S1 fill:#e3f2fd,stroke:#1976d2,stroke-width:3px,color:#000
    style NS1 fill:#e3f2fd,stroke:#1976d2,stroke-width:3px,color:#000
    style A1 fill:#fff9c4,stroke:#f57c00,stroke-width:3px,color:#000
    style P1 fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px,color:#000
    
    style S2 fill:#e3f2fd,stroke:#1976d2,stroke-width:3px,color:#000
    style ST fill:#c8e6c9,stroke:#388e3c,stroke-width:3px,color:#000
    style AP fill:#fff59d,stroke:#fbc02d,stroke-width:3px,color:#000
    style OP fill:#ffcc80,stroke:#f57c00,stroke-width:3px,color:#000
    style P2 fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px,color:#000
    
    style TRL fill:#fafafa,stroke:#666,stroke-width:2px
    style GRL fill:#fafafa,stroke:#666,stroke-width:2px
    
    linkStyle 4 stroke:#666,stroke-width:2px

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This formulation, inspired by the least-action principle in physics, leads to policies that are not only optimal but also physically grounded—preferring smooth, efficient transformations over abrupt changes.

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📖 Tutorial Papers

Part I: Reinforcement Fields — Particle-Based Learning

Status: 🔄 In progress (9/10 chapters complete)

Particle-based belief representation, energy landscapes, and functional learning over augmented state-action space.

Start Learning → | Research Roadmap →

Section	Chapters	Topics
Foundations	0, 1, 2, 3	Augmented space, particles, RKHS, energy
Field & Memory	4, 4a, 5, 6, 6a	Functional fields, Riesz theorem, belief states, MemoryUpdate, advanced memory
Algorithms	7	RF-SARSA (next)
Interpretation	8-10	Soft transitions, POMDP, synthesis

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Part II: Reinforcement Fields — Emergent Structure & Spectral Abstraction

Status: 📋 Planned (after Part I)

Spectral discovery of hierarchical concepts through functional clustering in RKHS.

Section	Chapters	Topics
Functional Clustering	11	Clustering in function space
Spectral Concepts	12	Concepts as eigenmodes
Hierarchical Control	13	Multi-level abstraction

Based on: Section V of the original paper

Reading time: ~10 hours total (both parts)

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Quantum-Inspired Extensions

Status: 🔬 Advanced topics (9 chapters complete)

Mathematical connections to quantum mechanics and novel probability formulations for ML.

Explore Advanced Topics →

Theme	Chapters	Topics
Foundations	01, 01a, 02	RKHS-QM parallel, state vs. wavefunction, amplitude interpretation
Complex RKHS	03	Complex-valued kernels, interference, phase semantics
Projections	04, 05, 06	Action/state fields, concept subspaces, belief dynamics
Learning & Memory	07, 08	Beyond GP, memory dynamics, principled consolidation

Novel Contributions:

• Amplitude-based RL: Complex-valued value functions with phase semantics
• MDL consolidation: Information-theoretic memory management
• Concept-based MoE: Hierarchical RL via subspace projections

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🔑 Key Innovations

Aspect	Classical RL	GRL
Action	Discrete index or vector	Parametric operator $\hat{O}(\theta)$
Action Space	Finite or bounded	Continuous manifold
Value Function	$Q(s, a)$	Reinforcement field $Q^+(s, \theta)$ over augmented space
Experience	Replay buffer	Particle memory in RKHS
Policy	Learned function	Inferred from energy landscape
Uncertainty	External (dropout, ensembles)	Emergent from particle sparsity

GRL as a Unifying Framework

Key Insight: Traditional RL algorithms (Q-learning, DQN, PPO, SAC, RLHF for LLMs) are special cases of GRL!

When you:

• Discretize actions → GRL recovers Q-learning
• Use neural networks → GRL recovers DQN
• Apply Boltzmann policies → GRL recovers REINFORCE/Actor-Critic
• Fine-tune LLMs → GRL generalizes RLHF

See: Recovering Classical RL from GRL →

Why GRL?

• Generalization: Subsumes existing methods as special cases
• Continuous actions: No discretization, full precision
• Smooth interpolation: Nearby parameters → similar behavior
• Compositional: Operators can be composed (operator algebra)
• Uncertainty: Sparse particles = high uncertainty (no ensembles needed)
• Interpretability: Energy landscapes, particle inspection
• Modern applications: Applies to RLHF, prompt optimization, neural architecture search

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🚀 Quick Start

Installation

# Clone the repository
git clone https://github.com/pleiadian53/GRL.git
cd GRL

# Create environment with mamba/conda
mamba env create -f environment.yml
mamba activate grl

# Install in development mode
pip install -e .

# Verify installation (auto-detects CPU/GPU/MPS)
python scripts/verify_installation.py

See INSTALL.md for detailed instructions.

First Steps

1. Read the tutorial: Start with Chapter 0: Overview
2. Explore concepts: Work through Chapter 1: Core Concepts
3. Understand algorithms: See the algorithm chapters (coming soon)
4. Implement: Follow the implementation guide

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📁 Project Structure

GRL/
├── src/grl/                    # Core library
│   ├── core/                   # Particle memory, kernels
│   ├── algorithms/             # MemoryUpdate, RF-SARSA
│   ├── envs/                   # Environments
│   └── visualization/          # Plotting tools
├── docs/                       # 📚 Public documentation
│   └── GRL0/                   # Tutorial paper (Reinforcement Fields)
│       ├── tutorials/          # Tutorial chapters (6/10 complete)
│       ├── paper/              # Paper-ready sections
│       └── implementation/     # Implementation specs
├── notebooks/                  # Jupyter notebooks
│   └── vector_field.ipynb     # Vector field demonstrations
├── examples/                   # Runnable examples
├── scripts/                    # Utility scripts
├── tests/                      # Unit tests
└── configs/                    # Configuration files

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📄 Documentation

Tutorial Papers: Reinforcement Fields (Two Parts)

Part I: Particle-Based Learning (6/10 chapters complete)

• Start Here — Overview
• Tutorials — Chapter-by-chapter learning
• Implementation — Technical specifications

Part II: Emergent Structure & Spectral Abstraction (Planned)

Additional Resources

• Installation Guide — Detailed setup instructions
• Interactive Notebooks — Jupyter demos with visualizations (best viewed on Pages)
  • View source — Raw notebooks in repository

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🔬 Research Papers

Original Paper (arXiv 2022)

Generalized Reinforcement Learning: Experience Particles, Action Operator, Reinforcement Field, Memory Association, and Decision Concepts

Po-Hsiang Chiu, Manfred Huber
arXiv:2208.04822 (2022) — 37 pages, 15 figures

The foundational work introducing particle-based belief states, reinforcement fields, and concept-driven learning.

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Tutorial Papers (This Repository)

Reinforcement Fields Framework — Enhanced exposition with modern formalization

Part I: Particle-Based Learning

• Functional fields over augmented state-action space
• Particle memory as belief state in RKHS
• MemoryUpdate and RF-SARSA algorithms
• Emergent soft state transitions, POMDP interpretation

Status: 🔄 Tutorial in progress (6/10 chapters complete)

Part II: Emergent Structure & Spectral Abstraction

• Functional clustering (clustering functions, not points)
• Spectral methods on kernel matrices
• Concepts as coherent subspaces of the reinforcement field
• Hierarchical policy organization

Status: 📋 Planned (after Part I)

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Planned Extensions

Paper	Title	Status	Progress
Paper A	Generalized Reinforcement Learning — Actions as Operators	🟢 Draft Complete	~70%
	Operator algebra, generalized Bellman equation, energy regularization		Complete draft, 3/7 figures, proofs outlined
Paper B	Operator Policies — Learning State-Space Operators with Neural Operator Networks (tentative)	⏳ Planned	~0%
	Neural operators, scalable training, operator-actor-critic		After Paper A
Paper C	Applications of GRL to Physics, Robotics, and Differentiable Control (tentative)	⏳ Planned	~0%
	Physics-based control, compositional behaviors, transfer learning		After Paper B

Timeline:

• Paper A: Target submission April 2026 (NeurIPS/ICML)
• Paper B: Target submission June 2026 (ICML/NeurIPS)
• Paper C: Target submission July 2026 (CoRL)

See: Research Roadmap for detailed timeline and additional research directions.

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📊 How GRL Works: Particle-Based Learning

flowchart LR
    A["🌍 <b>State</b><br/>s"] --> B["💾 <b>Query</b><br/>Memory Ω"]
    B --> C["📊 <b>Compute</b><br/>Field Q⁺"]
    C --> D["🎯 <b>Infer</b><br/>Action θ"]
    D --> E["⚡ <b>Execute</b><br/>Operator"]
    E --> F["👁️ <b>Observe</b><br/>s', r"]
    F --> G["✨ <b>Create</b><br/>Particle"]
    G --> H["🔄 <b>Update</b><br/>Memory"]
    H -->|Loop| B
    
    style A fill:#e3f2fd,stroke:#1976d2,stroke-width:3px,color:#000
    style B fill:#fff9c4,stroke:#f57c00,stroke-width:3px,color:#000
    style C fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px,color:#000
    style D fill:#fff59d,stroke:#fbc02d,stroke-width:3px,color:#000
    style E fill:#ffcc80,stroke:#f57c00,stroke-width:3px,color:#000
    style F fill:#c8e6c9,stroke:#388e3c,stroke-width:3px,color:#000
    style G fill:#f8bbd0,stroke:#c2185b,stroke-width:3px,color:#000
    style H fill:#b2dfdb,stroke:#00796b,stroke-width:3px,color:#000

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Code Example

from grl.core import ParticleMemory
from grl.core import RBFKernel
from grl.algorithms import MemoryUpdate, RFSarsa

# Create particle memory (the agent's belief state)
memory = ParticleMemory()

# Define similarity kernel
kernel = RBFKernel(lengthscale=1.0)

# Learning loop
for episode in range(num_episodes):
    state = env.reset()
    
    for step in range(max_steps):
        # Infer action from particle memory
        action = infer_action(memory, state, kernel)
        
        # Execute and observe
        next_state, reward, done = env.step(action)
        
        # Update particle memory (belief transition)
        memory = memory_update(memory, state, action, reward, kernel)
        
        state = next_state

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📝 Citation

Original arXiv Paper

The foundational work is available on arXiv:

Chiu, P.-H., & Huber, M. (2022). Generalized Reinforcement Learning: Experience Particles, Action Operator, Reinforcement Field, Memory Association, and Decision Concepts. arXiv:2208.04822.

@article{chiu2022generalized,
  title={Generalized Reinforcement Learning: Experience Particles, Action Operator, 
         Reinforcement Field, Memory Association, and Decision Concepts},
  author={Chiu, Po-Hsiang and Huber, Manfred},
  journal={arXiv preprint arXiv:2208.04822},
  year={2022},
  url={https://arxiv.org/abs/2208.04822}
}

Read on arXiv →

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Tutorial Papers (This Repository)

The tutorial series provides enhanced exposition and modern formalization:

Part I: Particle-Based Learning (In progress)

@article{chiu2026part1,
  title={Reinforcement Fields: Particle-Based Learning},
  author={Chiu, Po-Hsiang and Huber, Manfred},
  journal={In preparation},
  year={2026}
}

Part II: Emergent Structure & Spectral Abstraction (Planned)

@article{chiu2026part2,
  title={Reinforcement Fields: Emergent Structure and Spectral Abstraction},
  author={Chiu, Po-Hsiang and Huber, Manfred},
  journal={In preparation},
  year={2026}
}

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Operator Extensions (Future Work)

@article{chiu2026operators,
  title={Generalized Reinforcement Learning — Actions as Operators},
  author={Chiu, Po-Hsiang},
  journal={In preparation},
  year={2026+}
}

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📜 License

This project is licensed under the MIT License - see the LICENSE file for details.

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🌟 The GRL Framework

GRL (Generalized Reinforcement Learning) is a family of methods that rethink how actions are represented and learned.

Original paper: arXiv:2208.04822 (Chiu & Huber, 2022)

Reinforcement Fields (This Repository)

Two-Part Tutorial Series:

Part I: Particle-Based Learning

• Actions as continuous parameters in augmented state-action space
• Particle memory as belief state, kernel-induced value functions
• Learning through energy landscape navigation

Part II: Emergent Structure & Spectral Abstraction

• Concepts emerge from functional clustering in RKHS
• Spectral methods discover hierarchical structure
• Multi-level policy organization

Key Innovation: Learning emerges from particle dynamics in function space, not explicit policy optimization.

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Actions as Operators (Paper A — In Development)

Core Idea: Actions as parametric operators that transform state space, with operator algebra providing compositional structure.

Key Innovation: Operator manifolds replace fixed action spaces, enabling compositional behaviors and physical interpretability.

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🙏 Acknowledgments

Mathematical Foundations

Core Framework:

• Formulated in Reproducing Kernel Hilbert Spaces (RKHS) — the functional framework for particle-based belief states
• Kernel methods define the geometry and similarity structure of augmented state-action space
• Inspired by the least-action principle in classical mechanics

Quantum-Inspired Probability:

• Probability amplitudes instead of direct probabilities — RKHS inner products as amplitude overlaps
• Complex-valued RKHS enabling interference effects and phase semantics for temporal/contextual dynamics
• Wave function analogy — The reinforcement field as a superposition of particle basis states
• This formulation is novel to mainstream ML and opens new directions for probabilistic reasoning

See: Quantum-Inspired Extensions for technical details.

Conceptual Connections

• Energy-based models (EBMs) — Control as energy landscape navigation
• POMDPs and belief-based control — Particle ensembles as implicit belief states
• Score-based methods — Energy gradients guide policy inference

Implementation Tools

• Gaussian process regression can model scalar energy fields (but is not essential to the framework)
• Neural operators for learning parametric action transformations
• Diffusion models share the gradient-field perspective

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📚 Start the Tutorial →