GRL Research Roadmap¶

Last Updated: January 14, 2026
Purpose: High-level plan for GRL research, documentation, and implementation

Current Focus: GRL v0 (Baseline)¶

Status: 🔄 In Progress — Documentation & Formalization
Goal: Complete tutorial paper and reference implementation of the original GRL framework

Part I: Particle-Based Learning¶

Status: ✅ 7/10 chapters complete

Chapter	Title	Status
00	Overview	✅ Complete
01	Core Concepts	✅ Complete
02	RKHS Foundations	✅ Complete
03	Energy and Fitness	✅ Complete
04	Reinforcement Field	✅ Complete
04a	Riesz Representer	✅ Complete
05	Particle Memory	✅ Complete
06	MemoryUpdate Algorithm	✅ Complete
07	RF-SARSA Algorithm	⏳ Next
08	Soft State Transitions	⏳ Planned
09	POMDP Interpretation	⏳ Planned
10	Practical Implementation	⏳ Planned

Priority: Complete Chapters 07-10 by February 2026

Part II: Emergent Structure & Spectral Abstraction¶

Status: 🎯 Foundation laid, rewrite needed

Original Section V topics:

Functional clustering in RKHS
Spectral concept discovery
Hierarchical policy organization
Experience compression

New formalization available:

Chapter 5: Concept Projections and Measurements provides mathematical foundation

Next Steps:

Rewrite Section V content using concept subspace formalism
Add tutorial chapters on spectral clustering algorithms
Implement concept discovery in code

Priority: Start after Part I complete (March 2026)

Quantum-Inspired Extensions¶

Status: 🔬 Advanced topics — 9 chapters complete
Goal: Explore mathematical connections to QM and novel probability formulations

Completed Chapters¶

Chapter	Title	Key Contribution
01	RKHS-Quantum Parallel	Structural mapping between RKHS and QM Hilbert spaces
01a	Wavefunction Interpretation	State vector vs. wavefunction clarity
02	RKHS Basis and Amplitudes	Why GRL doesn't need Born rule normalization
03	Complex-Valued RKHS	Interference effects, phase semantics
04	Action and State Fields	Slicing \(Q^+\) into projections
05	Concept Projections	Formal subspace theory (foundation for Part II)
06	Agent State & Belief Evolution	What "the state" is in GRL
07	Learning Beyond GP	Alternative learning mechanisms
08	Memory Dynamics	Formation, consolidation, retrieval

Future Directions from These Chapters¶

1. Amplitude-Based Reinforcement Learning 🔥 High Priority

Motivation: QM's probability amplitude formulation hasn't been applied to RL

Proposal:

Complex-valued value functions: \(Q^+(z) \in \mathbb{C}\)
Policy from Born rule: \(\pi(a|s) \propto |Q^+(s,a)|^2\)
Phase semantics: temporal, directional, or contextual structure
Interference-based TD updates

Next Steps:

Expand Chapter 03 with detailed algorithms
Design toy problems where phase helps
Implement and benchmark

Target: Paper submission NeurIPS 2026 or ICML 2027

2. Information-Theoretic Memory Consolidation 🔥 High Priority

Motivation: Replace hard threshold \(\tau\) in MemoryUpdate with principled criteria

Proposal:

MDL objective: \(\min_{\Omega'} \text{TD-error}(Q^+(\Omega')) + \lambda |\Omega'|\)
Surprise-gated consolidation: store distinct if high TD-error, merge if low
Adaptive top-k neighbors (density-aware)

Next Steps:

Implement MDL consolidation algorithm
Compare: hard threshold vs. soft vs. top-k vs. MDL vs. surprise-gating
Meta-learn consolidation parameters

Target: Paper submission ICML 2026 or NeurIPS 2027

3. Mixture of Experts with Concept-Based Gating

Motivation: Multiple local fields for scalability and modularity

Proposal:

\(Q^+(z) = \sum_m g_m(z) Q_m(z)\)
Gate by concept activation: \(g_m(z) \propto \|P_{\mathcal{C}_m} k(z, \cdot)\|^2\)
Each expert specializes on a concept subspace

Next Steps:

Connect to Part II concept discovery
Implement hierarchical composition
Benchmark on multi-task environments

Target: Part of larger hierarchical RL paper (2027)

4. Hybrid Neural-Particle Architecture

Motivation: Combine scalability of deep nets with fast adaptation of particles

Proposal:

\(Q^+(z) = Q_\theta(z) + \beta \sum_{i \in \text{recent}} w_i k(z_i, z)\)
Neural net: long-term memory (slow updates)
Particle buffer: short-term memory (fast updates, bounded)

Next Steps:

Implement distillation from buffer to network
Large-scale continuous control experiments
Compare to pure GP and pure neural baselines

Target: Practical algorithms paper (2027)

GRL Extensions: Papers A, B, C¶

Status: Paper A ~70% complete, B & C planned
Goal: Extend GRL with operator formalism, scalable algorithms, and applications

Paper A: Theoretical Foundations (Operator Framework)¶

Status: 🟢 Draft Complete (~6,500 words), Figures in Progress
Progress: 70% — Theory done, figures 43%, proofs 40%, experiments 0%

Core Contribution:

Actions as parametric operators \(\hat{O}_\theta\) (not just parameter vectors)
Operator algebra: composition, Lie groups, hierarchical skills
Generalized Bellman equation with energy regularization
Unification: classical RL as special case

Current State:

✅ Complete:

Unified draft with all sections
3 critical figures implemented (paradigm, operator families, unification)
Proof outlines
Figure generation framework

⏳ Remaining Work:

4 additional figures (energy landscapes, composition, convergence, policy viz)
Expand proofs (Appendices A)
Operator catalog (Appendix B)
Related work section
Validation experiments

Timeline:

January-February 2026: Complete all figures, expand appendices
March 2026: Run experiments, related work, final polish
April 2026: Submit to NeurIPS/ICML 2026

Location: dev/papers/paper-a-theory/

Paper B: Algorithms & Implementation¶

Status: ⏳ Planned
Target: ICML/NeurIPS 2026 (Applied Track)

Planned Content:

Operator-Actor-Critic (OAC) algorithm
Neural operator architectures (DeepONet, FNO integration)
Training stability techniques
Benchmark results (continuous control, physics tasks)
Ablation studies

Dependencies:

Requires Paper A theory finalized
Requires implementation of src/grl/operators/ and src/grl/algorithms/

Timeline:

February-March 2026: Algorithm development
April-May 2026: Benchmarking
June 2026: Draft and submit

Paper C: Empirical Applications¶

Status: ⏳ Planned
Target: CoRL/IROS 2026 (Robotics/Applications)

Planned Content:

Real-world robotic manipulation
Fluid control problems
PDE-governed systems
Physics-based environments
Interpretability analysis (energy landscapes, concept activation)

Dependencies:

Requires Paper B algorithms
Requires mature implementation

Timeline:

April-June 2026: Application experiments
July 2026: Draft and submit (CoRL deadline)

Implementation Roadmap¶

Status: 🔄 Basic structure in place, algorithms pending
Goal: Reference implementation of GRL v0, classical RL recovery, and modern applications

Environment Simulation Package¶

Vision: Comprehensive environment package supporting:

Validation: Classical RL baselines (CartPole, Pendulum, MuJoCo)
Strategy: Modern applications (RLHF, prompt optimization, molecule design)
Innovation: GRL-native domains (physics simulation, field control)

Package Structure:

src/grl/envs/
├── validation/              # Tier 1: Reproduce classical RL
│   ├── nav2d.py            # 2D navigation (Priority 7) ⭐⭐
│   ├── cartpole.py         # DQN validation
│   ├── pendulum.py         # SAC validation
│   └── mujoco_envs.py      # Robotics (Ant, Humanoid)
│
├── strategic/               # Tier 2: Modern RL applications 🔥
│   ├── llm_finetuning.py   # RLHF for LLMs ⭐⭐⭐ HIGHEST PRIORITY
│   ├── prompt_opt.py       # Prompt optimization
│   ├── molecule_design.py  # Drug discovery
│   └── nas.py              # Neural Architecture Search
│
├── novel/                   # Tier 3: GRL-native applications
│   ├── physics_sim.py      # Force field control
│   ├── fluid_control.py    # PDE-governed systems
│   ├── image_editing.py    # Parametric transforms
│   └── multi_robot.py      # Multi-agent coordination
│
├── wrappers/                # Adapters for existing environments
│   ├── gym_wrapper.py      # OpenAI Gym → GRL
│   ├── gymnasium_wrapper.py# Gymnasium → GRL
│   ├── dm_control_wrapper.py # DeepMind Control → GRL
│   └── rlhf_wrapper.py     # TRL/transformers → GRL
│
└── scenarios/               # Predefined configurations
    ├── paper_scenarios.py   # Original paper scenarios
    ├── benchmark_suite.py   # Standard benchmarks
    └── tutorials.py         # Teaching examples

Key Design Principles:

Wrappers enable GRL on any existing RL environment
Strategic environments target commercially relevant problems
Scenarios provide reproducible experiments

Implementation Roadmap¶

Phase 1: GRL v0 Baseline (Current)¶

Target: March 2026

Modules:

Module	Status	Description
`grl/kernels/`	✅	Kernel functions (RBF, Matern, etc.)
`grl/particles/`	🔄	Particle memory management
`grl/fields/`	🔄	Reinforcement field computation
`grl/algorithms/memory_update.py`	⏳	MemoryUpdate algorithm
`grl/algorithms/rf_sarsa.py`	⏳	RF-SARSA algorithm
`grl/envs/`	🔄	Test environments

Priority Tasks:

Complete MemoryUpdate implementation
Implement RF-SARSA
Add sparse GP variants
Create tutorial notebooks

Phase 2: Scalable Learning (March-April 2026)¶

Goal: Implement alternative learning mechanisms from Chapter 07

Tasks:

Kernel ridge regression
Online SGD on weights
Sparse methods (LASSO, inducing points)
Hybrid (neural + particle)

Phase 3: Memory Dynamics (April-May 2026)¶

Goal: Implement principled memory consolidation from Chapter 08

Tasks:

Soft association (no hard threshold)
Top-k adaptive neighbors
MDL consolidation
Surprise-gated formation
Memory type tags (factual/experiential/working)

Phase 4: Operator Framework (May-July 2026)¶

Goal: Implement Paper A operator formalism

Tasks:

Operator base classes
Operator families (affine, field, kernel, neural)
Composition and algebra
Operator-Actor-Critic (OAC)

Phase 5: Applications (August 2026+)¶

Goal: Benchmarks and real-world demos

Tasks:

Continuous control tasks
Physics-based simulations
Robotic manipulation (if hardware available)
Transfer learning experiments

Documentation Structure¶

Goal: Multi-level documentation for different audiences

Public Documentation (`docs/`)¶

Tutorial Papers:

docs/GRL0/tutorials/ — Part I: Particle-Based Learning
docs/GRL0/quantum_inspired/ — Advanced topics (QM connections)
docs/GRL0/paper/ — Suggested edits for original paper
docs/GRL0/implementation/ — Implementation notes

Future:

docs/theory/ — Operator formalism theory (from Paper A)
docs/algorithms/ — Training algorithms (from Paper B)
docs/tutorials/ — Quick start guides

Private Development (`dev/`)¶

Current Work:

dev/GRL0/ — Private notes for GRL v0 development
dev/papers/ — Paper drafts (A, B, C)
dev/GRL_extensions/ — Extension ideas
dev/references/ — Original paper, related papers

Code Documentation¶

README files in src/grl/ subdirectories
Docstrings in code (NumPy style)
Tutorial notebooks in notebooks/
API reference (Sphinx, future)

Research Themes & Connections¶

Theme 1: Functional Learning¶

Across all work:

State as function \(Q^+ \in \mathcal{H}_k\)
Operators on function spaces
RKHS as mathematical foundation

Papers: GRL v0, Paper A, quantum-inspired extensions

Theme 2: Particle-Based Inference¶

Key insight: Weighted particles as basis for belief state

Papers: GRL v0, memory dynamics (Chapter 08)

Extensions:

Sparse approximations
Hierarchical particles
Nonparametric clustering

Theme 3: Energy-Based Learning¶

Key insight: Energy function \(E(z) = -Q^+(z)\) connects to physics

Papers: GRL v0 (Chapter 03), Paper A (least action principle)

Extensions:

Hamilton-Jacobi-Bellman PDEs
Conservative vector fields
Lagrangian mechanics for policy

Theme 4: Hierarchical Abstraction¶

Key insight: Concepts as subspaces in function space

Papers: GRL v0 Part II, concept projections (Chapter 05), MoE (Chapter 07)

Extensions:

Multi-scale representations
Transfer learning via shared basis
Compositional behaviors

Theme 5: Quantum-Inspired Probability¶

Key insight: Amplitude-based learning with phase and interference

Papers: Quantum-inspired chapters (01-04), potential standalone paper

Extensions:

Complex RKHS for RL
Born rule for action selection
Spectral methods for concept discovery

Strategic Applications: Demonstrating GRL's Generality¶

Goal: Show that GRL subsumes classical RL and applies to modern, commercially relevant problems.

Application 1: Recovering Classical RL 🔥 Critical for Adoption¶

Motivation: Researchers trust frameworks that generalize what they already know.

Objective: Demonstrate that Q-learning, DQN, PPO, SAC, RLHF are special cases of GRL.

Deliverables:

Document: Recovering Classical RL from GRL ✅ Complete
Implementation: Wrappers for Gym/Gymnasium environments
Validation: Reproduce classical results (±5% performance)
Q-learning on GridWorld
DQN on CartPole
SAC on Pendulum
PPO on continuous control

Timeline: Q2 2026

Impact:

Convinces classical RL researchers GRL is not alien
Provides clear migration path from classical to GRL
Enables GRL to leverage existing benchmarks

Application 2: RLHF for LLMs (Theoretical Connection + Future Direction)¶

Status: Theoretical connection established, implementation exploratory

Why This Matters:

Validation: RLHF (ChatGPT, Claude, Llama) is most commercially important RL application
Familiarity: Most ML researchers understand this problem
Generality: If GRL generalizes RLHF theoretically, it validates framework's breadth

Theoretical Formulation:

State: \(s_t\) = (prompt, partial response)
Action: \(\theta_t\) = token ID (discrete action space)
Augmented space: \((s_t, \theta_t)\)
Field: \(Q^+(s_t, \theta_t)\) = expected reward for token \(\theta_t\) in context \(s_t\)
Key insight: Standard RLHF (PPO) is GRL with discrete actions + neural network approximation

Documentation:

Recovering Classical RL from GRL — Section 6 covers RLHF

Potential Advantages (Theoretical):

Off-policy learning (replay buffer of human feedback)
Kernel generalization (transfer across prompts)
Uncertainty quantification (exploration where uncertain)
Interpretability (energy landscapes)

However: These are speculative without empirical validation.

Implementation Reality:

Challenges:

Infrastructure complexity (reward model, human feedback data)
Computational cost (expensive even for GPT-2)
Integration with existing tools (TRL, transformers, accelerate)
Validation difficulty (matching PPO requires extensive tuning)

Estimated Effort: 6-12 months of focused work with GPU resources

When to Pursue:

✅ After GRL validated on simpler environments
✅ If collaborators or funding available
✅ If clear path to demonstrating advantages

Realistic Alternative:

Write theoretical articles justifying the connection
Toy RLHF-like problem (not real LLM) as proof-of-concept
Wait for opportunities (industry collaboration, research grant)

Application 3: Additional Modern RL Domains¶

Prompt Optimization:

Parametric prompt generation (continuous in embedding space)
GRL learns smooth prompt space
Transfer across tasks

Molecule Design:

Parametric molecular operators
GRL discovers optimal molecules for drug properties
Physics-informed kernels

Neural Architecture Search:

Compositional architecture operators
GRL explores architecture space efficiently
Uncertainty-guided search

Timeline: Q4 2026+

Potential Novel Contributions (Publishable)¶

High-Priority Contributions¶

1. Amplitude-Based Reinforcement Learning 🔥 Top Priority

Novelty: First RL with complex-valued value functions
Venue: NeurIPS/ICML 2026-2027
Readiness: 30% (theory done, needs implementation)

2. Information-Theoretic Memory Consolidation

Novelty: MDL framework for experience replay
Venue: ICML/NeurIPS 2026-2027
Readiness: 40% (formulation clear, needs experiments)

3. Operator-Based GRL (Paper A)

Novelty: Actions as operators, not symbols
Venue: NeurIPS/ICML 2026
Readiness: 70% (draft complete, figures/experiments needed)

Medium-Priority Contributions¶

4. Theoretical Articles: Modern RL as Special Cases of GRL

Novelty: Justify that RLHF, prompt optimization, NAS, molecule design are GRL special cases
Venue: Blog posts, workshop papers, or sections in main papers
Readiness: 50% ("Recovering Classical RL" document provides template)
Impact: Demonstrates GRL's generality without requiring full implementations

5. Concept Subspaces for Hierarchical RL

Novelty: Rigorous RKHS subspace formalism
Venue: ICLR/AISTATS 2027
Readiness: 50% (math done, algorithms needed)

6. Surprise-Modulated Episodic Memory

Novelty: Bio-inspired consolidation
Venue: CogSci/Neural Computation 2027
Readiness: 60% (theory clear, needs validation)

7. Hybrid Neural-Particle RL

Novelty: Combining deep learning with GP memory
Venue: ICLR/ICML 2027
Readiness: 30% (concept clear, full implementation needed)

Strategic Applications (Future Possibilities, No Timeline)¶

8. GRL for LLM Fine-tuning (RLHF)

Novelty: Application of functional RL to most commercially important RL problem
Venue: ICLR/NeurIPS (if pursued)
Readiness: 20% (theoretical connection clear, implementation requires major resources)
Status: Exploratory — pursue only if collaborators/funding available
Alternative: Write theoretical articles + toy proof-of-concept

9. Other Strategic Applications

Prompt optimization, molecule design, neural architecture search
Status: Theoretical connections to be documented
Implementation: Pick 1-2 if resources available
Primary Value: Demonstrate GRL generalizes modern RL methods

Timeline Summary¶

2026 Q1 (January-March)¶

Focus: Complete GRL v0 documentation and baseline implementation

✅ Finish Part I tutorial chapters (07-10)
✅ Implement MemoryUpdate and RF-SARSA
🔄 Run first experiments
🔄 Complete Paper A figures and proofs

2026 Q2 (April-June)¶

Focus: Paper A submission, Classical RL recovery, scalable algorithms

Submit Paper A (April deadline)
Implement wrappers for Gym/Gymnasium (recover classical RL)
Reproduce DQN on CartPole (validation)
Reproduce SAC on Pendulum (validation)
Document: "Recovering Classical RL from GRL" 🔥 Strategic
Implement alternative learning mechanisms (Chapter 07)
Implement memory dynamics (Chapter 08)
Draft Paper B algorithms
Start benchmark experiments

2026 Q3 (July-September)¶

Focus: Paper B submission, novel contributions, extensions

Submit Paper B (June ICML or September NeurIPS)
Explore amplitude-based RL (if promising after Part I complete)
Implement MDL consolidation (principled memory dynamics)
Concept-based MoE (mixture of experts via subspaces)
Start operator framework implementation
Run application experiments for Paper C
Write theoretical articles: How RLHF/prompt-opt/NAS are special cases of GRL

2026 Q4 (October-December)¶

Focus: Paper C submission, novel contributions (amplitude/MDL)

Submit Paper C (CoRL deadline ~July)
Develop amplitude-based RL fully
Implement MDL consolidation
Draft standalone papers on extensions

2027+¶

Focus: Consolidate results, broader impact

Package releases and documentation
Workshop papers and tutorials
Integration with popular RL libraries
Real-world applications

Success Metrics¶

Short-Term (6 months)¶

Complete GRL v0 tutorial paper (Parts I & II)
Reference implementation working on 3+ environments
Submit Paper A to top venue
At least 10 GitHub stars

Medium-Term (12 months)¶

Paper A accepted or under review
Papers B & C submitted
2-3 additional papers on extensions (amplitude, MDL, concepts)
50+ GitHub stars, some external users

Long-Term (24+ months)¶

3+ papers published at top venues
GRL adopted by other researchers
Integration with popular libraries (Stable-Baselines3, RLlib)
Tutorial at major conference (NeurIPS, ICML)
Real-world deployment (robotics, control systems)

Open Questions & Research Opportunities¶

Theoretical Questions¶

Sample complexity: How does GRL compare to classical RL theoretically?
Convergence rates: Can we prove faster convergence in certain settings?
Operator algebra: What's the right group structure for operator composition?
Phase semantics: What should complex phase represent in amplitude-based RL?

Algorithmic Questions¶

Scalability: Best way to handle millions of particles?
Consolidation criterion: MDL vs. surprise-gating vs. other?
Mixture of experts: How to partition concept subspaces automatically?
Transfer learning: Can concept basis enable zero-shot transfer?

Application Questions¶

Best domains: Where does GRL shine vs. classical RL?
Interpretability: Can energy landscapes help explain decisions?
Safety: Can concept subspaces encode constraints?
Multi-agent: How to extend GRL to multi-agent settings?

Resources & References¶

Key Papers (Original Work)¶

Chiu & Huber (2022). Generalized Reinforcement Learning. arXiv:2208.04822.

Inspirations¶

Kernel Methods:

Rasmussen & Williams (2006). Gaussian Processes for Machine Learning.

Operator Learning:

Lu et al. (2021). Learning Nonlinear Operators via DeepONet. Nature Machine Intelligence.
Li et al. (2021). Fourier Neural Operator. ICLR.

Quantum-Inspired ML:

Cheng et al. (2018). Quantum Generative Adversarial Learning. PRL.
Havlíček et al. (2019). Supervised Learning with Quantum-Enhanced Feature Spaces. Nature.

Memory & Agent Systems:

Cao et al. (2024). Memory in the Age of AI Agents. arXiv:2512.13564.

Contact & Collaboration¶

Documentation: docs/
Code: src/grl/
Papers: dev/papers/
Issues: GitHub Issues (coming soon)

This roadmap is a living document and will be updated as research progresses.

Last Updated: January 14, 2026

GRL Research Roadmap¶

Current Focus: GRL v0 (Baseline)¶

Part I: Particle-Based Learning¶

Part II: Emergent Structure & Spectral Abstraction¶

Quantum-Inspired Extensions¶

Completed Chapters¶

Future Directions from These Chapters¶

GRL Extensions: Papers A, B, C¶

Paper A: Theoretical Foundations (Operator Framework)¶

Paper B: Algorithms & Implementation¶

Paper C: Empirical Applications¶

Implementation Roadmap¶

Environment Simulation Package¶

Implementation Roadmap¶

Phase 1: GRL v0 Baseline (Current)¶

Phase 2: Scalable Learning (March-April 2026)¶

Phase 3: Memory Dynamics (April-May 2026)¶

Phase 4: Operator Framework (May-July 2026)¶

Phase 5: Applications (August 2026+)¶

Documentation Structure¶

Public Documentation (docs/)¶

Private Development (dev/)¶

Code Documentation¶

Research Themes & Connections¶

Theme 1: Functional Learning¶

Theme 2: Particle-Based Inference¶

Theme 3: Energy-Based Learning¶

Theme 4: Hierarchical Abstraction¶

Theme 5: Quantum-Inspired Probability¶

Strategic Applications: Demonstrating GRL's Generality¶

Application 1: Recovering Classical RL 🔥 Critical for Adoption¶

Application 2: RLHF for LLMs (Theoretical Connection + Future Direction)¶

Application 3: Additional Modern RL Domains¶

Potential Novel Contributions (Publishable)¶

High-Priority Contributions¶

Medium-Priority Contributions¶

Strategic Applications (Future Possibilities, No Timeline)¶

Timeline Summary¶

2026 Q1 (January-March)¶

2026 Q2 (April-June)¶

2026 Q3 (July-September)¶

2026 Q4 (October-December)¶

2027+¶

Success Metrics¶

Short-Term (6 months)¶

Medium-Term (12 months)¶

Long-Term (24+ months)¶

Open Questions & Research Opportunities¶

Theoretical Questions¶

Algorithmic Questions¶

Application Questions¶

Resources & References¶

Key Papers (Original Work)¶

Inspirations¶

Contact & Collaboration¶

Public Documentation (`docs/`)¶

Private Development (`dev/`)¶