TopoGRPO: Topological Group Relative Policy Optimization

Extending Policy Optimization through Topological Decomposition

A Novel Approach to Reinforcement Learning in Complex Environments

Motivation

Traditional reinforcement learning algorithms assume uniform treatment of the state-action space, which can be inefficient in complex environments.

Bellman Optimality Equation:

$$V^*(s) = \max_{a \in \mathcal{A}} \left( R(s,a) + \gamma \sum_{s' \in \mathcal{S}} P(s'|s,a) V^*(s') \right)$$

This formulation implicitly assumes the decision process is topologically trivial—that any path between states is equivalent to any other.

Key Insight: Complex environments may have non-trivial topological structure that requires multiple local policies coordinated by a meta-policy.

Theoretical Foundation

Topological Complexity

The topological complexity TC(X) of a space X is the minimum number of local sections needed to cover the space.

If TC(X) = 1: Single global policy sufficient

If TC(X) > 1: Multiple local policies required

Cohomological Obstructions

Memory requirements in partially observable environments correspond to non-vanishing cohomology groups:

$$H^k(\mathcal{C}(M)) \neq 0 \implies k\text{-step memory requirement}$$

where $\mathcal{C}(M)$ is the transition complex of the POMDP $M$.

TopoGRPO Algorithm

Algorithm: TOPO-GRPO

Input: Environment, iterations N, trajectories T, discount γ

  1. Initialize local policy π_θ, meta-policy π_θ₀, embedder φ
  2. For iteration = 1 to N:
  3.     Collect trajectories using current policies
  4.     Embed history sequences: X = φ(histories)
  5.     Compute charts via clustering: C = DBSCAN(X)
  6.     Assign charts to trajectories
  7.     Compute localized advantages within charts
  8.     Update policies using combined loss function
  9. Return: Trained policies π_θ, π_θ₀

Core Components

  • Local Policy π_θ(a|s,c): Maps states and chart assignments to action distributions. Learns specialized behaviors for different regions of the state space.
  • Meta-Policy π_θ₀(c|x): Selects appropriate charts based on embedded history sequences. Provides global coordination between local behaviors.
  • History Embedder φ(h): Neural network (LSTM/Transformer) that maps sequences of (state, action, reward) tuples to fixed-size embeddings.
  • Chart Computer: DBSCAN clustering algorithm that discovers topological structure in the embedded experience manifold.

Topological Decomposition

Chart 1
Chart 2
Chart 3

Chart Discovery Process

  1. Extract overlapping history windows from collected trajectories
  2. Embed each window using neural history embedder φ
  3. Apply DBSCAN clustering to discover natural groupings
  4. Assign chart labels to transitions based on clustering results
  5. Handle noise points by assigning to separate chart

This process automatically discovers the underlying topological structure of the experience manifold without manual specification.

Localized Advantage Computation

Standard Approach: Global normalization

$$\hat{A}_t = \frac{G_t - \mu_{global}}{\sigma_{global}}$$

TopoGRPO Approach: Chart-localized normalization

$$\hat{A}_t^{(c)} = \frac{G_t - \mu_c}{\sigma_c}$$

where μ_c and σ_c are computed using only returns from chart c.

Advantage: By normalizing within topological regions, we exploit the assumption that reward structure is locally simpler, leading to more stable advantage estimates with reduced variance.

Combined Loss Function

$$\mathcal{L}_{total} = \mathcal{L}_{local} + \alpha \mathcal{L}_{meta} + \beta \mathcal{L}_{KL}$$

Local Policy Loss (PPO)

$$\mathcal{L}_{local} = -\mathbb{E}\left[\min\left(r_t(\theta)\hat{A}_t^{(c)}, \text{clip}(r_t(\theta), 1-\epsilon, 1+\epsilon)\hat{A}_t^{(c)}\right)\right]$$

Meta-Policy Loss (REINFORCE)

$$\mathcal{L}_{meta} = -\mathbb{E}\left[\hat{A}_t^{(c)} \log \pi_{\theta_0}(c_t|x_t)\right]$$

KL Divergence Penalty

$$\mathcal{L}_{KL} = \mathbb{E}\left[D_{KL}(\pi_\theta(\cdot|s,c) \| \pi_{ref}(\cdot|s,c))\right]$$

Implementation Details

Network Architectures

  • Local Policy: Feed-forward network with state and chart embeddings as input
  • Meta-Policy: Feed-forward network mapping history embeddings to chart probabilities
  • History Embedder: LSTM with 2 layers, hidden dimension 256

Training Parameters

# Hyperparameters gamma = 0.99 # Discount factor alpha = 0.3 # Meta-policy loss weight beta = 0.01 # KL divergence weight eps_clip = 0.2 # PPO clipping parameter lr_local = 1e-4 # Local policy learning rate lr_meta = 3e-4 # Meta-policy learning rate chart_eps = 0.4 # DBSCAN epsilon chart_min_samples = 3 # DBSCAN minimum samples

Experimental Results

CartPole-v1 Environment

Method Average Reward Standard Deviation Improvement
Random Policy 17.8 5.64 Baseline
TopoGRPO 44.4 21.04 2.5x

Chart Discovery Statistics

  • Charts Discovered: 1-4 charts dynamically during training
  • Noise Points: Effectively handled by separate chart assignment
  • Training Stability: No NaN issues with proper gradient clipping

Computational Complexity

Phase Complexity Notes
Data Collection O(T × S × H) T trajectories, S steps, H history length
History Embedding O(W × H × d) W windows, H history, d embedding dim
Chart Computation O(W²) DBSCAN clustering bottleneck
Advantage Computation O(N) Linear in number of transitions
Policy Updates O(B × E) Batch size × epochs

Overall Complexity: O(I × W²) where I is iterations and W is number of history windows.

Future Research Directions

Algorithmic Improvements

  • Prime-Number Hashing: O(log n) topological sketching for scalability
  • Persistent Homology: Track topological features across multiple scales
  • Hyperbolic Meta-Policies: Natural representation of hierarchical structures

Theoretical Developments

  • Dixmier-Douady Loss: Gerbe-theoretic policy consistency guarantees
  • Convergence Analysis: Formal guarantees for topological decomposition
  • Sample Complexity: Bounds for chart discovery and policy learning

Applications

  • Continuous Control: Extension to continuous action spaces
  • Multi-Agent Systems: Topological coordination mechanisms
  • Foundation Models: Application to large-scale language model training

Conclusion

TopoGRPO represents a novel approach to reinforcement learning that:

  • Automatically discovers topological structure in complex environments
  • Learns specialized policies for different regions of the state space
  • Provides global coordination through meta-policy mechanisms
  • Achieves improved performance through localized advantage computation
Key Contribution: This work demonstrates that complex reinforcement learning problems are not merely "larger versions of simple problems," but possess intrinsic geometric structure that can be exploited for more efficient learning.

The integration of topological principles with policy optimization opens new avenues for understanding and solving complex sequential decision problems.

Thank You

Questions?

Implementation: Available in PyTorch

Paper: "On the Topological Obstructions of Bellman Recursion"