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Hierarchical Policy Gradient

Hierarchical Policy Gradient

Extends traditional policy gradient methods to handle temporal abstraction and hierarchical task decomposition with multi-level policies.

Family: Hierarchical Reinforcement Learning Status: ๐Ÿ“‹ Planned

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Overview

Hierarchical Policy Gradient extends the traditional policy gradient framework to handle temporal

abstraction and hierarchical task decomposition. The algorithm learns policies at multiple levels: a high-level meta-policy that selects subgoals or options, and low-level policies that execute primitive actions to achieve these subgoals.

This hierarchical approach enables the agent to solve complex, long-horizon tasks by breaking them down into manageable subproblems. The meta-policy learns to sequence subgoals effectively, while the low-level policies learn to achieve specific subgoals efficiently. Hierarchical Policy Gradient is particularly powerful in domains where tasks have natural hierarchical structure, such as robotics manipulation, navigation, and game playing.

Mathematical Formulation

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Problem Definition

Given:

  • State space: S
  • Subgoal space: G
  • Action space: A
  • Meta-policy: ฯ€_meta(g_t|s_t)
  • Low-level policy: ฯ€_low(a_t|s_t, g_t)
  • Reward function: R(s,a,s')

Find hierarchical policies that maximize expected cumulative reward:

ฯ€_h(a_t|s_t) = โˆ‘_{g_t} ฯ€_meta(g_t|s_t) ยท ฯ€_low(a_t|s_t, g_t)

Key Properties

Hierarchical Policy Gradient Theorem

โˆ‡_ฮธ J(ฮธ) = E_{ฯ„ ~ ฯ€_h}[โˆ‘_{t=0}^T โˆ‡_ฮธ log ฯ€_meta(g_t|s_t) R(ฯ„) + โˆ‡_ฮธ log ฯ€_low(a_t|s_t, g_t) R(ฯ„)]

Policy gradient decomposes into meta and low-level components

Hierarchical Advantage Function

A_h(s_t, g_t, a_t) = Q_h(s_t, g_t, a_t) - V_h(s_t)

Advantage function for hierarchical policy updates

Meta-Policy Update

โˆ‡_ฮธ J_meta = E[โˆ‘_t โˆ‡_ฮธ log ฯ€_meta(g_t|s_t) A_meta(s_t, g_t)]

Gradient update for meta-policy

Key Properties

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  • Temporal Abstraction

    High-level policies operate over longer time horizons

  • Subgoal Decomposition

    Complex tasks broken into manageable subproblems

  • Hierarchical Learning

    Policies at different levels learn simultaneously

  • Transfer Learning

    Low-level policies can be reused across different tasks

Implementation Approaches

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Standard hierarchical policy gradient with meta and low-level policy networks

Complexity:

  • Time: O(batch_size ร— (meta_params + low_params))
  • Space: O(batch_size ร— (state_size + subgoal_size))

Advantages

  • Extends familiar policy gradient framework to hierarchical settings

  • Temporal abstraction enables learning at different time scales

  • Subgoal decomposition makes complex tasks manageable

  • Transfer learning allows reuse of low-level policies

Disadvantages

  • Requires careful coordination between meta and low-level policies

  • Subgoal achievement detection can be challenging

  • Two networks increase complexity and training time

  • Policy gradient methods can have high variance

Complete Implementation

The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:

Complexity Analysis

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Time & Space Complexity Comparison

Approach Time Complexity Space Complexity Notes
Basic Hierarchical Policy Gradient O(batch_size ร— (meta_params + low_params)) O(batch_size ร— (state_size + subgoal_size)) Two-network architecture requires coordination and careful training

Use Cases & Applications

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Application Categories

Robotics and Control

  • Robot Manipulation: Complex manipulation tasks with hierarchical subgoals

  • Autonomous Navigation: Multi-level navigation with waypoint subgoals

  • Industrial Automation: Process control with hierarchical objectives

  • Swarm Robotics: Coordinated behavior with hierarchical task decomposition

Game AI and Strategy

  • Strategy Games: Multi-level decision making with tactical and strategic goals

  • Puzzle Games: Complex puzzles broken into simpler subproblems

  • Adventure Games: Quest completion with hierarchical objectives

  • Simulation Games: Resource management with hierarchical planning

Real-World Applications

  • Autonomous Vehicles: Multi-level driving with navigation and control subgoals

  • Healthcare: Treatment planning with hierarchical medical objectives

  • Finance: Portfolio management with hierarchical investment strategies

  • Network Control: Traffic management with hierarchical routing policies

Educational Value

  • Hierarchical Learning: Understanding multi-level decision making

  • Subgoal Decomposition: Learning to break complex tasks into simpler parts

  • Temporal Abstraction: Understanding different time scales in learning

  • Transfer Learning: Learning reusable skills across different tasks

Educational Value

  • Hierarchical Learning: Perfect introduction to multi-level decision making

  • Subgoal Decomposition: Shows how to break complex tasks into manageable parts

  • Temporal Abstraction: Demonstrates learning at different time scales

  • Transfer Learning: Illustrates how skills can be reused across tasks

References & Further Reading

:material-library: Core Papers

:material-file-document:
Foundational work on hierarchical reinforcement learning
:material-file-document:
Hierarchical reinforcement learning with value function decomposition

:material-book: Hierarchical RL Textbooks

:material-file-document:
Comprehensive introduction to reinforcement learning including hierarchical methods
:material-file-document:
Algorithms for reinforcement learning with hierarchical approaches

:material-web: Online Resources

:material-link:
Hierarchical Reinforcement Learning
Wikipedia article on hierarchical reinforcement learning
:material-link:
Policy Gradient Methods
OpenAI Spinning Up tutorial on policy gradient methods

:material-code-tags: Implementation & Practice

:material-link:
PyTorch Documentation
PyTorch deep learning framework documentation
:material-link:
OpenAI Gym
RL environments for testing hierarchical algorithms
:material-link:
Stable Baselines3
High-quality RL algorithm implementations

Interactive Learning

Try implementing the different approaches yourself! This progression will give you deep insight into the algorithm's principles and applications.

Pro Tip: Start with the simplest implementation and gradually work your way up to more complex variants.

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Related Algorithms in Hierarchical Reinforcement Learning:

  • Hierarchical Q-Learning - Extends traditional Q-Learning to handle temporal abstraction and hierarchical task decomposition with multi-level Q-functions.

  • Hierarchical Task Networks (HTNs) - A hierarchical reinforcement learning approach that decomposes complex tasks into hierarchical structures of subtasks for planning and execution.

  • Option-Critic - A hierarchical reinforcement learning algorithm that learns options (temporally extended actions) end-to-end using policy gradient methods.

  • Hierarchical Actor-Critic (HAC) - An advanced hierarchical reinforcement learning algorithm that extends the actor-critic framework with temporal abstraction and hierarchical structure.

  • Feudal Networks (FuN) - A hierarchical reinforcement learning algorithm that implements a manager-worker architecture for temporal abstraction and goal-based learning.