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Hierarchical Q-Learning

Hierarchical Q-Learning

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

Family: Hierarchical Reinforcement Learning Status: 📋 Planned

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Overview

Hierarchical Q-Learning extends the traditional Q-Learning framework to handle temporal abstraction

and hierarchical task decomposition. The algorithm learns Q-functions at multiple levels: a high-level Q-function that estimates the value of subgoals, and low-level Q-functions that estimate the value of actions given specific subgoals.

This hierarchical approach enables the agent to solve complex, long-horizon tasks by breaking them down into manageable subproblems. The high-level Q-function learns to sequence subgoals effectively, while the low-level Q-functions learn to achieve specific subgoals efficiently. Hierarchical Q-Learning 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 Q-function: Q_meta(s, g)
  • Low-level Q-function: Q_low(s, g, a)
  • Reward function: R(s,a,s')

Find hierarchical Q-functions that maximize expected cumulative reward:

Q_h(s_t, g_t, a_t) = Q_meta(s_t, g_t) + Q_low(s_t, g_t, a_t)

Key Properties

Hierarchical Q-Function Decomposition

Q_h(s_t, g_t, a_t) = Q_meta(s_t, g_t) + Q_low(s_t, g_t, a_t)

Q-function decomposes into meta and low-level components

Hierarchical Q-Learning Update

Q_h(s_t, g_t, a_t) ← Q_h(s_t, g_t, a_t) + α[r_t + γ max_{g'} Q_meta(s_{t+1}, g') - Q_h(s_t, g_t, a_t)]

Update rule for hierarchical Q-functions

Subgoal Selection

g_t = argmax_g Q_meta(s_t, g)

Subgoal selection based on meta Q-function

Key Properties

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  • Temporal Abstraction

    High-level Q-functions operate over longer time horizons

  • Subgoal Decomposition

    Complex tasks broken into manageable subproblems

  • Hierarchical Learning

    Q-functions at different levels learn simultaneously

  • Transfer Learning

    Low-level Q-functions can be reused across different tasks

Implementation Approaches

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Standard hierarchical Q-Learning with meta and low-level Q-tables

Complexity:

  • Time: O(|S| × |G| × |A| × episodes)
  • Space: O(|S| × |G| × |A|)

Advantages

  • Extends familiar Q-Learning 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 Q-functions

Disadvantages

  • Requires discrete state-action spaces

  • Memory requirements grow with state and subgoal space sizes

  • Subgoal achievement detection can be challenging

  • Coordination between meta and low-level Q-functions is complex

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 Q-Learning O( S ×

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 with MAXQ decomposition
: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:
MAXQ Value Function Decomposition
Original MAXQ paper and implementation details

:material-code-tags: Implementation & Practice

:material-link:
OpenAI Gym
RL environments for testing hierarchical algorithms
:material-link:
Stable Baselines3
High-quality RL algorithm implementations
:material-link:
Ray RLlib
Scalable RL library for production use

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 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.

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

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