Policy Gradient
Policy Gradient
A policy-based reinforcement learning algorithm that directly optimizes the policy using gradient ascent on expected returns.
Family: Reinforcement Learning Status: 📋 Planned
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Overview
Policy Gradient methods are a class of reinforcement learning algorithms that directly optimize
the policy function using gradient ascent on the expected return. Unlike value-based methods like Q-Learning, policy gradient methods learn the policy directly without needing to learn value functions first.
These methods are particularly useful for continuous action spaces, stochastic policies, and scenarios where the optimal policy might be stochastic. They work by computing gradients of the expected return with respect to the policy parameters and updating the policy in the direction that increases expected return.
The most common policy gradient algorithm is REINFORCE, which uses the policy gradient theorem to derive unbiased gradient estimates from sampled trajectories.
Mathematical Formulation¶
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Problem Definition
Given:
- State space: S
- Action space: A
- Policy: π(a|s;θ) parameterized by θ
- Reward function: R(s,a,s')
- Discount factor: γ ∈ [0,1]
Find policy parameters θ that maximize expected return:
J(θ) = E[∑{t=0}^∞ γ^t R | π(·|·;θ)]
Using gradient ascent: θ ← θ + α ∇_θ J(θ)
Key Properties
Policy Gradient Theorem
∇_θ J(θ) = E[∑_{t=0}^∞ γ^t ∇_θ log π(a_t|s_t;θ) A^π(s_t,a_t)]
Gradient of expected return with respect to policy parameters
REINFORCE Update
θ ← θ + α ∑_{t=0}^T γ^t R_t ∇_θ log π(a_t|s_t;θ)
Policy parameter update using sampled returns
Baseline Subtraction
θ ← θ + α ∑_{t=0}^T γ^t (R_t - b(s_t)) ∇_θ log π(a_t|s_t;θ)
Reduces variance by subtracting baseline from returns
Key Properties¶
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Policy-Based
Directly optimizes the policy function
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Continuous Actions
Naturally handles continuous action spaces
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Stochastic Policies
Can learn stochastic optimal policies
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High Variance
Gradient estimates have high variance
Implementation Approaches¶
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Basic policy gradient algorithm using Monte Carlo returns
Complexity:
- Time: O(episodes × steps_per_episode × policy_forward_pass)
- Space: O(policy_parameters)
Advantages
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Directly optimizes policy
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Handles continuous action spaces naturally
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Can learn stochastic policies
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Theoretically sound convergence guarantees
Disadvantages
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High variance in gradient estimates
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Sample inefficient
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Slow convergence
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Requires complete episodes
Policy gradient with value function baseline for variance reduction
Complexity:
- Time: O(episodes × steps_per_episode × (policy_forward_pass + value_forward_pass))
- Space: O(policy_parameters + value_parameters)
Advantages
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Reduces variance compared to REINFORCE
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More sample efficient
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Can learn online (no need for complete episodes)
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Combines benefits of policy and value methods
Disadvantages
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More complex implementation
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Requires tuning of two networks
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Can be unstable during training
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Still has higher variance than value-based methods
Complete Implementation
The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:
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Main implementation with REINFORCE and Actor-Critic variants:
src/algokit/reinforcement_learning/policy_gradient.py -
Comprehensive test suite including convergence tests:
tests/unit/reinforcement_learning/test_policy_gradient.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| REINFORCE | O(episodes × steps_per_episode × policy_forward_pass) | O(policy_parameters) | Time complexity dominated by policy network forward passes. Space complexity includes policy parameters only |
Use Cases & Applications¶
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Application Categories
Continuous Control
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Robot Control: Learning continuous control policies for robots
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Autonomous Vehicles: Learning steering and acceleration policies
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Game Playing: Learning continuous control in games
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Physics Simulation: Learning control policies in physics engines
Stochastic Environments
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Financial Trading: Learning stochastic trading policies
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Resource Allocation: Learning allocation policies under uncertainty
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Game Theory: Learning mixed strategies in games
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Multi-Agent Systems: Learning policies in competitive environments
High-Dimensional Action Spaces
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Robotics: Learning control for high-DOF robots
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Animation: Learning motion policies for characters
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Music Generation: Learning policies for music composition
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Text Generation: Learning policies for text generation
Real-World Applications
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Recommendation Systems: Learning stochastic recommendation policies
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Ad Placement: Learning placement policies with uncertainty
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Energy Management: Learning energy allocation policies
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Traffic Control: Learning traffic light control policies
Educational Value
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Policy-Based Methods: Understanding direct policy optimization
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Gradient Methods: Understanding gradient-based optimization in RL
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Variance Reduction: Understanding techniques to reduce variance
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Continuous Control: Understanding continuous action spaces
Educational Value
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Policy-Based Methods: Understanding direct policy optimization approaches
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Gradient Methods: Learning gradient-based optimization in RL
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Variance Reduction: Understanding techniques to reduce variance in estimates
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Continuous Control: Understanding how to handle continuous action spaces
References & Further Reading¶
:material-book: Core Textbooks
:material-library: Policy Gradient
:material-library: Actor-Critic
:material-web: Online Resources
:material-code-tags: Implementation & Practice
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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Navigation¶
Related Algorithms in Reinforcement Learning:
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SARSA - An on-policy temporal difference learning algorithm that learns action-value functions by following the current policy.
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Actor-Critic - A hybrid reinforcement learning algorithm that combines policy gradient methods with value function approximation for improved learning efficiency.
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Q-Learning - A model-free reinforcement learning algorithm that learns optimal action-value functions through temporal difference learning.
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Proximal Policy Optimization (PPO) - A state-of-the-art policy gradient algorithm that uses clipped objective to ensure stable policy updates with improved sample efficiency.
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Deep Q-Network (DQN) - A deep reinforcement learning algorithm that uses neural networks to approximate Q-functions for high-dimensional state spaces.