Edition 23: Introduction to Policy Gradient Methods

Dear RL Enthusiasts,

Welcome back to RL Zone!

In this series, we will continue to explore reinforcement learning (RL) concepts guided by the great textbook Reinforcement Learning: An Introduction by Richard S. Sutton and Andrew G. Barto.

Summary of previous edition

In the last edition, we explored various off-policy learning techniques with function approximation, discussing how agents can achieve stable learning in large, complex environments.

Edition 23: Introduction to Policy Gradient Methods

In this edition, we shift our focus to policy gradient methods, a powerful class of algorithms where agents learn policies directly through gradient-based optimization. This chapter opens up new possibilities for solving complex reinforcement learning problems, particularly in environments with continuous state or action spaces.

What Are Policy Gradient Methods?

In reinforcement learning, policy gradient methods directly optimize the policy by adjusting its parameters using the gradient of the expected reward. Unlike value-based methods such as Q-learning, which learn a value function and then derive a policy from it, policy gradient methods operate directly in the space of policies. This makes them particularly useful for problems where the policy cannot be easily derived from a value function, such as tasks with continuous action spaces.

A policy π(a ∣ s, θ) is a probability distribution over actions given a state, parameterized by θ. The goal of policy gradient methods is to find the optimal parameters θ that maximize the expected return, typically defined as:

J(θ) = E?? [G?]

Where:

• G? is the return (the sum of discounted rewards starting from time t).
• The expectation is taken over the trajectory of states and actions under policy π(a ∣ s, θ).

Gradient Ascent on Expected Return

The core idea of policy gradient methods is to perform gradient ascent on the expected return J(θ), by adjusting the policy parameters in the direction that increases the expected return. The policy parameters θ are updated according to the gradient of the expected return:

θ ← θ + α ?? J(θ)

Where:

• α is the learning rate.
• ?? J(θ) is the gradient of the expected return with respect to the policy parameters θ.

By following the gradient, the agent continually improves its policy, increasing the likelihood of actions that lead to higher returns.

Policy Gradient Theorem

The policy gradient theorem provides the foundation for computing the gradient of the expected return. It states that the gradient of the expected return can be expressed as:

?? J(θ) = E?? [?? log π(a ∣ s, θ) q?(s, a)]

This theorem shows that the gradient of the expected return depends on two key components:

1. The log-likelihood gradient ?? log π(a ∣ s, θ) captures how the probability of selecting an action changes with respect to the policy parameters.
2. The action-value function qπ(s,a), which represents the expected return from taking action a in state s and following policy π thereafter.

This formulation simplifies the process of calculating the gradient, allowing agents to update their policies in a way that directly maximizes the expected return.

REINFORCE Algorithm

One of the simplest and most well-known policy gradient algorithms is REINFORCE. The REINFORCE algorithm uses the policy gradient theorem to update the policy parameters based on the return from sampled trajectories.

The update rule for REINFORCE is:

θ ← θ + α ?? log π(a? ∣ s?, θ) G?

Where:

• G? is the actual return observed after taking action a? in state s?.
• By scaling the gradient of the log-likelihood by the return G?, REINFORCE increases the probability of actions that lead to higher rewards.

Variance Reduction with Baseline

While REINFORCE is a powerful and simple algorithm, it can suffer from high variance, particularly when the returns G? vary widely. To reduce the variance of the updates, policy gradient methods often introduce a baseline b(s), which is a function of the state used to normalize the return. The baseline does not change the expected value of the gradient but helps to reduce variance in practice.

The update rule with a baseline becomes:

θ ← θ + α ?? log π(a? ∣ s?, θ) (G? ? b(s?))

A common choice for the baseline is the state-value function vπ(s), which represents the expected return from state s under policy π. By using vπ(s) as the baseline, the update focuses on improving actions that lead to better-than-expected outcomes, relative to the baseline performance of the policy.

Actor-Critic Methods

Actor-critic methods combine the strengths of policy gradient and value-based methods. In these methods, the actor is responsible for updating the policy parameters, while the critic estimates the value function (either the state-value function vπ(s) or the action-value function qπ(s, a)) to provide a baseline for variance reduction.

The actor-critic architecture is advantageous because it allows for more stable updates, as the critic provides the actor with a smoother estimate of the return. The update rule for the actor-critic method is:

1. Critic update: The critic updates its estimate of the value function based on the observed return.
2. Actor update: The actor updates the policy parameters using the gradient of the log-likelihood, scaled by the advantage A(s?, a?) = G? ? vπ(s?), where the advantage represents how much better the action performed compared to the expected value.

The actor-critic method helps address the high variance issue present in REINFORCE, leading to more stable and efficient learning.

Advantages of Policy Gradient Methods

Policy gradient methods offer several advantages over value-based methods:

1. Direct Policy Optimization: By directly optimizing the policy, policy gradient methods can handle problems with continuous or high-dimensional action spaces, where deriving a policy from a value function would be difficult or impractical.
2. Stochastic Policies: Policy gradient methods naturally allow for stochastic policies, which are important for exploration in environments where a deterministic policy may not perform well.
3. Smooth Updates: The use of gradient ascent ensures that the policy is updated gradually, reducing the risk of large, destabilizing updates.

However, policy gradient methods also come with challenges, such as high variance and the need for careful tuning of hyperparameters, such as the learning rate.

Summary

In this edition, we introduced policy gradient methods, a class of algorithms that directly optimize policies by performing gradient ascent on the expected return. We discussed the policy gradient theorem, which provides a foundation for calculating policy updates, and explored the REINFORCE algorithm, one of the simplest policy gradient methods. We also introduced techniques for variance reduction, such as using a baseline, and explored actor-critic methods, which combine policy optimization with value-based learning for more stable updates.

In the next edition of RL Zone, we will continue exploring Chapter 12, focusing on more advanced policy gradient methods, including natural policy gradients and techniques for improving the stability and efficiency of policy optimization.

Stay tuned for more insights into the fascinating world of reinforcement learning!

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