Practice: Implementing REINFORCE

Alright, let's put theory into practice and implement the REINFORCE algorithm. As we discussed, REINFORCE (also known as Monte Carlo Policy Gradient) works by directly adjusting the parameters $\theta$ of our policy $\pi_\theta(a|s)$ to maximize expected future rewards. The core idea is to sample complete episodes, calculate the return $G_t$ for each step, and then update the policy parameters in the direction suggested by the policy gradient theorem. Specifically, we want to increase the log probability of actions taken at state $s_t$ if they led to a high return $G_t$, and decrease it otherwise.

The update rule, in its simplest form (without a baseline), looks like this:

$\theta \leftarrow \theta + \alpha \nabla_\theta \log \pi_\theta(a_t|s_t) G_t$

Here, $\alpha$ is the learning rate, $\nabla_\theta \log \pi_\theta(a_t|s_t)$ is the gradient of the log probability of the action taken, and $G_t$ is the return from time step $t$ onwards.

We'll use the classic CartPole-v1 environment from the Gymnasium library. In this environment, the goal is to balance a pole on a cart by moving the cart left or right. The state is represented by four continuous values (cart position, cart velocity, pole angle, pole angular velocity), and the action space is discrete (0 for left, 1 for right). This is a good fit for REINFORCE because the policy needs to map continuous states to discrete actions.

Setting Up the Environment and Policy Network

First, ensure you have Gymnasium and a deep learning library like PyTorch installed.

import gymnasium as gym
import numpy as np
import torch
import torch.nn as nn
import torch.optim as optim
import torch.nn.functional as F
from collections import namedtuple, deque
import matplotlib.pyplot as plt

# Check if GPU is available
device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
print(f"Using device: {device}")

# Create the environment
env = gym.make('CartPole-v1')

# Get state and action dimensions
state_dim = env.observation_space.shape[0]
action_dim = env.action_space.n

print(f"State dimensions: {state_dim}")
print(f"Action dimensions: {action_dim}")

Now, let's define our policy network. A simple Multi-Layer Perceptron (MLP) should suffice for CartPole. This network will take the state as input and output the probabilities for each possible action (left or right).

class PolicyNetwork(nn.Module):
    def __init__(self, state_dim, action_dim, hidden_dim=128):
        super(PolicyNetwork, self).__init__()
        self.fc1 = nn.Linear(state_dim, hidden_dim)
        self.fc2 = nn.Linear(hidden_dim, action_dim)

    def forward(self, state):
        x = F.relu(self.fc1(state))
        # Output raw scores (logits) for actions
        action_scores = self.fc2(x)
        # Convert scores to probabilities using softmax
        # Use dim=-1 to apply softmax across the action dimension
        return F.softmax(action_scores, dim=-1)

# Instantiate the policy network and optimizer
policy_net = PolicyNetwork(state_dim, action_dim).to(device)
optimizer = optim.Adam(policy_net.parameters(), lr=1e-3) # Learning rate

The REINFORCE Training Loop

The core of REINFORCE involves interacting with the environment to collect trajectories (sequences of states, actions, and rewards) and then using these trajectories to update the policy network.

1. Collect Trajectories: Run one or more complete episodes using the current policy. For each step, store the state, the action taken, and the reward received.
2. Calculate Returns: For each step $t$ in an episode, calculate the discounted return $G_t = \sum_{k=t}^{T} \gamma^{k-t} r_k$, where $T$ is the episode length and $\gamma$ is the discount factor (often set to 0.99 or 1.0 for episodic tasks).
3. Compute Policy Loss: Calculate the loss term for the episode. This is typically the negative sum of $\log \pi_\theta(a_t|s_t) G_t$ over all steps $t$. We use the negative because optimizers perform gradient descent, while we want to perform gradient ascent on the expected return.
4. Update Policy Network: Perform backpropagation on the loss and update the network parameters $\theta$ using the optimizer.

Let's structure the training loop:

# Define a structure to store trajectory data
SavedAction = namedtuple('SavedAction', ['log_prob', 'value']) # Value here means Return G_t

def select_action(state):
    """Selects an action based on the policy network's output probabilities."""
    state = torch.from_numpy(state).float().unsqueeze(0).to(device)
    probs = policy_net(state)
    # Create a categorical distribution over the list of probabilities of actions
    m = torch.distributions.Categorical(probs)
    # Sample an action from the distribution
    action = m.sample()
    # Store the log probability of the sampled action
    policy_net.saved_action = SavedAction(m.log_prob(action), 0) # Placeholder for G_t
    return action.item()

def calculate_returns(rewards, gamma=0.99):
    """Calculates discounted returns for an episode."""
    R = 0
    returns = []
    # Iterate backwards through the rewards
    for r in reversed(rewards):
        R = r + gamma * R
        returns.insert(0, R) # Prepend to maintain order
    # Normalize returns (optional but often helpful)
    returns = torch.tensor(returns, device=device)
    returns = (returns - returns.mean()) / (returns.std() + np.finfo(np.float32).eps.item())
    return returns

def finish_episode(episode_rewards):
    """Performs the REINFORCE update at the end of an episode."""
    policy_loss = []
    returns = calculate_returns(episode_rewards)

    # Retrieve saved log probabilities and associate them with calculated returns
    for (log_prob, _), G_t in zip(policy_net.all_saved_actions, returns):
         # REINFORCE objective: maximize log_prob * G_t
         # Loss is the negative objective
        policy_loss.append(-log_prob * G_t)

    # Sum the losses for the episode
    optimizer.zero_grad() # Reset gradients
    loss = torch.stack(policy_loss).sum() # Combine losses
    loss.backward() # Compute gradients
    optimizer.step() # Update network weights

    # Clear saved actions for the next episode
    del policy_net.all_saved_actions[:]


# Training parameters
num_episodes = 1000
gamma = 0.99
log_interval = 50 # Print status every 50 episodes
max_steps_per_episode = 1000 # Prevent excessively long episodes

all_episode_rewards = []
episode_durations = []

# Main training loop
for i_episode in range(num_episodes):
    state, _ = env.reset()
    episode_rewards = []
    policy_net.all_saved_actions = [] # Store log_probs for the episode

    for t in range(max_steps_per_episode):
        action = select_action(state)
        next_state, reward, terminated, truncated, _ = env.step(action)
        done = terminated or truncated

        policy_net.all_saved_actions.append(policy_net.saved_action)
        episode_rewards.append(reward)
        state = next_state

        if done:
            break

    # Episode finished, update policy
    finish_episode(episode_rewards)

    # Logging
    total_reward = sum(episode_rewards)
    all_episode_rewards.append(total_reward)
    episode_durations.append(t + 1)

    if i_episode % log_interval == 0:
        avg_reward = np.mean(all_episode_rewards[-log_interval:])
        print(f'Episode {i_episode}\tAverage Reward (last {log_interval}): {avg_reward:.2f}\tLast Duration: {t+1}')

    # Optional: Stop training if the problem is solved
    # CartPole-v1 is considered solved if avg reward > 475 over 100 consecutive episodes
    if len(all_episode_rewards) > 100:
         if np.mean(all_episode_rewards[-100:]) > 475:
              print(f'\nSolved in {i_episode} episodes!')
              break

env.close()

Visualizing Results

After training, it's useful to plot the rewards per episode to see if the agent learned effectively.

# Plotting the results
plt.figure(figsize=(12, 6))
plt.plot(all_episode_rewards)
plt.title('Episode Rewards over Time (REINFORCE)')
plt.xlabel('Episode')
plt.ylabel('Total Reward')

# Calculate and plot rolling average
rolling_avg = np.convolve(all_episode_rewards, np.ones(100)/100, mode='valid')
plt.plot(np.arange(99, len(all_episode_rewards)), rolling_avg, label='100-episode rolling average', color='orange')
plt.legend()
plt.grid(True)
plt.show()

{"data":[{"type":"scatter","mode":"lines","name":"Episode Reward","x":[i for i in range(len(all_episode_rewards))], "y": all_episode_rewards, "line":{"color": "#339af0"}}, {"type":"scatter","mode":"lines","name":"100-episode Rolling Average","x":[i for i in range(99, len(all_episode_rewards))], "y": list(np.convolve(all_episode_rewards, np.ones(100)/100, mode='valid')), "line":{"color": "#fd7e14"}}], "layout":{"title":"Episode Rewards over Time (REINFORCE)", "xaxis":{"title":"Episode"}, "yaxis":{"title":"Total Reward"}, "template":"plotly_white"}}

Plot showing the total reward obtained in each episode during training, along with a 100-episode rolling average to visualize the learning trend.

Discussion

This implementation demonstrates the basic REINFORCE algorithm. You should observe the agent's performance gradually improving, indicated by longer episode durations and higher total rewards.

• High Variance: One characteristic you might notice (or experience if you run this multiple times) is the high variance in learning. Sometimes REINFORCE learns quickly, other times it might struggle or take longer. This is because the gradient estimate relies on Monte Carlo sampling (full episodes), which can be noisy.
• Importance of Normalization: Normalizing the returns (returns = (returns - returns.mean()) / (returns.std() + eps)) is a common technique that helps stabilize training. It centers the returns around zero, meaning actions leading to above-average returns increase in probability, while those leading to below-average returns decrease.
• Baseline: As mentioned earlier in the chapter, subtracting a baseline (like the state value estimate $V(s_t)$) from the return $G_t$ can significantly reduce variance without introducing bias. The update would involve $(G_t - b(s_t))$ instead of just $G_t$. Implementing this would typically lead to an Actor-Critic method, where one network (the critic) estimates the baseline $V(s_t)$ and another (the actor) updates the policy based on the advantage $G_t - V(s_t)$.

This hands-on example provides a foundation for understanding and implementing policy gradient methods. Feel free to experiment with different hyperparameters (learning rate, network architecture, discount factor) or try implementing a baseline to see how it affects performance.