Expected SARSA in Reinforcement Learning

Prerequisites: SARSA

SARSA and Q-Learning technique in Reinforcement Learning are algorithms that uses Temporal Difference(TD) Update to improve the agent’s behaviour. Expected SARSA technique is an alternative for improving the agent’s policy. It is very similar to SARSA and Q-Learning, and differs in the action value function it follows.
We know that SARSA is an on-policy techique, Q-learning is an off-policy technique, but Expected SARSA can be use either as an on-policy or off-policy. This is where Expected SARSA is much more flexible compared to both these algorithms.

Let’s compare the action-value function of all the three algorithms and find out what is different in Expected SARSA.

SARSA: $Q(S_{t}, A_{t}) = Q(S_{t}, A_{t}) + \alpha (R_{t+1}+\gamma Q(S_{t+1}, A_{t+1})-Q(S_{t}, A_{t}))$
Q-Learning:
$Q(s_{t}, a_{t}) = Q(s_{t}, a_{t}) + \alpha (r_{t+1}+\gamma max_{a}Q(s_{t+1}, a)-Q(s_{t}, a_{t}))$
Expected SARSA:
$Q(s_{t}, a_{t}) = Q(s_{t}, a_{t}) + \alpha (r_{t+1}+\gamma \sum_{a} \pi (a | s_{t+1}) Q(s_{t+1}, a)-Q(s_{t}, a_{t}))$

We see that Expected SARSA takes the weighted sum of all possible next actions with respect to the probability of taking that action. If the Expected Return is greedy with respect to the expected return, then this equation gets transformed to Q-Learning. Otherwise Expected SARSA is on-policy and computes the expected return for all actions, rather than randomly selecting an action like SARSA.

Keeping the theory and the formulae in mind, let us compare all the three algorithms, with an experiment. We shall implement a Cliff Walker as our environment provided by the gym library

Code: Python code to create the class Agent which will be inherited by the other agents to avoid duplicate code.

# Agent.py 
  
import numpy as np 
  
class Agent: 
    """ 
    The Base class that is implemented by 
    other classes to avoid the duplicape 'choose_action' 
    method 
    """
    def choose_action(self, state):  
        action = 0
        if np.random.uniform(0, 1) < self.epsilon:  
            action = self.action_space.sample() 
        else: 
            action = np.argmax(self.Q[state, :])  
        return action 

Code: Python code to create the SARSA Agent.

# SarsaAgent.py 
  
import numpy as np 
from Agent import Agent 
  
class SarsaAgent(Agent): 
    """ 
    The Agent that uses SARSA update to improve it's behaviour 
    """
    def __init__(self, epsilon, alpha, gamma, num_state, num_actions, action_space): 
        """ 
        Contructor 
        Args: 
            epsilon: The degree of exploration 
            gamma: The discount factor 
            num_state: The number of states 
            num_actions: The number of actions 
            action_space: To call the random action 
        """
        self.epsilon = epsilon 
        self.alpha = alpha 
        self.gamma = gamma 
        self.num_state = num_state 
        self.num_actions = num_actions 
  
        self.Q = np.zeros((self.num_state, self.num_actions)) 
        self.action_space = action_space 
  
    def update(self, prev_state, next_state, reward, prev_action, next_action):  
        """ 
        Update the action value function using the SARSA update. 
        Q(S, A) = Q(S, A) + alpha(reward + (gamma * Q(S_, A_) - Q(S, A)) 
        Args: 
            prev_state: The previous state 
            next_state: The next state 
            reward: The reward for taking the respective action 
            prev_action: The previous action 
            next_action: The next action 
        Returns: 
            None 
        """
        predict = self.Q[prev_state, prev_action] 
        target = reward + self.gamma * self.Q[next_state, next_action] 
        self.Q[prev_state, prev_action] += self.alpha * (target - predict) 

Code: Python code to create the Q-Learning Agent.

# QLearningAgent.py 
  
import numpy as np 
from Agent import Agent 
  
class QLearningAgent(Agent): 
    def __init__(self, epsilon, alpha, gamma, num_state, num_actions, action_space): 
        """ 
        Contructor 
        Args: 
            epsilon: The degree of exploration 
            gamma: The discount factor 
            num_state: The number of states 
            num_actions: The number of actions 
            action_space: To call the random action 
        """
        self.epsilon = epsilon 
        self.alpha = alpha 
        self.gamma = gamma 
        self.num_state = num_state 
        self.num_actions = num_actions 
  
        self.Q = np.zeros((self.num_state, self.num_actions)) 
        self.action_space = action_space 
    def update(self, state, state2, reward, action, action2): 
        """ 
        Update the action value function using the Q-Learning update. 
        Q(S, A) = Q(S, A) + alpha(reward + (gamma * Q(S_, A_) - Q(S, A)) 
        Args: 
            prev_state: The previous state 
            next_state: The next state 
            reward: The reward for taking the respective action 
            prev_action: The previous action 
            next_action: The next action 
        Returns: 
            None 
        """
        predict = self.Q[state, action] 
        target = reward + self.gamma * np.max(self.Q[state2, :]) 
        self.Q[state, action] += self.alpha * (target - predict) 

Code: Python code to create the Expected SARSA Agent. In this experiment we are using the following equation for the policy.

$\pi (a | s_{t+1}) = \begin{cases} \dfrac{\epsilon}{A} &\text{if a = Greedy Action}\\ 1 - \epsilon + \dfrac{\epsilon}{\text{Number of Greedy Action}} &\text{if a = Non-Greedy Action}\\ \end{cases}$

# ExpectedSarsaAgent.py 
  
import numpy as np 
from Agent import Agent 
  
class ExpectedSarsaAgent(Agent): 
    def __init__(self, epsilon, alpha, gamma, num_state, num_actions, action_space): 
        """ 
        Contructor 
        Args: 
            epsilon: The degree of exploration 
            gamma: The discount factor 
            num_state: The number of states 
            num_actions: The number of actions 
            action_space: To call the random action 
        """
        self.epsilon = epsilon 
        self.alpha = alpha 
        self.gamma = gamma 
        self.num_state = num_state 
        self.num_actions = num_actions 
  
        self.Q = np.zeros((self.num_state, self.num_actions)) 
        self.action_space = action_space 
    def update(self, prev_state, next_state, reward, prev_action, next_action): 
        """ 
        Update the action value function using the Expected SARSA update. 
        Q(S, A) = Q(S, A) + alpha(reward + (pi * Q(S_, A_) - Q(S, A)) 
        Args: 
            prev_state: The previous state 
            next_state: The next state 
            reward: The reward for taking the respective action 
            prev_action: The previous action 
            next_action: The next action 
        Returns: 
            None 
        """
        predict = self.Q[prev_state, prev_action] 
  
        expected_q = 0
        q_max = np.max(self.Q[next_state, :]) 
        greedy_actions = 0
        for i in range(self.num_actions): 
            if self.Q[next_state][i] == q_max: 
                greedy_actions += 1
      
        non_greedy_action_probability = self.epsilon / self.num_actions 
        greedy_action_probability = ((1 - self.epsilon) / greedy_actions) + non_greedy_action_probability 
  
        for i in range(self.num_actions): 
            if self.Q[next_state][i] == q_max: 
                expected_q += self.Q[next_state][i] * greedy_action_probability 
            else: 
                expected_q += self.Q[next_state][i] * non_greedy_action_probability 
  
        target = reward + self.gamma * expected_q 
        self.Q[prev_state, prev_action] += self.alpha * (target - predict) 

Python code to create an environment and Test all the three algorithms.

# main.py 
  
import gym 
import numpy as np 
  
from ExpectedSarsaAgent import ExpectedSarsaAgent 
from QLearningAgent import QLearningAgent 
from SarsaAgent import SarsaAgent 
from matplotlib import pyplot as plt 
  
# Using the gym library to create the environment 
env = gym.make('CliffWalking-v0') 
  
# Defining all the required parameters 
epsilon = 0.1
total_episodes = 500
max_steps = 100
alpha = 0.5
gamma = 1
""" 
    The two parameters below is used to calculate 
    the reward by each algorithm 
"""
episodeReward = 0
totalReward = { 
    'SarsaAgent': [], 
    'QLearningAgent': [], 
    'ExpectedSarsaAgent': [] 
} 
  
# Defining all the three agents 
expectedSarsaAgent = ExpectedSarsaAgent( 
    epsilon, alpha, gamma, env.observation_space.n,  
    env.action_space.n, env.action_space) 
qLearningAgent = QLearningAgent( 
    epsilon, alpha, gamma, env.observation_space.n,  
    env.action_space.n, env.action_space) 
sarsaAgent = SarsaAgent( 
    epsilon, alpha, gamma, env.observation_space.n,  
    env.action_space.n, env.action_space) 
  
# Now we run all the episodes and calculate the reward obtained by 
# each agent at the end of the episode 
  
agents = [expectedSarsaAgent, qLearningAgent, sarsaAgent] 
  
for agent in agents: 
    for _ in range(total_episodes): 
        # Initialize the necesary parameters before  
        # the start of the episode 
        t = 0
        state1 = env.reset()  
        action1 = agent.choose_action(state1)  
        episodeReward = 0
        while t < max_steps: 
  
            # Getting the next state, reward, and other parameters 
            state2, reward, done, info = env.step(action1)  
      
            # Choosing the next action  
            action2 = agent.choose_action(state2)  
              
            # Learning the Q-value  
            agent.update(state1, state2, reward, action1, action2)  
      
            state1 = state2  
            action1 = action2  
              
            # Updating the respective vaLues  
            t += 1
            episodeReward += reward 
              
            # If at the end of learning process  
            if done:  
                break
        # Append the sum of reward at the end of the episode 
        totalReward[type(agent).__name__].append(episodeReward) 
env.close() 
  
# Calculate the mean of sum of returns for each episode 
meanReturn = { 
    'SARSA-Agent': np.mean(totalReward['SarsaAgent']), 
    'Q-Learning-Agent': np.mean(totalReward['QLearningAgent']), 
    'Expected-SARSA-Agent': np.mean(totalReward['ExpectedSarsaAgent']) 
} 
  
# Print the results 
print(f"SARSA Average Sum of Reward: {meanReturn['SARSA-Agent']}") 
print(f"Q-Learning Average Sum of Return: {meanReturn['Q-Learning-Agent']}") 
print(f"Expected Sarsa Average Sum of Return: {meanReturn['Expected-SARSA-Agent']}") 

Output:

Conclusion:
We have seen that Expected SARSA performs reasonably well in certain problems. It considers all possible outcomes before selecting a particular action. The fact that Expected SARSA can be used either as an off or on policy, is what makes this algorithm so dynamic.

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