# 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-Learning: • Expected SARSA: 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. # 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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