Answers.Body: <p>As an aside, the act of taking knowledge of outcomes and using it to assign values to previous state or actions is not called backpropagation in the literature. It is called <em>backup</em>, as in <a href="http://incompleteideas.net/book/ebook/node34.html#eq:Bellman-Vpi" rel="nofollow noreferrer">backup diagram</a>. I will use that terminology in the answer. You may find the distinction useful in future when you have systems that are part of reinforcement learning that do also use backpropagation (as part of training a neural network).</p>
  <p>Single-step Q learning is simple to implement and can be effective. After a single episode of TicTacToe, it may look like your second example, with only the state immediately preceding a reward storing any value. However this is not generally true. Q learning has some interesting &quot;search&quot; behaviour in environments where state can be revisited, and where non-zero rewards may occur other than at episode termination. Finding the exit to a maze, with reward of -1 per timestep, can be a good example of this, with the immediate updates per timestep in Q learning offering a large advantage over waiting until the end of the episode (as in Monte Carlo control).</p>
  <p>However, there can be more effective versions using multiple steps, and there are several variants of this idea. A common one, and a core idea in RL, is <a href="http://www.incompleteideas.net/book/ebook/node72.html" rel="nofollow noreferrer">eligibility traces</a>, leading to the algorithm Q(<span class="math-container">$\lambda$</span>) which is a generalisation of the idea of assigning reward further back in time, but with less certainty, based on the parameter <span class="math-container">$\lambda$</span> which varies between <span class="math-container">$0$</span> for considering immediate effects only, to considering everything to the end of an episode. Single-step Q learning can be called Q(0) in reference to this algorith, and Q(1) has similar behavior to Monte Carlo control.</p>
  <p>Working with Q(<span class="math-container">$\lambda$</span>) means storing a history of actions and states either directly or more commonly summarised in the trace (which also decides how much to backup), in order to assign value updates via a backup. It could lead to an update in TicTacToe similar to your first example, after a single episode. In <em>some</em> environments this can significantly increase the sample efficiency from gathering experience by the agent. However, don't underestimate the power of simple fast updates and collecting more samples, especially in simple simulations.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-01-28T11:46:01.073Z
Answers.Text: Is it possible to have a Q-Table that looks like the first example after a single episode when just using TD learning? Tbh, I never heard of ET before, but that is because I'm new to RL. I always thought that if there is only one episode of training that then the last move has a q-value and all its previous moves that led to the last move have a q-value that is calculated based on the last moves q-value. And one thing I also don't understand is if wether the q-values are updated from last move to first move or the other way around.
Answers.UserDisplayName: Hans123
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-01-28T12:46:39.893Z
Answers.Text: Furthermore, I still don't really get one more thing:
  
  Let's say, I'm implementing tabular q-learning for tictactoe. Would the updating mechanism be in a forward or backward manner?  (i.e., from the start of the episode to the end), the first state-action pair that would have its Q-value updated would be the first one encountered in the episode. In the case of Tic-Tac-Toe, this would typically be the empty board state if the agent is the first to move. Or in a backward manner? In this case, the last state-action pair in the episode would be the first to have its Q-value updated.
Answers.UserDisplayName: Hans123
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-01-28T17:06:43.440Z
Answers.Text: @Hans You seem to be asking more and different questions in comments, and to have edited your first question to add code and asking for help with it in the main question. It would be better instead to leave the original question as-is, and only ask clarifying questions about this answer here. If you want to follow up with more detail, then it is better to ask a new question.
Answers.UserDisplayName: Neil Slater
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-01-27T22:52:19.030Z
Answers.LastActivityDate: 2024-01-27T22:57:51.690Z
Answers.LastEditDate: 2024-01-27T22:57:51.690Z
Answers.UserDisplayName: Neil Slater
Body: <p>I am implementing a Tabular Q-Learning algorithm in Python and have questions regarding the use of an 'action_history' variable.</p>
  <ol>
  <li><p>Necessity of 'action_history': In Q-Learning, the Q-value update is typically based on the immediate action and its outcome. Is there a benefit in maintaining a history of all actions taken in an episode to backpropagate the updated Q-value to these actions? Would this approach enhance learning, or does it contradict Q-Learning principles?</p>
  </li>
  <li><p>Q-value Update Mechanism: If an 'action_history' is beneficial, how should the Q-value be updated across multiple past actions? Is there a standard method or formula for distributing the updated Q-value to previous actions?</p>
  </li>
  <li><p>Q-Table Appearance After One Episode: If I set the learning to only one episode, how would the Q-table typically look? Would only the last action have a non-zero Q-value, with others being updated in subsequent episodes? For example, would it look like this:</p>
  </li>
  </ol>
  <pre><code>    &quot;('XO__XOXO_', 8)&quot;: 0.2,
      &quot;('X___XOXO_', 1)&quot;: 0.038000000000000006,
      &quot;('X____OXO_', 4)&quot;: 0.007220000000000002,
      &quot;('X_____XO_', 5)&quot;: 0.0013718000000000003,
      &quot;('______XO_', 0)&quot;: 0.00026064200000000006,
      &quot;('______X__', 7)&quot;: 4.952198e-05,
      &quot;('_________', 6)&quot;: 9.4091762e-06
  }
  </code></pre>
  <p>or would it look like:</p>
  <pre><code>{
      &quot;('XO__XOXO_', 8)&quot;: 0.2,
      &quot;('X___XOXO_', 1)&quot;: 0,
      &quot;('X____OXO_', 4)&quot;: 0,
      &quot;('X_____XO_', 5)&quot;: 0,
      &quot;('______XO_', 0)&quot;: 0,
      &quot;('______X__', 7)&quot;: 0,
      &quot;('_________', 6)&quot;: 0
  }
  </code></pre>
  <p>I'm looking to understand the standard practices and the theoretical justification behind these aspects of Q-Learning.</p>
  <p>Edit: I will add my Python code, in which I'm trying to implement tabular q-learning for 3x3 tictactoe. The q-table for a single episode looks like the following. I don't know how I need to update the value of the moves that led to the last move:</p>
  <pre><code>{
      &quot;('_________', 3)&quot;: 0.002,
      &quot;('___X_____', 6)&quot;: 0.002,
      &quot;('___X__O__', 5)&quot;: 0.002,
      &quot;('___X_XO__', 7)&quot;: 0.002,
      &quot;('___X_XOO_', 1)&quot;: 0.002,
      &quot;('_X_X_XOO_', 8)&quot;: 0.2
  }
  </code></pre>
  <pre><code>class Mark(enum.StrEnum):
      CROSS = &quot;X&quot;
      NAUGHT = &quot;O&quot;
      EMPTY = &quot;_&quot;
  
  
  class Reward(enum.Enum):
      WIN = 1
      LOSE = -1
      TIE = -0.065
      NON_TERMINAL = 0.01
  
  
  # Constants for Q-Learning
  EPSILON = 0.9  # Exploration factor
  ALPHA = 0.2  # Learning rate
  GAMMA = 0.95  # Discount factor
  
  TOTAL_EPISODES = 1  # Total number of games the agent will play
  
  BOARD = np.array([Mark.EMPTY] * BOARD_SIZE)
  
  
  def empty_count(board: np.ndarray) -&gt; int:
      return np.count_nonzero(board == Mark.EMPTY)
  
  
  def update_q_table(board, action, reward, new_board):
      board_key = &quot;&quot;.join(board)
      new_board_key = &quot;&quot;.join(new_board)
  
      old_value = Q_TABLE_DICT.get((board_key, action), 0)
  
      if game_over(new_board):
          # If it's a terminal state, there are no future rewards to consider
          next_max = 0
      else:
          # Estimate the optimal future value
          next_max = max(
              Q_TABLE_DICT.get((new_board_key, a), 0) for a in possible_moves(new_board)
          )
  
      temporal_difference = reward + (GAMMA * next_max) - old_value
      # Update the Q-value for the current state-action pair using the bellman equation
      new_q_value = old_value + (ALPHA * temporal_difference)
      Q_TABLE_DICT[(board_key, action)] = new_q_value
  
  def train_q_learning_agent():
      for episode in range(TOTAL_EPISODES):
          board = np.array([Mark.EMPTY] * BOARD_SIZE)  # Reset the board
          current_mark = Mark.CROSS  # Start with X (or choose randomly)
  
          while not game_over(board):
              # Q-learning agent (X) makes a move
              # if current_mark == Mark.CROSS:
              action = choose_action_q_learning(board, training=True)
              new_board = make_move_to(board, action, current_mark)
              reward = get_reward(new_board, current_mark)
              update_q_table(board, action, reward, new_board)
  
              board = new_board
              current_mark = Mark.NAUGHT if current_mark == Mark.CROSS else Mark.CROSS
          if episode == 1200:
              EPSILON = 0.1
  
  
  def choose_action_q_learning(board, training=True) -&gt; int:
      if training and random.uniform(0, 1) &lt; EPSILON:
          # Exploration: choose a random action
          return np.random.choice(possible_moves(board))
      else:
          # Exploitation: choose the best action based on current Q-table
          board_key = &quot;&quot;.join(board)
          q_values = {
              action: Q_TABLE_DICT.get((board_key, action), 0)
              for action in possible_moves(board)
          }
          return max(q_values, key=q_values.get)
  </code></pre>
  
Tags: <reinforcement-learning><q-learning>
Title: Tabular Q-Learning: Is a variable for "action_history" needed for backpropagating the q-value for all previous actions?
UserDisplayName: Hans123
ThreadText: # Tabular Q-Learning: Is a variable for "action_history" needed for backpropagating the q-value for all previous actions?
  
  (Asked by: username_0 on 2024-01-27)
  
  I am implementing a Tabular Q-Learning algorithm in Python and have questions regarding the use of an 'action_history' variable.
  
  Necessity of 'action_history': In Q-Learning, the Q-value update is typically based on the immediate action and its outcome. Is there a benefit in maintaining a history of all actions taken in an episode to backpropagate the updated Q-value to these actions? Would this approach enhance learning, or does it contradict Q-Learning principles?
  
  Q-value Update Mechanism: If an 'action_history' is beneficial, how should the Q-value be updated across multiple past actions? Is there a standard method or formula for distributing the updated Q-value to previous actions?
  
  Q-Table Appearance After One Episode: If I set the learning to only one episode, how would the Q-table typically look? Would only the last action have a non-zero Q-value, with others being updated in subsequent episodes? For example, would it look like this:
  
  
      "('XO__XOXO_', 8)": 0.2,
      "('X___XOXO_', 1)": 0.038000000000000006,
      "('X____OXO_', 4)": 0.007220000000000002,
      "('X_____XO_', 5)": 0.0013718000000000003,
      "('______XO_', 0)": 0.00026064200000000006,
      "('______X__', 7)": 4.952198e-05,
      "('_________', 6)": 9.4091762e-06
  }
  
  or would it look like:
  {
      "('XO__XOXO_', 8)": 0.2,
      "('X___XOXO_', 1)": 0,
      "('X____OXO_', 4)": 0,
      "('X_____XO_', 5)": 0,
      "('______XO_', 0)": 0,
      "('______X__', 7)": 0,
      "('_________', 6)": 0
  }
  
  I'm looking to understand the standard practices and the theoretical justification behind these aspects of Q-Learning.
  Edit: I will add my Python code, in which I'm trying to implement tabular q-learning for 3x3 tictactoe. The q-table for a single episode looks like the following. I don't know how I need to update the value of the moves that led to the last move:
  {
      "('_________', 3)": 0.002,
      "('___X_____', 6)": 0.002,
      "('___X__O__', 5)": 0.002,
      "('___X_XO__', 7)": 0.002,
      "('___X_XOO_', 1)": 0.002,
      "('_X_X_XOO_', 8)": 0.2
  }
  
  class Mark(enum.StrEnum):
      CROSS = "X"
      NAUGHT = "O"
      EMPTY = "_"
  
  
  class Reward(enum.Enum):
      WIN = 1
      LOSE = -1
      TIE = -0.065
      NON_TERMINAL = 0.01
  
  
  # Constants for Q-Learning
  EPSILON = 0.9  # Exploration factor
  ALPHA = 0.2  # Learning rate
  GAMMA = 0.95  # Discount factor
  
  TOTAL_EPISODES = 1  # Total number of games the agent will play
  
  BOARD = np.array([Mark.EMPTY] * BOARD_SIZE)
  
  
  def empty_count(board: np.ndarray) -> int:
      return np.count_nonzero(board == Mark.EMPTY)
  
  
  def update_q_table(board, action, reward, new_board):
      board_key = "".join(board)
      new_board_key = "".join(new_board)
  
      old_value = Q_TABLE_DICT.get((board_key, action), 0)
  
      if game_over(new_board):
          # If it's a terminal state, there are no future rewards to consider
          next_max = 0
      else:
          # Estimate the optimal future value
          next_max = max(
              Q_TABLE_DICT.get((new_board_key, a), 0) for a in possible_moves(new_board)
          )
  
      temporal_difference = reward + (GAMMA * next_max) - old_value
      # Update the Q-value for the current state-action pair using the bellman equation
      new_q_value = old_value + (ALPHA * temporal_difference)
      Q_TABLE_DICT[(board_key, action)] = new_q_value
  
  def train_q_learning_agent():
      for episode in range(TOTAL_EPISODES):
          board = np.array([Mark.EMPTY] * BOARD_SIZE)  # Reset the board
          current_mark = Mark.CROSS  # Start with X (or choose randomly)
  
          while not game_over(board):
              # Q-learning agent (X) makes a move
              # if current_mark == Mark.CROSS:
              action = choose_action_q_learning(board, training=True)
              new_board = make_move_to(board, action, current_mark)
              reward = get_reward(new_board, current_mark)
              update_q_table(board, action, reward, new_board)
  
              board = new_board
              current_mark = Mark.NAUGHT if current_mark == Mark.CROSS else Mark.CROSS
          if episode == 1200:
              EPSILON = 0.1
  
  
  def choose_action_q_learning(board, training=True) -> int:
      if training and random.uniform(0, 1) < EPSILON:
          # Exploration: choose a random action
          return np.random.choice(possible_moves(board))
      else:
          # Exploitation: choose the best action based on current Q-table
          board_key = "".join(board)
          q_values = {
              action: Q_TABLE_DICT.get((board_key, action), 0)
              for action in possible_moves(board)
          }
          return max(q_values, key=q_values.get)
  
  **Tags:** reinforcement-learning, q-learning
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2024-01-27. Score: 1
  
  As an aside, the act of taking knowledge of outcomes and using it to assign values to previous state or actions is not called backpropagation in the literature. It is called *backup*, as in [backup diagram](http://incompleteideas.net/book/ebook/node34.html#eq:Bellman-Vpi). I will use that terminology in the answer. You may find the distinction useful in future when you have systems that are part of reinforcement learning that do also use backpropagation (as part of training a neural network).
  
  Single-step Q learning is simple to implement and can be effective. After a single episode of TicTacToe, it may look like your second example, with only the state immediately preceding a reward storing any value. However this is not generally true. Q learning has some interesting "search" behaviour in environments where state can be revisited, and where non-zero rewards may occur other than at episode termination. Finding the exit to a maze, with reward of -1 per timestep, can be a good example of this, with the immediate updates per timestep in Q learning offering a large advantage over waiting until the end of the episode (as in Monte Carlo control).
  
  However, there can be more effective versions using multiple steps, and there are several variants of this idea. A common one, and a core idea in RL, is [eligibility traces](http://www.incompleteideas.net/book/ebook/node72.html), leading to the algorithm Q($\lambda$) which is a generalisation of the idea of assigning reward further back in time, but with less certainty, based on the parameter $\lambda$ which varies between $0$ for considering immediate effects only, to considering everything to the end of an episode. Single-step Q learning can be called Q(0) in reference to this algorith, and Q(1) has similar behavior to Monte Carlo control.
  
  Working with Q($\lambda$) means storing a history of actions and states either directly or more commonly summarised in the trace (which also decides how much to backup), in order to assign value updates via a backup. It could lead to an update in TicTacToe similar to your first example, after a single episode. In *some* environments this can significantly increase the sample efficiency from gathering experience by the agent. However, don't underestimate the power of simple fast updates and collecting more samples, especially in simple simulations.
  
  > **Comments:**
  > **username_0**: Is it possible to have a Q-Table that looks like the first example after a single episode when just using TD learning? Tbh, I never heard of ET before, but that is because I'm new to RL. I always thought that if there is only one episode of training that then the last move has a q-value and all its previous moves that led to the last move have a q-value that is calculated based on the last moves q-value. And one thing I also don't understand is if wether the q-values are updated from last move to first move or the other way around.  
  > **username_0**: Furthermore, I still don't really get one more thing:  Let's say, I'm implementing tabular q-learning for tictactoe. Would the updating mechanism be in a forward or backward manner?  (i.e., from the start of the episode to the end), the first state-action pair that would have its Q-value updated would be the first one encountered in the episode. In the case of Tic-Tac-Toe, this would typically be the empty board state if the agent is the first to move. Or in a backward manner? In this case, the last state-action pair in the episode would be the first to have its Q-value updated.  
  > **username_1**: @Hans You seem to be asking more and different questions in comments, and to have edited your first question to add code and asking for help with it in the main question. It would be better instead to leave the original question as-is, and only ask clarifying questions about this answer here. If you want to follow up with more detail, then it is better to ask a new question.  
  
  ---
  
  