RL (chatgpt)

October 12, 2023

(original) https://chat.openai.com/share/cd0a2f47-fafd-4e73-a545-193e0b69bb65

You:

i wanna know about reinforcement learning

let’s start from bigger picture, but with concrete example.

ChatGPT:

Certainly! Reinforcement Learning (RL) is a type of machine learning where an agent learns to make decisions by taking actions in an environment to maximize some notion of cumulative reward.

Let’s illustrate this with a classic example: The FrozenLake Environment.

Scenario: Imagine a frozen lake that’s divided into a grid. The agent starts in the top-left corner and its goal is to reach the bottom-right corner where there’s a frisbee. Some of the grid cells are safe (frozen) while others have holes. If the agent steps into a hole, it falls and has to start over. The agent doesn’t initially know which cells are safe or where the holes are.

Representation:

• S : starting point
• F : frozen surface (safe)
• H : hole (fall in, start over)
• G : goal (frisbee)

Example grid:

SFFF
FHFH
FFFH
HFFG

Components:

• Agent: The player navigating the frozen lake.
• Environment: The frozen lake itself, with all its holes and the frisbee.
• State: A specific cell in the grid, representing the agent’s current position.
• Action: The agent can move in any of the four directions: up, down, left, right.
• Reward:
  • If the agent reaches the goal (frisbee), it gets a reward of +1.
  • If the agent falls into a hole, it gets a reward of 0.
  • For every other step, the agent gets a small negative reward (e.g., -0.01) to encourage quicker solutions.
• Policy: A strategy that the agent employs to determine the next action based on the current state. Initially, the agent might take random actions, but as it learns from rewards and penalties, it refines this policy to take better actions that maximize its cumulative reward.

How it works:

• The agent starts at the beginning (S) and takes an action (e.g., move right).
• The environment responds by updating the agent’s state (e.g., moves to a new cell) and providing a reward.
• The agent uses this reward feedback to adjust its strategy (policy).
• Repeat steps 1-3 until the agent reaches the goal or falls into a hole.

Over many episodes (iterations of the game), the agent learns which paths are optimal (or at least better than others) by associating actions taken in specific states with the cumulative reward they produce. The learning process can be achieved using various algorithms, with Q-learning being a popular choice for this kind of problem.

In Summary: In the FrozenLake environment, reinforcement learning helps the agent learn the best path to the frisbee while avoiding holes, even if it doesn’t know the lake’s layout in advance. The agent refines its strategy over time using feedback from its interactions with the environment.

You:

ok, can we implement q-learning for this problem, with pseudo (python like) code, I’m not expecting a code which runs, expecting a code that help me understand the “learning” process, you can focus on “learning” step, only I guess how it updates the policy would be learning step.. (let me know if there’s other important ones)

ChatGPT:

Absolutely! Q-learning is a model-free reinforcement learning algorithm that aims to find the best action to take given a current state. This is done by maintaining a table of Q-values for every state-action pair.

For the FrozenLake problem, the Q-learning process can be broken down as follows:

• Initialize a Q-table with zeros.
• Explore actions: For each state, select any one among all possible actions based on a greedy policy derived from Q (e.g., epsilon-greedy).
• For each action taken, update the Q-value for the state-action pair based on the reward received and the maximum predicted Q-value for the next state. This is done using the Q-learning formula: Q ( s t a t e , a c t i o n ) ← ( 1 − α ) × Q ( s t a t e , a c t i o n )

α × ( r e w a r d + γ × max ⁡ a Q ( n e w _ s t a t e , a ) ) Q(state, action) \leftarrow (1-\alpha) \times Q(state, action) + \alpha \times (reward + \gamma \times \max_{a} Q(new_state, a)) Q(state, action) ← (1 − α) × Q(state, action) + α × (re wa rd + γ × max a​ Q(ne w _state, a)) Where:

• α \alpha α is the learning rate.
• γ \gamma γ is the discount factor.

Let’s dive into a pseudo-code:

# Initialize Q-table with zeros
Q = initialize_zeros_table(states, actions)

# Hyperparameters
alpha = 0.8  # learning rate
gamma = 0.95  # discount factor
epsilon = 0.1  # exploration rate

# Learning process
for episode in range(total_episodes):
    state = start_state
    
    while not is_terminal(state):  # continue till goal or hole
        # Choose action using an epsilon-greedy policy derived from Q
        if random_value < epsilon:
            action = random_choice(all_possible_actions)
        else:
            action = argmax(Q[state, :])
        
        # Take action and observe new state and reward
        new_state, reward = take_action(state, action)
        
        # Update Q-value using the Q-learning formula
        Q[state, action] = (1 - alpha) * Q[state, action] + alpha * (reward + gamma * max(Q[new_state, :]))
        
        # Move to the new state
        state = new_state

# Policy is now implicitly stored in Q. For any state, the best action is the one with the highest Q-value.

Key points:

• Epsilon-greedy strategy: At each step, with a probability of ϵ \epsilon ϵ, we choose a random action (exploration), and the rest of the time (with probability 1 − ϵ 1 - \epsilon 1 − ϵ), we choose the action that has the maximum current Q-value (exploitation).
• Updating Q-values: After taking an action, we use the Q-learning formula to update our Q-value. This is based on the reward we get for the current action plus the maximum Q-value for the new state (which represents our estimate of future rewards). The balance between the current Q-value and our new estimate is controlled by the learning rate α \alpha α.

That’s the core of the Q-learning algorithm! By repeatedly updating the Q-values in this manner, the Q-table will eventually converge, giving us a good policy for navigating the FrozenLake.

You:

Q holds Q[state, action_i] = reward for all possible action_i at state, but it seems, for any state, only maximal action is considered, and only single action_i is updated as well..

let me clarify, state holds only board’s position (not any more info is considered, such as at which step we are in) (and Q[state, action_] holds initially random rewards or uniform rewards) and updating Q for position (x,y) happens with iterating many episodes

so it’s like randomly walking (but with previous walking affecting current walk by rewards)

question then is, how does it know when to stop? does the process guarantee convergence?

ChatGPT:

You’ve made a couple of insightful observations about Q-learning, and I’ll address your points and questions.

• Updating Single Q-Value: Indeed, in each step of an episode, only a single Q-value corresponding to the state and action taken is updated. The intuition is that the agent is learning the value of taking that specific action in that specific state from its experience.
• Exploration vs. Exploitation: The epsilon-greedy strategy ensures a balance between exploration (trying out different actions) and exploitation (using the knowledge gained so far). This randomness in choosing actions, especially early on when the Q-values are not well-established, allows the agent to explore different paths and not get stuck in a sub-optimal policy.
• Stopping Criteria: In the pseudo-code, the learning process stops after a fixed number of episodes (total_episodes). In practice, you might want to set a convergence criterion, such as when the difference in Q-values between episodes (or over a moving average of several episodes) is below a certain threshold, indicating that learning has stabilized. Another common practice is to reduce the value of ϵ \epsilon ϵ over time, ensuring that the agent explores more in the early stages and exploits more in the later stages.
• Convergence: The Q-learning algorithm has been proven to converge to the optimal policy as long as every state-action pair is visited an infinite number of times and the learning rate α \alpha α meets certain conditions. One common condition is that it should be reduced over time but not too rapidly. In simpler terms, given enough time and episodes, Q-learning will find the best policy. In practical scenarios, especially in finite and discrete environments like FrozenLake, Q-learning usually converges well.

However, it’s essential to mention that while Q-learning has theoretical guarantees under certain conditions, its performance can be sensitive to hyperparameters like $\alpha$α, $\gamma$γ, and $\epsilon$ϵ. Additionally, the initialization of the Q-table and the specifics of the environment can influence the speed and quality of convergence.

Lastly, real-world problems often involve more complex and continuous state spaces where tabular Q-learning (as shown in the pseudo-code) might not be feasible. In such cases, function approximation techniques, like neural networks in Deep Q-Learning, are used to estimate Q-values.

You:

is gamma playing the role of “the agent gets a small negative reward (e.g., -0.01)” ?

ChatGPT:

Not exactly, but the two concepts are related in terms of the impact they have on the agent’s behavior. Let’s clarify:

• Gamma ( γ \gamma γ) - The Discount Factor:
  • It represents the agent’s consideration for future rewards. It’s a value between 0 and 1.
  • If γ \gamma γ is close to 1, the agent values future rewards nearly as much as immediate rewards, making it more “foresighted”. This means the agent may be willing to take a longer path if it believes a bigger reward is at the end.
  • If γ \gamma γ is close to 0, the agent is more “myopic” and primarily cares about immediate rewards.
  • In the Q-value update formula: Q ( s t a t e , a c t i o n ) ← ( 1 − α ) × Q ( s t a t e , a c t i o n )
  α × ( r e w a r d + γ × max ⁡ a Q ( n e w _ s t a t e , a ) ) Q(state, action) \leftarrow (1-\alpha) \times Q(state, action) + \alpha \times (reward + \gamma \times \max_{a} Q(new_state, a)) Q(state, action) ← (1 − α) × Q(state, action) + α × (re wa rd + γ × max a​ Q(ne w _state, a)) γ \gamma γ determines how much of the estimated future reward (from the next state onward) should influence the current Q-value update.
• Small Negative Reward:
  • This is an actual reward given by the environment to encourage or discourage certain behaviors.
  • For instance, in the FrozenLake problem, a small negative reward for each step encourages the agent to find the shortest path to the goal. Without this, the agent might not have a strong incentive to reach the goal quickly, especially if γ \gamma γ is close to 1.
  • The small negative reward affects the immediate reward in the Q-value update formula, whereas γ \gamma γ affects the future reward estimation.

So, while both $\gamma$γ and the small negative reward influence the agent’s behavior in terms of its consideration for immediate vs. future rewards, they operate at different levels. $\gamma$γ determines the weight given to future rewards in the Q-value update, while the small negative reward provides immediate feedback to the agent about its actions.

You:

new_state, reward = take_action(state, action)

expand this a bit, how is reward calculated here, is it that reward = Q[state, action]

ChatGPT:

Let’s clarify the take_action function. This function simulates the agent taking an action in the environment and returns the result of that action.

The reward is not directly fetched from the Q-table. Instead, it’s determined by the environment based on the agent’s action in a given state. Remember, the Q-table is the agent’s evolving understanding (or estimate) of those rewards, not the source of the rewards themselves.

Here’s an expanded pseudo-code for the take_action function in the context of the FrozenLake problem:

def take_action(state, action):
    # Given the current state and an action, the function determines the 
    # new state and the associated reward for taking that action.

    # Depending on the state (position on the lake) and the chosen action,
    # the agent can move up, down, left, or right.
    
    if action == "UP":
        new_state = move_up(state)
    elif action == "DOWN":
        new_state = move_down(state)
    elif action == "LEFT":
        new_state = move_left(state)
    elif action == "RIGHT":
        new_state = move_right(state)

    # Once the new state is determined based on the action, 
    # the environment provides a reward based on the new state.

    if new_state == GOAL:
        reward = 1  # Reached the frisbee
    elif new_state == HOLE:
        reward = 0  # Fell into a hole
    else:
        reward = -0.01  # Took a step without reaching the goal

    return new_state, reward

So, the environment (in this case, the frozen lake) dictates the reward based on the outcome of the action:

• If the agent reaches the goal (frisbee), it gets a reward of +1.
• If the agent falls into a hole, it gets a reward of 0 (or some negative value, if you want to penalize it).
• For every other step, the agent gets a small negative reward (e.g., -0.01) to encourage quicker solutions.

The Q-values, on the other hand, are the agent’s learned estimates of expected cumulative rewards for taking certain actions in given states. They get updated based on actual rewards received and the maximum estimated Q-value of the next state.

You:

ok

    Q[state, action] = (1 - alpha) * Q[state, action] + alpha * (reward * (1-gamma) + gamma * max(Q[new_state, :]))

if gamma is weighting factor, should n’t there be 1-gamma as well?

ChatGPT:

I understand the confusion. Let’s dissect the Q-learning update equation again.

The classic Q-learning update formula is: $Q(state, action) \leftarrow (1-\alpha) \times Q(state, action) + \alpha \times (reward + \gamma \times \max$Q(state,action)←(1−α)×Q(state,action)+α×(reward+γ×maxa​Q(new_state,a))

Here’s what each part is doing:

• ( 1 − α ) × Q ( s t a t e , a c t i o n ) (1-\alpha) \times Q(state, action) (1 − α) × Q(state, action): This represents a portion of the old Q-value we retain.
• α × r e w a r d \alpha \times reward α × re wa rd: This is the learned value from the immediate reward for taking the action.
• α × γ × max ⁡ a Q ( n e w _ s t a t e , a ) \alpha \times \gamma \times \max_{a} Q(new_state, a) α × γ × max a​ Q(ne w _state, a): This is the learned value from the estimated future rewards.

The term $\gamma$γ is the discount factor for the future rewards. We don’t need $1-\gamma$1−γ because the immediate reward (reward) is already being fully considered. The $\gamma$γ term is essentially saying, “I’ll discount future rewards by this factor because they’re uncertain and in the future.”

Including a $(1-\gamma)$(1−γ) term in the way you’ve suggested would imply that we’re discounting the immediate reward, which isn’t standard in Q-learning. The immediate reward is typically fully considered (i.e., not discounted) because it’s a known, certain value that has resulted directly from the agent’s action.

So, to reiterate, in the standard Q-learning formula:

• The immediate reward is fully considered.
• Future rewards are discounted by γ \gamma γ.

The equation you provided seems to be a variant or modification of the standard Q-learning update. While there’s a vast landscape of RL algorithms and adaptations, the classic Q-learning doesn’t use the $(1-\gamma)$(1−γ) term for the immediate reward.

You:

ok, let’s implement this, let’s start with defining the structure for Q and initializing it

ChatGPT:

Certainly! For simplicity, let’s consider the FrozenLake 4x4 grid example:

SFFF
FHFH
FFFH
HFFG

We’ll need a Q-table where:

• The number of rows represents the total number of states (in this case, 16 for each cell in the 4x4 grid).
• The number of columns represents the total number of actions (in this case, 4 for up, down, left, and right).

Let’s go step by step:

• Defining the Environment:

• n_states is 16 (4x4 grid).

• n_actions is 4 (up, down, left, right).

• Initializing the Q-table:

The Q-table can be initialized with all zeros. However, for certain problems or to expedite learning, other initialization strategies can be considered.

Here’s a simple Python implementation:

import numpy as np

# Define the environment
n_states = 16  # For the 4x4 grid
n_actions = 4  # Up, Down, Left, Right

# Initialize the Q-table with zeros
Q = np.zeros((n_states, n_actions))

print(Q)

This will produce a 16x4 matrix filled with zeros.

Note: In practice, the agent doesn’t necessarily know n_states and n_actions in advance, especially in more complex environments. However, for the FrozenLake problem and this illustration, we’re making certain assumptions for clarity.

Once you’ve initialized the Q-table, you can then proceed with the Q-learning algorithm to update the Q-values based on the agent’s interactions with the environment.

You:

ok let’s proceed to next step, do it incrementally

ChatGPT:

Absolutely! Let’s build this incrementally. Now that we’ve defined and initialized our Q-table, the next steps are to define the environment’s dynamics and start the learning process.

For simplicity, I’ll represent the FrozenLake as a list of strings. Each character will represent a cell: ‘S’ for start, ‘F’ for frozen (safe), ‘H’ for hole, and ‘G’ for goal.

• Define the Environment’s Dynamics:

frozen_lake = [
    'SFFF',
    'FHFH',
    'FFFH',
    'HFFG'
]

Now, let’s define some helper functions:

• State and Action Functions:

• Given a state (i.e., a 2D coordinate in our 4x4 grid), determine the next state for a given action.

• Handle edge cases, for example, if the agent tries to move left from the leftmost column, it should remain in its current state.

def state_to_coord(state):
    """Convert a state number to a coordinate (i, j) on the grid."""
    return divmod(state, 4)

def coord_to_state(coord):
    """Convert a grid coordinate (i, j) to a state number."""
    i, j = coord
    return i * 4 + j

def next_state(state, action):
    """Determine the next state given an action."""
    i, j = state_to_coord(state)
    if action == 0:  # UP
        i = max(i - 1, 0)
    elif action == 1:  # DOWN
        i = min(i + 1, 3)
    elif action == 2:  # LEFT
        j = max(j - 1, 0)
    elif action == 3:  # RIGHT
        j = min(j + 1, 3)
    return coord_to_state((i, j))

• Reward Function:

• Return the reward based on the next state.

def get_reward(state):
    i, j = state_to_coord(state)
    cell = frozen_lake[i][j]
    if cell == 'G':
        return 1
    elif cell == 'H':
        return 0
    else:
        return -0.01

• Epsilon-Greedy Policy:

• Choose an action for a given state using the epsilon-greedy strategy.

def choose_action(state, epsilon):
    if np.random.uniform(0, 1) < epsilon:
        return np.random.choice(n_actions)  # Explore: choose a random action
    else:
        return np.argmax(Q[state, :])  # Exploit: choose the action with max Q-value

With these helper functions, we’ve set up the basic dynamics of our FrozenLake environment. The next step will be to loop through episodes, where in each episode the agent will explore the environment and update the Q-values.

Would you like to proceed with the Q-learning loop, or would you like to review or modify anything in the setup so far?

You:

how example ‘state’ would look like for the state functions?

ChatGPT:

The state in our current representation is a single integer number that represents the position of the agent on the FrozenLake grid.

For our 4x4 grid:

SFFF    0 1 2 3
FHFH -> 4 5 6 7
FFFH    8 9 10 11
HFFG    12 13 14 15

• The top-left corner (S) is state 0.
• The next cell to the right is state 1, and so on.
• The bottom-right corner (G) is state 15.

So, for example:

• state = 0 would represent the starting point (S).
• state = 7 would represent the second hole from the top in the rightmost column.
• state = 15 would represent the goal (G).

Our helper functions state_to_coord and coord_to_state help in converting between this single integer representation and the 2D coordinate on the grid. For instance:

• state_to_coord(7) would return (1, 3), meaning row 1 and column 3 (0-indexed) on the grid.
• coord_to_state((1, 3)) would return 7, which is the integer representation of the state.

This representation simplifies indexing into the Q-table and working with the environment dynamics.

You:

let’s proceed with next step

ChatGPT:

Great! Let’s move on to the main Q-learning loop. This is where the agent interacts with the environment, learns from its experiences, and updates the Q-values.

Q-learning Algorithm:

• For a set number of episodes, do the following:
• Initialize the starting state.
• While the state is not terminal (i.e., neither the goal nor a hole), choose an action based on the current policy (epsilon-greedy), take that action, observe the reward and the next state, and then update the Q-table.

Here’s how the loop might look:

# Hyperparameters
alpha = 0.8  # learning rate
gamma = 0.95  # discount factor
epsilon = 0.1  # exploration rate
total_episodes = 10000  # number of episodes to train on

# Q-learning Loop
for episode in range(total_episodes):
    state = 0  # start at the beginning of the lake
    while frozen_lake[state_to_coord(state)] != 'G' and frozen_lake[state_to_coord(state)] != 'H':  # not at goal or hole
        # Choose action using epsilon-greedy strategy
        action = choose_action(state, epsilon)
        
        # Take action and observe new state and reward
        new_state = next_state(state, action)
        reward = get_reward(new_state)
        
        # Update Q-value using the Q-learning update rule
        Q[state, action] = (1 - alpha) * Q[state, action] + \
                           alpha * (reward + gamma * np.max(Q[new_state, :]))
        
        # Move to the next state
        state = new_state

    # Optionally reduce epsilon over episodes to reduce exploration as we learn
    # epsilon = epsilon * 0.995  # for example

print("Training complete!")

After this loop completes, the Q-table (Q) should have updated values based on the agent’s experiences in the environment. The agent’s policy can then be derived from this table: for any given state, the best action is the one with the highest Q-value.

Remember, this is a simplified representation of the Q-learning process tailored for the FrozenLake problem. There are nuances and optimizations that can be added, such as dynamically adjusting epsilon to balance exploration vs. exploitation as the agent learns more about the environment.

You:

frozen_lake[state_to_coord(state)] , i don’t think python list takes tuple as 2d index

ChatGPT:

You’re right. My apologies for the oversight. In Python, to index into a 2D list (or a list of lists) like frozen_lake, we need to use two separate indices.

Let’s correct the mistake.

Replace:

frozen_lake[state_to_coord(state)]

With:

i, j = state_to_coord(state)
frozen_lake[i][j]

So the loop’s condition will look like:

i, j = state_to_coord(state)
while frozen_lake[i][j] != 'G' and frozen_lake[i][j] != 'H':  # not at goal or hole

Thank you for pointing that out, and I appreciate your patience.

You:

it’s rather cumbersome, is there a way to use that shorthand notation? maybe numpy 2d array? no?

ChatGPT:

Yes, you’re absolutely right. If we use a NumPy 2D array for the frozen_lake representation, we can use tuple indexing. This will make the notation cleaner and more concise.

Here’s how we can refactor our code:

• Convert frozen_lake to a NumPy array:

frozen_lake = np.array([
    ['S', 'F', 'F', 'F'],
    ['F', 'H', 'F', 'H'],
    ['F', 'F', 'F', 'H'],
    ['H', 'F', 'F', 'G']
])

• Access using tuple indexing:

With this NumPy 2D array, you can use:

while frozen_lake[state_to_coord(state)] != 'G' and frozen_lake[state_to_coord(state)] != 'H':

This way, our state_to_coord function directly provides the tuple needed for 2D indexing in the NumPy array. This makes the code more readable and less error-prone.

Ensure you’ve imported NumPy at the beginning of your script:

import numpy as np

By using a NumPy array, you also open up the possibility of leveraging more advanced operations and optimizations available in the NumPy library.

You:

given

frozen_lake = [ ‘SFFF’, ‘FHFH’, ‘FFFH’, ‘HFFG’ ]

can you build that np array?

ChatGPT:

Certainly! You can convert the given frozen_lake list of strings into a NumPy 2D array using the following approach:

import numpy as np

frozen_lake_list = [
    'SFFF',
    'FHFH',
    'FFFH',
    'HFFG'
]

# Convert the list of strings to a 2D numpy array
frozen_lake_np = np.array([list(row) for row in frozen_lake_list])

print(frozen_lake_np)

This will produce:

[['S' 'F' 'F' 'F']
 ['F' 'H' 'F' 'H']
 ['F' 'F' 'F' 'H']
 ['H' 'F' 'F' 'G']]

Now, frozen_lake_np is a 2D numpy array, and you can use tuple indexing as discussed earlier.

You:

we are missing the “using the learned Q” in any way

ChatGPT:

Indeed, once the Q-learning process is complete, the next step is to use the learned Q-values to derive a policy for the agent and then to test or deploy this policy.

Deriving the Policy: #

The policy, often denoted as $\pi$π, is a mapping from states to actions that tells the agent the best action to take in each state. With our Q-table in hand, deriving the policy is straightforward: for each state, choose the action with the highest Q-value.

policy = np.argmax(Q, axis=1)

Now, policy is an array where each entry represents the best action for the corresponding state.

Testing the Policy: #

To test the learned policy, you’d simulate episodes similar to the learning phase but without any exploration (i.e., always choose the action as per the policy) and without updating the Q-values. Here’s how you might do this:

def test_policy(policy, num_episodes=1000):
    success_count = 0

    for episode in range(num_episodes):
        state = 0  # start at the beginning of the lake
        while frozen_lake_np[state_to_coord(state)] != 'G' and frozen_lake_np[state_to_coord(state)] != 'H':
            action = policy[state]  # choose action from policy
            state = next_state(state, action)
        
        if frozen_lake_np[state_to_coord(state)] == 'G':
            success_count += 1  # Increment success count if goal reached

    success_rate = success_count / num_episodes
    return success_rate

success_rate = test_policy(policy)
print(f"Policy success rate over {num_episodes} episodes: {success_rate * 100:.2f}%")

The success rate gives an estimate of how well the learned policy performs in reaching the goal over multiple episodes.

Remember, in a stochastic environment or if there’s slippage (as is sometimes introduced in FrozenLake variants), even an optimal policy might not have a 100% success rate.

You:

don’t we need to update the state where ‘H’ is?

ChatGPT:

Yes, you’re right. If the agent reaches a state corresponding to a ‘H’ (hole), the episode should end since the agent has “fallen in”. Similarly, when the agent reaches ‘G’ (goal), the episode ends as the agent has achieved its objective. Both these states are terminal states in the FrozenLake environment.

Let’s correct the testing function to incorporate this:

def test_policy(policy, num_episodes=1000):
    success_count = 0

    for episode in range(num_episodes):
        state = 0  # start at the beginning of the lake
        while True:  # Continue until a terminal state (H or G) is reached
            action = policy[state]  # choose action from policy
            state = next_state(state, action)
            
            # Check for terminal states
            cell_content = frozen_lake_np[state_to_coord(state)]
            if cell_content == 'G':
                success_count += 1  # Increment success count if goal reached
                break
            elif cell_content == 'H':
                break  # end episode if fallen into a hole

    success_rate = success_count / num_episodes
    return success_rate

success_rate = test_policy(policy)
print(f"Policy success rate over {num_episodes} episodes: {success_rate * 100:.2f}%")

This way, the episodes will correctly terminate whenever the agent encounters a hole or reaches the goal, providing a more accurate assessment of the policy’s success rate.

You:

while training as well?

ChatGPT:

Yes, absolutely. During the training phase, episodes should also terminate upon reaching either a hole (‘H’) or the goal (‘G’). I apologize for missing that detail earlier.

Let’s update the training loop to incorporate the termination conditions for these states:

# Q-learning Loop
for episode in range(total_episodes):
    state = 0  # start at the beginning of the lake
    while True:  # Continue until a terminal state (H or G) is reached
        # Choose action using epsilon-greedy strategy
        action = choose_action(state, epsilon)
        
        # Take action and observe new state and reward
        new_state = next_state(state, action)
        reward = get_reward(new_state)
        
        # Update Q-value using the Q-learning update rule
        Q[state, action] = (1 - alpha) * Q[state, action] + \
                           alpha * (reward + gamma * np.max(Q[new_state, :]))
        
        # Move to the next state
        state = new_state
        
        # Check for terminal states
        cell_content = frozen_lake_np[state_to_coord(state)]
        if cell_content in ['G', 'H']:
            break

Now, during the training phase, each episode will correctly terminate when the agent reaches a hole or the goal. This provides a more accurate simulation of the environment’s dynamics and should lead to better learning outcomes.

You:

i mean, don’t u need to update the state where H is? even though you don’t proceed further?

ChatGPT:

Ah, I see your point now. Indeed, the Q-value for the state-action pair that led to the hole (‘H’) should be updated even if we don’t proceed further. This allows the agent to learn from the negative experience of falling into the hole. The update step does handle that, but for clarity:

• The agent takes an action from some state.
• This action leads the agent into the next state, which might be a hole.
• The Q-value for the original state-action pair is updated based on the reward (which might be negative for falling into a hole).
• Only after the Q-value update do we check if the next state is terminal (like a hole or goal) and break out of the loop if it is.

In the provided loop, this is the sequence of events:

# Take action and observe new state and reward
new_state = next_state(state, action)
reward = get_reward(new_state)

# Update Q-value using the Q-learning update rule
Q[state, action] = (1 - alpha) * Q[state, action] + \
                   alpha * (reward + gamma * np.max(Q[new_state, :]))

# Move to the next state
state = new_state

# Check for terminal states
cell_content = frozen_lake_np[state_to_coord(state)]
if cell_content in ['G', 'H']:
    break

The Q-value is updated based on the new_state before checking if the new_state is terminal. This ensures that the agent learns from reaching terminal states like the holes. If this wasn’t the case, the agent wouldn’t be able to learn the negative consequence of actions leading to holes, which is crucial for navigating the FrozenLake environment effectively.

You:

when new_state’s cell_content is ‘H’, the new_state is not updated, only current_state is updated,

ChatGPT:

I understand your point now. You’re saying that the Q-value for the action leading to a hole (‘H’) is updated, but the Q-value for the hole state itself (i.e., for actions taken from the hole state) isn’t updated since the loop terminates upon reaching the hole.

In the context of the FrozenLake problem, this is generally fine because the hole states (‘H’) are terminal states. An agent shouldn’t be taking any actions from terminal states (neither holes nor the goal). So, in essence, the Q-values for actions from terminal states should remain at their initialized values (typically zeros) because they’re not actionable states.

However, if you still want to update the Q-values for the hole state itself, you could do so, but it’s not customary in this context. This would involve setting Q-values for all actions from the hole state to the negative reward of falling into a hole, but again, since the agent won’t take any further actions from this state, these Q-values are moot.

Here’s how you could do it, though:

# After updating the Q-value for the original state-action pair
cell_content = frozen_lake_np[state_to_coord(state)]
if cell_content == 'H':
    Q[state, :] = reward  # set Q-values for all actions from the hole state to the negative reward

But to reiterate, in typical implementations of the FrozenLake problem, this isn’t necessary since actions from terminal states are irrelevant.

You:

i’m saying it, because without that, H state remains 0 , and other state gets negative value, and the training won’t reach goal.

ChatGPT:

Got it. You’ve made a valid observation. The Q-values for the states leading to the hole might not be sufficiently penalized if the hole’s Q-value itself remains zero, especially if the discount factor $\gamma$γ is close to 1. This is particularly important if falling into the hole is given a reward of 0 (or any non-negative value) rather than a distinct negative value. The absence of a significant negative reinforcement when falling into a hole can hinder learning.

To ensure the agent learns effectively to avoid holes, the Q-values for the hole states should be explicitly penalized. Let’s update the Q-values for the new_state when it’s a hole:

# Take action and observe new state and reward
new_state = next_state(state, action)
reward = get_reward(new_state)

# Update Q-value using the Q-learning update rule for the current state-action pair
Q[state, action] = (1 - alpha) * Q[state, action] + \
                   alpha * (reward + gamma * np.max(Q[new_state, :]))

# If the new state is a hole, set its Q-values to a significant negative value
cell_content = frozen_lake_np[state_to_coord(new_state)]
if cell_content == 'H':
    Q[new_state, :] = -1  # or any other distinct negative value

# Move to the next state
state = new_state

By setting the Q-values for the hole states to a significant negative value, the states that lead to the holes will also get appropriately penalized during the Q-learning updates, improving the agent’s learning process and increasing its chances of reaching the goal.

Thank you for pointing it out, and I appreciate your patience in iterating on this.

You:

we updated Q only, and at the end made policy,

while you mentioned, algorithms that updates policy while training. how would one update policy in this example in training?

ChatGPT:

There are algorithms in reinforcement learning that operate on policy-based methods instead of value-based methods like Q-learning. One of the simplest policy-based methods is the policy gradient method, but implementing it for the FrozenLake problem would be more complex than our Q-learning approach.

However, if you’re looking for an intermediary approach that updates the policy along with values, you might be interested in SARSA (State-Action-Reward-State-Action). SARSA is an on-policy TD-learning algorithm, while Q-learning is off-policy. In SARSA, you update the Q-value based on the action you actually take next (guided by the current policy), rather than the maximum Q-value of the next state.

Here’s a high-level difference:

• Q-learning: Q ( s t a t e , a c t i o n ) ← Q ( s t a t e , a c t i o n )

α × ( r e w a r d + γ × max ⁡ a Q ( n e w _ s t a t e , a ) − Q ( s t a t e , a c t i o n ) ) Q(state, action) \leftarrow Q(state, action) + \alpha \times (reward + \gamma \times \max_{a} Q(new_state, a) - Q(state, action)) Q(state, action) ← Q(state, action) + α × (re wa rd + γ × max a​ Q(ne w _state, a) − Q(state, action))

• SARSA: Q ( s t a t e , a c t i o n ) ← Q ( s t a t e , a c t i o n )

α × ( r e w a r d + γ × Q ( n e w _ s t a t e , n e w _ a c t i o n ) − Q ( s t a t e , a c t i o n ) ) Q(state, action) \leftarrow Q(state, action) + \alpha \times (reward + \gamma \times Q(new_state, new_action) - Q(state, action)) Q(state, action) ← Q(state, action) + α × (re wa rd + γ × Q(ne w _state, ne w _action) − Q(state, action)) where new_action is the action taken in the new_state based on the current policy (typically epsilon-greedy).

To implement SARSA for the FrozenLake problem:

• Initialize your Q-table as before.
• For each episode:
  • Start in the initial state.
  • Choose an action using the epsilon-greedy policy.
  • For each step in the episode:
    • Take the action, observe the reward and the new state.
    • Choose the next action using the epsilon-greedy policy based on the new_state.
    • Update the Q-value using the SARSA update rule.
    • Set state to new_state and action to new_action and continue.

This method updates the policy (via the Q-values) at each step based on the action that the current policy would take, rather than the best action as in Q-learning.

If you’re interested in other algorithms that explicitly represent and update the policy, methods like REINFORCE or Actor-Critic would be the way to go, but they require a more complex setup and often utilize neural networks to parameterize the policy.

You:

ok, i’m interested in more complex setup , it might be the updating the policy setup, to move forward to next step of learning RL,

or, we could think of next level example of RL after the frozen_lake, what would be good one? (or we could proceed frozen_lake with policy updating)

ChatGPT:

If you’re looking to delve deeper into reinforcement learning, updating policies, and working with more complex setups, there are several avenues you can explore:

• Policy Gradient Methods on FrozenLake: You can extend the FrozenLake example using policy gradient methods like REINFORCE. This involves directly parameterizing and updating a policy function, often using neural networks. This is more challenging because you’d need to handle gradient computations and would likely need frameworks like TensorFlow or PyTorch.
• CartPole with Deep Q-Networks (DQN): The CartPole problem is a classic in reinforcement learning. The goal is to balance a pole on a moving cart. This problem requires a continuous state representation, making it a great introduction to using neural networks in RL. Here, instead of a Q-table, you’d use a neural network to approximate Q-values—a method commonly referred to as Deep Q-Learning or DQN.
• Actor-Critic Methods: This is a more advanced method that combines value-based and policy-based learning. You maintain both a value function (the critic) and a policy (the actor). The critic helps in updating the policy. Actor-Critic methods can be applied to a range of environments, including CartPole or even more complex problems.
• Pong or Breakout with Proximal Policy Optimization (PPO): Moving into more complex environments, using game-playing as a testbed is a common practice in RL. PPO is a popular algorithm in practice and has seen success in many complex tasks. It’s a type of policy optimization method.
• Custom Environments: OpenAI’s gym provides tools to create custom environments. This way, you can define and solve your own RL problems.
• Simulated Robotic Tasks with ROS and Gazebo: For those interested in robotics, combining reinforcement learning with robot simulators can be a challenging yet rewarding avenue. Tasks can range from simple pick-and-place operations to complex navigation and interaction tasks.

If you’re keen to stick with the FrozenLake and want a direct next step, I’d recommend trying the Policy Gradient Method on it. It’s a natural progression from the value-based method we used, and it will provide a good introduction to policy optimization techniques.

Would you like a brief guide or code snippets for any specific method? Or would you like more detailed resources/recommendations on any of these topics?

You:

let’s go to policy gradient method,

if desired, I can setup gym or any other library that can speed up the learning / implementing process

ChatGPT:

Fantastic choice! Policy gradient methods, especially in combination with neural networks, form the backbone of many advanced reinforcement learning algorithms. Let’s start with a basic introduction to policy gradients and then move on to an implementation for the FrozenLake environment using gym and PyTorch (or TensorFlow, if you have a preference).

Policy Gradient Methods: A Brief Overview #

Policy gradient methods optimize the policy directly. Instead of learning a value function which then informs a policy, these methods directly adjust the policy to maximize expected rewards. The fundamental idea is to push up the probabilities of actions that lead to higher rewards and push down the probabilities of actions that lead to lower rewards.

The most basic policy gradient algorithm is called REINFORCE.

Here’s a very high-level overview of REINFORCE:

• Use the policy π θ ( a ∣ s ) \pi_{\theta}(a|s) π θ​ (a∣s) (parameterized by θ \theta θ) to sample an action in a given state.
• Execute the action in the environment and observe the reward.
• Calculate the gradient of the log probability of the action times the reward: ∇ θ log ⁡ π θ ( a ∣ s ) × R \nabla_{\theta} \log \pi_{\theta}(a|s) \times R ∇ θ​ lo g π θ​ (a∣s) × R.
• Adjust the weights θ \theta θ in the direction of the gradient to maximize expected rewards.

Implementation for FrozenLake using gym and PyTorch #

• Setup: Install and import necessary libraries.

!pip install gym torch

import gym
import torch
import torch.nn as nn
import torch.optim as optim

• Policy Network: We’ll represent our policy with a simple neural network.

class PolicyNet(nn.Module):
    def init(self, n_states, n_actions):
        super(PolicyNet, self).init()
        self.fc = nn.Linear(n_states, n_actions)
        
    def forward(self, x):
        return torch.softmax(self.fc(x), dim=-1)

• Training Loop:

env = gym.make('FrozenLake-v0', is_slippery=False)
policy = PolicyNet(env

You:

i prefer tensorflow over pytorch (unless you have reasons to suggest pytorch over the other)

ChatGPT:

No problem! TensorFlow is an excellent choice. I’ll provide the implementation using TensorFlow. Given your preference, we’ll use TensorFlow 2.x, which integrates the Keras API, making it easier to define and train models.

Policy Gradient for FrozenLake using gym and TensorFlow #

• Setup: Install and import necessary libraries.

!pip install gym tensorflow

import gym
import numpy as np
import tensorflow as tf
from tensorflow import keras

• Policy Network: We’ll represent our policy with a simple neural network using Keras.

n_states = env.observation_space.n
n_actions = env.action_space.n

policy_model = keras.models.Sequential([
    keras.layers.Dense(n_actions, activation='softmax', input_shape=[n_states])
])

• Sampling Action: Given the current state, this function will return an action sampled according to the policy.

def sample_action(policy, state):
    state_one_hot = tf.one_hot(state, depth=n_states)
    action_probs = policy_model.predict(state_one_hot[np.newaxis])
    return np.random.choice(n_actions, p=action_probs[0])

• Training Loop:

optimizer = keras.optimizers.Adam(learning_rate=0.01)
loss_fn = keras.losses.sparseCategoricalCrossentropy()

for episode in range(500):
    episode_rewards = []
    episode_grads = []
    state = env.reset()
    done = False

    while not done:
        with tf.GradientTape() as tape:
            state_one_hot = tf.one_hot(state, depth=n_states)
            action_probs = policy_model(state_one_hot[np.newaxis])
            action = sample_action(action_probs)
            loss = -tf.math.log(action_probs[0][action])
        
        grads = tape.gradient(loss, policy_model.trainable_variables)
        episode_grads.append(grads)
        
        next_state, reward, done, _ = env.step(action)
        episode_rewards.append(reward)
        
        state = next_state

    total_episode_reward = sum(episode_rewards)
    for grads in episode_grads:
        optimizer.apply_gradients(zip([grad * total_episode_reward for grad in grads], policy_model.trainable_variables))

• Testing the Policy:

def test_policy(policy_model, num_episodes=100):
    total_rewards = 0.0
    for episode in range(num_episodes):
        state = env.reset()
        episode_rewards = 0.0
        done = False
        while not done:
            state_one_hot = tf.one_hot(state, depth=n_states)
            action_probs = policy_model(state_one_hot[np.newaxis])
            action = np.argmax(action_probs)
            state, reward, done, _ = env.step(action)
            episode_rewards += reward
        total_rewards += episode_rewards
    return total_rewards / num_episodes

print("Policy success rate:", test_policy(policy_model) * 100, "%")

Here, we use the REINFORCE algorithm to update the policy directly based on the total rewards received in each episode. This TensorFlow implementation should provide you with a solid understanding of the basic policy gradient method.

As always, further optimization and tuning can improve the performance, and exploring more advanced variants of policy gradients can be a great next step in your RL journey.

You:

sorry it seems pytorch gained a lot of traction recently, i’ll go with pytorch

ChatGPT:

Absolutely, PyTorch has indeed gained significant traction in the research community and offers a very intuitive and dynamic computational graph, which many find easier for prototyping and experimenting. Let’s continue with the PyTorch implementation for REINFORCE on the FrozenLake problem.

Policy Gradient for FrozenLake using gym and PyTorch #

• Setup: Install and import necessary libraries.

!pip install gym torch

import gym
import numpy as np
import torch
import torch.nn as nn
import torch.optim as optim

• Policy Network:

class PolicyNet(nn.Module):
    def init(self, n_states, n_actions):
        super(PolicyNet, self).init()
        self.fc = nn.Linear(n_states, n_actions)
        
    def forward(self, x):
        return torch.softmax(self.fc(x), dim=-1)

• Sampling Action:

def sample_action(policy, state):
    state_one_hot = torch.eye(n_states)[state]
    action_probs = policy(state_one_hot)
    return torch.multinomial(action_probs, 1).item()

• Training Loop:

env = gym.make('FrozenLake-v0', is_slippery=False)
n_states = env.observation_space.n
n_actions = env.action_space.n

policy = PolicyNet(n_states, n_actions)
optimizer = optim.Adam(policy.parameters(), lr=0.01)

for episode in range(500):
    episode_rewards = []
    episode_log_probs = []
    state = env.reset()
    done = False

    while not done:
        state_one_hot = torch.eye(n_states)[state]
        action_probs = policy(state_one_hot)
        action = sample_action(action_probs)
        log_prob = torch.log(action_probs[action])
        
        next_state, reward, done, _ = env.step(action)
        
        episode_rewards.append(reward)
        episode_log_probs.append(log_prob)
        
        state = next_state

    total_episode_reward = sum(episode_rewards)
    
    loss = []
    for log_prob in episode_log_probs:
        loss.append(-log_prob * total_episode_reward)
    loss = torch.stack(loss).sum()

    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

• Testing the Policy:

def test_policy(policy, num_episodes=100):
    total_rewards = 0.0
    for episode in range(num_episodes):
        state = env.reset()
        episode_rewards = 0.0
        done = False
        while not done:
            state_one_hot = torch.eye(n_states)[state]
            action_probs = policy(state_one_hot)
            action = torch.argmax(action_probs).item()
            state, reward, done, _ = env.step(action)
            episode_rewards += reward
        total_rewards += episode_rewards
    return total_rewards / num_episodes

print("Policy success rate:", test_policy(policy) * 100, "%")

This PyTorch code applies the REINFORCE algorithm to the FrozenLake problem. The dynamic nature of PyTorch’s computational graph and its intuitive tensor operations make it a powerful framework for reinforcement learning.

You:

so, the overall structure is similar to the plain Q learning algorithm. i guess crucial different step is env.step(action)

but before we go into that, tell me what is action_probs, action, log_prob conceptually (with previous plain python example, i guess action_probs, are list of probabilities for each action, action is chosen by multinomial (but possibly different from previous version, so i need explanation on this), log_prob is just log ?

ChatGPT:

Absolutely, the overarching structure of most reinforcement learning algorithms tends to be similar, where the agent interacts with the environment, observes outcomes, and then updates its strategy. However, the details of how the strategy (or policy) is updated can vary significantly.

Let’s break down the concepts:

• action_probs (action_probs): This is a probability distribution over all possible actions in the current state as outputted by the policy network. For the FrozenLake environment with 4 actions (UP, DOWN, LEFT, RIGHT), action_probs is a list (or tensor) of 4 probabilities summing to 1.

In our Q-learning example, the agent chose an action based on an epsilon-greedy strategy derived from the Q-values. In the policy gradient method, the agent directly outputs the probabilities of each action through the policy network.

For instance, if action_probs for a state is [0.1, 0.5, 0.2, 0.2], it means that in the current state and with the current policy, the agent believes it should move UP with a 10% probability, DOWN with a 50% probability, LEFT with a 20% probability, and RIGHT with a 20% probability.

• action (action): Once we have action_probs, the next step is to decide which action to actually take. Instead of always taking the action with the highest probability (greedy approach), we sample an action based on the probabilities in action_probs. This introduces stochasticity in the agent’s actions, which is crucial for exploration, especially in the early phases of learning.

The function torch.multinomial(action_probs, 1) does this sampling for us. It takes a weighted random choice over the action space. Using the previous example of action_probs, the action DOWN would be chosen with a 50% chance, while the action UP would be chosen with only a 10% chance.

• log_prob (log_prob): Once an action is sampled from action_probs, we compute the logarithm of its probability. The reason for this is twofold:
  • Mathematically, policy gradient methods involve optimizing the expected cumulative reward weighted by the log probabilities of the actions taken. The logarithm helps in emphasizing actions that lead to higher rewards and discouraging those that lead to lower rewards.
  • From an optimization standpoint, using logarithms can lead to more stable and efficient updates, especially when dealing with very small probabilities.

In our plain Python Q-learning example, the agent used Q-values to determine the best action and then learned by comparing the predicted Q-value with the experienced reward. In the policy gradient method, the agent is directly adjusting the probabilities of actions to maximize the expected reward. The use of log probabilities in the learning process helps in adjusting these probabilities in the right direction based on the received rewards.

You:

action_probs = policy(state_one_hot) action = torch.multinomial(action_probs, 1).item()

action_probs are non zero but action is always 0

ChatGPT:

If the action is consistently returning 0 when sampling from action_probs, a few things could be causing this:

• Initialization Issue: The initial weights of the policy network might be set in such a way that one action’s logit is significantly higher than the others. When passed through a softmax activation (to get action_probs), this could result in an overwhelming probability for one action, making it likely to be sampled every time. You might want to double-check the initialization method for your network’s weights.
• Learning Rate: An overly high learning rate might cause the policy to converge prematurely to a suboptimal action or oscillate. Try reducing the learning rate.
• Numerical Stability: Sometimes numerical issues, especially when working with probabilities and logarithms, can cause unexpected behavior. It’s useful to:
  • Add a small epsilon (like 1 e − 8 1e-8 1e −
  8. when taking logarithms to avoid log(0).
  • Check if the output of the softmax (in action_probs) truly sums to 1 for each state.
  • Ensure there’s no inadvertent modification to the action_probs tensor after it’s computed.
• Environment Dynamics: If you’re using the non-slippery version of FrozenLake (with is_slippery=False), the environment is deterministic. Even so, the action sampling should still have variability, especially in the early episodes.
• Debugging Tips:
  • Print the action_probs for multiple states and see if there’s a pattern or if one action’s probability is consistently higher.
  • Ensure the gradients are being computed and applied correctly during training.
  • Double-check the architecture and forward pass of your policy network.
• Model Complexity: Ensure your policy model is appropriate for the task. For the FrozenLake problem, the model is simple. However, if there are hidden layers or additional complexities, it might be worth simplifying or revisiting the architecture.

If you’ve validated all of the above and are still facing the issue, there might be other specifics to consider. Please share any additional observations, and I’ll do my best to assist further!

You:

next_state, reward, done, _ = env.step(action)

env.step seems to return same value over and over

ChatGPT:

If env.step(action) seems to be returning the same next_state and reward over and over for different action values, there are a few potential reasons and ways to diagnose the issue:

• Environment Dynamics:
  • For the non-slippery version of FrozenLake (is_slippery=False),

You:

i’m using your code, so slippery is false, and action, state is not changing .. in the loop

ChatGPT:

Alright, let’s diagnose this step by step.

• Start State: The agent always starts at the top-left corner of the FrozenLake grid, which corresponds to state 0.
• Action Determination: Let’s check the action probabilities and the actions being taken.
  • At the start of each episode, and in the first state (top-left corner), print out the action_probs tensor. This will let us see the probabilities being assigned to each action.

   Copy code
   print(action_probs)

• Next, print out the chosen action:

   Copy code
   print(action)

If action is always 0, even when action_probs seems to give non-zero probabilities for other actions, then there might be an issue with the action sampling process. However, if action_probs is heavily biased towards a single action, it could be a policy network or initialization issue.

• Effect of Actions: The next thing to check is the effect of the chosen action.
  • After you get next_state from env.step(action), print out the next_state:

   Copy code
   print(next_state)

For the non-slippery version of the FrozenLake environment:

• Action 0 should result in moving left (when possible).
• Action 1 should move the agent down.
• Action 2 should move the agent right.
• Action 3 should move the agent up.

If the agent starts at the top-left corner and the chosen action is always 0, the agent will keep hitting the wall on its left and remain in the same spot.

To Diagnose Further:

• If the action is always the same (like always 0), then we need to inspect why the policy network or action sampling process is biased in this way.
• If the action varies but the next_state doesn’t change, we need to understand if the actions being taken aren’t causing any movement (like continuously moving into walls).

By printing and understanding the outputs for action_probs, action, and next_state, we should get a clearer picture of what’s happening.

You:

action_probs tensor([0.2824, 0.3099, 0.1699, 0.2377], grad_fn=)

next_state 0 state 0

ChatGPT:

Thank you for providing the action_probs values.

Given the probabilities you provided, the agent is choosing different actions with a reasonable chance (28.24% for LEFT, 30.99% for DOWN, 16.99% for RIGHT, and 23.77% for UP). If action is indeed being sampled from this distribution, we would expect the agent to move fairly often, especially since there’s a ~31% chance of moving DOWN, which would be a valid move from the start state.

However, the fact that next_state is still 0 after taking an action suggests a couple of possibilities:

• Sampling Issue: The action sampling process via torch.multinomial is not sampling the actions as expected.
• Environment Dynamics: Given that is_slippery=False, the environment should behave deterministically. But if there’s any misconfiguration or issue with the gym environment, it could affect the dynamics.

To pinpoint the issue, let’s do the following:

• Print the sampled action:

After this line:

Copy code
action = torch.multinomial(action_probs, 1).item()

Add a print statement:

Copy code
print("Sampled Action:", action)

We want to check if the sampled action corresponds to the given probabilities.

• Directly Set the Action:

Temporarily override the action to a specific value (like 1 for DOWN) and check if the state changes:

Copy code
action = 1  # This should move the agent DOWN from the starting state
next_state, reward, done, _ = env.step(action)
print("Next State after forced DOWN:", next_state)

If the agent still doesn’t move after explicitly setting the action, then there might be an issue with the gym environment. If the agent moves as expected when the action is set explicitly, but not when sampled, then the problem lies in the action sampling mechanism.

You:

after setting action=1

sampled action gives different values

Sampled Action: 0 Sampled Action: 1 Sampled Action: 3

ChatGPT:

Great, this indicates that the action sampling mechanism with torch.multinomial is working as intended because it’s sampling various actions based on their probabilities.

Now, the crux of the matter lies in understanding why the agent’s state isn’t changing even when a valid action is sampled.

Let’s try a few more diagnostic steps:

• Explicit Movement: Let’s try moving in all directions explicitly from the start state and observe the next state:

Copy code
state = env.reset()

for action in range(4):
    next_state, _, _, _ = env.step(action)
    print(f"Moving with action {action}, Next state: {next_state}")
    env.reset()  # Reset environment after each step

This should help us confirm if the environment responds as expected to all actions from the starting state.

• Environment Version: There might be updates or changes to the gym environment. Ensure you’re using a version that behaves as expected. Consider reinstalling or updating the library or creating a fresh virtual environment to rule out any inconsistencies.
• Custom Step Logic: If the environment isn’t behaving as expected even with explicit actions, you can implement a custom step function for the FrozenLake environment. This would be a more involved step, but it provides full control over the environment dynamics.
• Observation Space: Double-check if there’s any wrapper or modification to the environment that could change the observation space. Normally, env.step(action) should change the state of the environment, but if there’s any change or wrapper that resets the state or affects the dynamics, it could lead to the observed behavior.

By running the explicit movement diagnostic and checking the environment’s version and setup, we should be closer to identifying the root cause.

You:

Moving with action 0, Next state: 0 Moving with action 1, Next state: 4 Moving with action 2, Next state: 1 Moving with action 3, Next state: 0

ChatGPT:

The results from the explicit movement diagnostic make sense:

• Action 0 (LEFT) in state 0 leads to state 0 (hitting the wall and staying in place).
• Action 1 (DOWN) in state 0 leads to state 4 (moving down one row).
• Action 2 (RIGHT) in state 0 leads to state 1 (moving to the right).
• Action 3 (UP) in state 0 leads to state 0 (can’t move up from the top row, so it stays in place).

The environment seems to be behaving correctly when we’re setting actions explicitly.

So, the issue seems to arise in the context of your training loop. Specifically, somewhere between sampling the action, executing it, and then transitioning to the next state.

To diagnose further:

• Training Loop Examination: Within your training loop, right after the action is sampled and before the env.step(action) call, print out the action and then the resulting next_state:

Copy code
action = torch.multinomial(action_probs, 1).item()
print("Sampled Action:", action)
next_state, _, _, _ = env.step(action)
print("Resulting State:", next_state)

This will give a clear picture of how the environment is reacting to each sampled action during training.

• State Reset: Ensure that you’re resetting the environment (with env.reset()) at the beginning of each episode. If the environment isn’t being reset, the agent could keep starting in states where movements have limited effects (e.g., near walls or in terminal states).
• Environment State: Ensure there’s no accidental manipulation of the environment state (like setting it to a fixed value) anywhere in the training loop.

By closely examining the training loop and the agent-environment interaction within that context, we should be able to pinpoint the issue. The fact that explicit actions are working as expected suggests that the environment is set up correctly, and the issue might lie in how actions are being processed or the state is being managed within the training loop.

You:

well didn’t we find the problem already, it’s the initial value of probs, that leads to left action at start, which gets stuck.

somehow every episode is run with same initial probabilities (NN weights i guess)

ChatGPT:

Ah, I see what you’re getting at. If the policy network consistently initializes in such a way that it prefers the LEFT action (especially early in training) and does not learn to adjust away from that suboptimal policy, the agent can get stuck, especially in the deterministic setting of is_slippery=False.

Here are some possible solutions and checks:

• Network Initialization: The default initialization of weights in PyTorch’s Linear layer is Kaiming uniform. This might lead to biases in the initial action distributions. You can experiment with other initializations like Xavier (Glorot) or uniform distributions within a narrow range to see if they produce a more balanced initial action distribution.
• Learning Rate: If the learning rate is too small, the policy network might not be updating significantly between episodes. This would mean the initial bias in action selection persists across episodes. Try adjusting the learning rate.
• Reward Signals: Ensure that the agent is getting a clear reward signal that allows it to learn. If the agent isn’t receiving any positive reward or doesn’t get enough episodes where it happens to explore the right path by chance, it might not learn to adjust its policy.
• Epsilon-Greedy Exploration: While policy gradient methods naturally incorporate exploration through the stochastic nature of the policy, in environments where certain actions can result in no progress (like continuously going left in FrozenLake), it can be beneficial to add an epsilon-greedy exploration strategy on top of the policy network’s output, especially in the early phases of training.
• Baseline or Advantage: Pure REINFORCE can have high variance. Using a baseline or advantage function can reduce this variance, making learning more stable. This involves a bit more work, as you’ll need another network (or method) to estimate state values, but it can significantly stabilize and speed up training.
• Network Architecture: While the FrozenLake problem is simple, sometimes adding a bit of complexity to the network (like an additional hidden layer) can help in learning a good policy, especially if combined with techniques like batch normalization.

In summary, if every episode starts with the same initial probabilities that heavily favor the LEFT action and this doesn’t adjust quickly, the agent can indeed get stuck. Solutions involve tweaking network initialization, adjusting learning rates, ensuring strong reward signals, and potentially augmenting the exploration strategy or the learning algorithm itself.

You:

yeah increaing learning rate solves the problem

ChatGPT:

Great to hear!

A higher learning rate allows for larger updates to the network’s weights during the backpropagation process. This means that the policy can adjust more rapidly in response to the received reward signals.

In the case of the FrozenLake environment with is_slippery=False, the environment is deterministic. Thus, once the agent starts getting a sense of the right actions to take (by occasionally stumbling upon the right path), it can quickly adjust its policy with a higher learning rate to reinforce that path.

However, keep in mind the following as you continue to work with RL:

• Balance: A higher learning rate can speed up learning, but it can also introduce oscillations or instability. It’s essential to find a balance, especially as you work with more complex environments or networks.
• Adaptive Learning Rates: Modern optimizers like Adam adjust the learning rate on-the-fly, which can often help in finding a good learning pace without manual tuning.
• Exploration vs. Exploitation: Remember the importance of balancing exploration and exploitation. In environments with more complex or stochastic dynamics, exploration strategies become crucial.
• Environment Dynamics: The optimal learning rate can vary based on the dynamics of the environment. Deterministic environments like the non-slippery FrozenLake might require different settings than stochastic ones.

It’s great that increasing the learning rate worked for this problem. As you delve deeper into RL and tackle more complex problems, you’ll find that hyperparameter tuning, including learning rate adjustments, can play a pivotal role in algorithm performance.

You:

ok i need some explanation,

the lerning should happen inside policynet,

code seems to run whole episode , then update policy, and policy needs to differentiate [state , action-taken] to do learning i guess. but i don’t see it happening, learning is using an aggregated data.. (i think the connection is through optimizer and loss, i can see optimizer is connected to policy, but how is loss connected..)

ChatGPT:

Certainly! Let’s break down the learning process in the context of the REINFORCE algorithm using policy gradients.

• What Are We Learning? In Q-learning, we learned a value function (Q-values) which represents the expected future reward for taking an action in a given state. In policy gradient methods, we’re learning a policy directly. The policy, in this case, is a neural network (the PolicyNet) that, given a state, outputs a probability distribution over actions.
• How Are We Learning? The goal in REINFORCE is to maximize the expected cumulative reward by adjusting the probabilities of the actions taken. The intuition is:
  • If an action results in a good outcome (positive reward), we want to increase its probability in that state.
  • If an action results in a bad outcome (negative reward), we want to decrease its probability.
• The Loss & Gradient: The objective function in REINFORCE is the expected cumulative reward, weighted by the log probabilities of the actions taken. We aim to maximize this objective. However, in deep learning frameworks like PyTorch, we typically minimize a loss. So, the REINFORCE “loss” (which we aim to minimize) is the negative of this objective: Loss = − ∑ log ⁡ ( π θ ( a t ∣ s t ) ) × R t \text{Loss} = -\sum \log(\pi_{\theta}(a_t|s_t)) \times R_t Loss = − ∑ lo g( π θ​ (a t​ ∣s t​ )) × R t​ where:
  • π θ ( a t ∣ s t ) \pi_{\theta}(a_t|s_t) π θ​ (a t​ ∣s t​ ) is the probability of taking action a t a_t a t​ in state s t s_t s t​ under policy π \pi π parameterized by θ \theta θ.
  • R t R_t R t​ is the cumulative reward from time t t t to the end of the episode.
• How the Loss Connects to the Policy: The PolicyNet produces the action probabilities. When we compute the loss (using the log probabilities of the taken actions and the rewards), the gradients of this loss with respect to the network’s parameters ( θ \theta θ) indicate how we should adjust the parameters to increase the probabilities of actions that lead to higher rewards and decrease the probabilities of actions that lead to lower rewards.
• Aggregated Data: You’re right; the algorithm aggregates data over the episode. This is because REINFORCE is a Monte Carlo method, meaning it waits until the end of the episode to compute the returns R t R_t R t​ and update the policy. Each step in the episode contributes to the loss, which is why we sum over the log probabilities weighted by rewards.
• Backpropagation & Update: Once the loss for an episode is computed, we perform backpropagation to compute the gradients with respect to the network’s parameters. The optimizer (like Adam) then uses these gradients to adjust the parameters. This connection between the loss and the policy network is established through the computation graph in PyTorch. When you call loss.backward(), PyTorch automatically computes the gradients for all tensors (that have requires_grad=True) involved in computing that loss.

To summarize:

• We run an episode using the current policy to sample actions.
• We compute the return/reward for each timestep in the episode.
• We compute the loss based on the log probabilities of the actions taken and the returns.
• We backpropagate through this loss to get gradients for the PolicyNet.
• We use an optimizer to adjust the weights of the PolicyNet based on these gradients.
• This process gradually adjusts the policy towards one that achieves higher expected rewards.

You:

let’s clarify the basics first

“If an action results in a good outcome (positive reward), we want to increase its probability in that state.”

i see rewards and loss are aggregated (total_episode_reward, loss = torch.stack(loss).sum())

so we only have reward and loss for entire episode, how do we dicern(?) state, (increase probability for desired state or action) when we have only aggregated data , when I think the learning is happening at the optimizer.zero_grad() loss.backward(), optimzer.step() (which are after rewards/loss are summed so, individual values are lost)

ChatGPT:

Certainly, let’s dissect this further.

The crux of policy gradient methods is to adjust the policy based on the trajectory of the agent in an episode, specifically through the sequence of states, actions, and rewards experienced. The key insight here is that while the rewards and loss might be aggregated over an episode, the gradient of the loss with respect to the network’s parameters inherently carries information about the individual state-action pairs.

Here’s how:

• Gradient Computation: When we call loss.backward(), PyTorch computes the gradients for every parameter in the network based on the loss value. These gradients tell us how each parameter should be adjusted to minimize the loss.
• Trajectory Information: The loss is a sum over the log probabilities of actions taken in every timestep, each weighted by its corresponding return. When we compute the gradient of this summed loss, the gradient inherently considers every term in the sum, effectively considering every state-action pair in the trajectory.
• Adjusting Probabilities: Suppose, in a particular state, an action was taken that led to a high return. The log probability of this action will be weighted by this high return in the loss. When computing the gradient, the parameters of the network that contributed to this action’s probability in this state will have gradients that encourage increasing this action’s probability in future episodes. Conversely, if an action in a state resulted in a low return, the gradient will encourage decreasing this action’s probability.
• Parameter Update: Once we have these gradients, optimizer.step() adjusts each parameter based on its gradient. This means parameters that influenced action probabilities in states leading to high returns will be adjusted to increase those probabilities, and vice versa.
• Aggregation: The reason for aggregating (summing) the rewards and losses is computational efficiency and alignment with the REINFORCE algorithm. By computing a single scalar loss value for the entire episode and then computing gradients based on this, we effectively backpropagate the outcome of the entire episode through the trajectory of state-action pairs.
• Why Not Update After Every Step?: In theory, we could compute a separate loss for each state-action pair and update the network after every step. However, this would be computationally inefficient. Also, policy gradient methods like REINFORCE rely on the idea of adjusting the policy based on the cumulative future reward (return) from each state, which inherently requires considering the full trajectory (or at least the trajectory from a state to the end of an episode).

To summarize, even though the rewards and losses are aggregated, the gradient computation inherently takes into account each state-action pair’s contribution to the loss. Thus, the learning process does differentiate between different states and actions and adjusts the policy appropriately.

You:

ok, so policy(state) call , is forward step, and it knows all the state transitions (because it knows every state we visited)

.. so reward and loss are, distributed over all the state / action(the link betwen state) , it might not be acurate as doing update every step, but we get to do more episode iterations..

ChatGPT:

Exactly, you’ve grasped the essence of it!

To reiterate and clarify:

• Forward Step with policy(state): Every time we call policy(state), we’re doing a forward pass through the neural network for that particular state. This produces action probabilities based on the current parameters of the policy network.
• Storing Trajectory Information: As the agent interacts with the environment, we store information about states, actions, and rewards. The log probabilities of the actions (as produced by the policy network) and the returns/rewards are used to compute the loss.
• Gradient Carries Information: When we compute the gradient of the loss, it takes into account how each state-action pair contributed to the loss. The computed gradients reflect the desired changes in the network’s parameters based on the trajectory (sequence of state-action-reward) in the episode.
• Distributed Learning: The aggregated reward and loss for an episode effectively distribute the outcome of the episode over all the state-action pairs in the trajectory. This aggregated approach in REINFORCE (and many other policy gradient methods) aims to adjust the policy based on long-term outcomes (i.e., how actions influence not just immediate rewards but also future rewards).
• Efficiency: As you rightly pointed out, updating the policy after each episode (instead of after every step) allows for more efficient computation, as we can process many steps with a single backpropagation and optimization step. This also aligns with the idea of considering long-term rewards. It might sacrifice a bit of granularity, but it’s more computationally efficient and can still lead to effective policy learning.
• More Episode Iterations: By being more efficient with our updates, we can indeed afford to run more episodes, which means more exploration and more opportunities for the agent to learn from various trajectories.

In more advanced reinforcement learning algorithms, there are methods like Actor-Critic that combine the ideas of value-based methods (like Q-learning) and policy-based methods (like REINFORCE) to get the best of both worlds and often achieve more stable and faster learning. But understanding the foundational concepts, as you are doing now with REINFORCE, is crucial to grasping those more advanced methods!

You:

i guess i need to go over basic NN concepts. but before going into that, can you give slightly mathematical but conceptual explanation on the frozen-lake example,

what is ‘forward step’ doing what is ‘backward’ step doing, and what is optimizer.step

so basic NN concept in this example with bit of conceptual mathematical explanation

ChatGPT:

Certainly! Let’s dive into the fundamental concepts of neural networks in the context of the FrozenLake example, focusing on the forward pass, the backward pass, and the optimizer’s role.

The Neural Network (Policy Network) #

For the FrozenLake environment, our neural network (often called the “Policy Network” in this context) aims to approximate the optimal policy. The policy is a function that takes in a state and produces a probability distribution over possible actions.

Given the state $s$s, our policy network (let’s call it $\pi_\theta$πθ​ where $\theta$θ represents the network parameters or weights) produces probabilities for each action:

$\pi_\theta(s) = [P(a_0|s), P(a_1|s), …, P(a_{n-1}|s)]$πθ​(s)=[P(a0​∣s),P(a1​∣s),…,P(an−1​∣s)]

Here, $n$n is the number of possible actions.

Forward Step #

The forward step refers to the process of passing a state $s$s through the policy network to get the action probabilities.

Mathematically, for a simple linear layer followed by a softmax, it’s:

$z = s \times W + b$z=s×W+b $\pi_\theta(s) = \text{Softmax}(z)$πθ​(s)=Softmax(z)

Where:

• s s s is the input state.
• W W W represents the weights of the network.
• b b b represents the biases.
• z z z is the raw output (often called logits).
• Softmax converts these logits into probabilities.

In the context of PyTorch:

action_probs = policy(state)

This line is essentially performing the forward step, where state is passed through the network to produce action_probs.

Backward Step #

The backward step, or backpropagation, is the process of computing the gradient of the loss with respect to the network’s parameters.

Given a loss function $L(\theta)$L(θ), the backward step computes:

$\frac{\partial L}{\partial \theta}$∂θ∂L​

This gradient tells us how the loss would change for a small change in each parameter, which gives us a direction in which to adjust our parameters to minimize the loss.

In the context of PyTorch, when we do:

loss.backward()

We’re computing the gradient of loss with respect to all tensors (like the weights and biases of our policy network) that have requires_grad=True.

Optimizer Step #

Once we have the gradients, we need to adjust our parameters in the direction that minimizes the loss. This is the job of the optimizer.

For an optimizer like SGD (Stochastic Gradient Descent), the update rule is:

$\theta_{\text{new}} = \theta_{\text{old}} - \alpha \frac{\partial L}{\partial \theta}$θnew​=θold​−α∂θ∂L​

Where:

• α \alpha α is the learning rate.
• ∂ L ∂ θ \frac{\partial L}{\partial \theta} ∂ θ ∂L​ is the gradient computed during the backward step.

More advanced optimizers, like Adam, have more sophisticated update rules that consider past gradients and other factors, but the essence is the same: adjust the parameters to minimize the loss.

In PyTorch:

optimizer.step()

This line updates the parameters of our policy network based on the gradients computed in the backward step and using the rules defined by the optimizer.

In Summary: #

• Forward Step: Pass the input through the network to get an output (action probabilities).
• Backward Step: Compute how changing the network’s parameters would affect the loss.
• Optimizer Step: Adjust the network’s parameters to decrease the loss.

In the context of the FrozenLake example, the forward step provides us with action probabilities, the backward step tells us how changing the network’s parameters would affect our performance (loss), and the optimizer step adjusts the parameters to hopefully improve our agent’s policy over time.

You:

can you give more concrete explanation, i know it’s like going upheal (or downheal) where we don’t know how mountain looks like, we incrementally adjust our direction to find the top of mountain.

what i mean by concrete is, that, given with this example, i could imagine myself a climber (who gets the input, and adjust the direction) there would be inputs, and adjusting step, (but it won’t be pictorical, but data will be just vector of numbers, but still, since it’s only 4 dimensional vector, i guess it’s feasible to artifically generate feasible data that shows how adjusting step looks like)

ChatGPT:

Absolutely! Let’s imagine you’re climbing a mountain in a 4-dimensional space, representing our FrozenLake example. Each dimension corresponds to the probability of one of the four actions (LEFT, DOWN, RIGHT, UP). Your goal is to find the peak, which corresponds to the optimal action probabilities for each state.

The Mountain Landscape (Loss Surface):

• Think of the entire landscape as the combination of all possible sets of probabilities for the actions.
• The height at any point in this landscape is the opposite of the reward your agent would get, on average, by following the policy defined by that point. So, higher elevations mean worse policies, and lower elevations mean better policies. Your goal is to find the lowest point.

You (The Climber) - Forward Step:

• At your current position, you decide which direction to move in by using a flashlight (your neural network) that shows you the slope of the ground around you (action probabilities).
• Your flashlight (policy network) takes in your current state (e.g., a specific position in the FrozenLake grid) and tells you which direction has the steepest slope (gives action probabilities).
• You then move a tiny step in one of these directions (take an action) and see how it feels (get a reward).

Your GPS Device - Backward Step:

• After your journey (an episode), you look back and evaluate how good or bad your steps were. If a particular direction consistently led you to slippery spots or dead-ends (holes or the start position), you know that was a bad decision.
• You have a GPS device that can tell you how much to adjust your flashlight’s guidance (compute gradients) based on your journey’s outcome (cumulative reward).
• It basically helps you realize: “Whenever I was in ’this’ state, and my flashlight told me to go ’that’ direction, things didn’t turn out well. So, let’s adjust it.”

Your Backpack’s Tools - Optimizer Step:

• Based on your GPS device’s feedback, you have tools in your backpack that allow you to tinker with your flashlight (adjust the neural network weights) so that next time, in similar situations, it’ll guide you slightly differently.
• The exact way you adjust your flashlight is based on the tools you have. If you have basic tools (SGD), you just make straightforward adjustments. If you have advanced tools (Adam), you remember your past adjustments and combine that knowledge with your current situation.

Visualizing in 4D:

• Imagine a 3D plot where the x, y, and z axes represent the probabilities of taking actions LEFT, DOWN, and RIGHT, respectively. The vertical color gradient (from cold to warm) represents the probability of UP and also indicates the reward magnitude.
• Your position on this plot represents the current policy for a specific state. The landscape’s height and color at each point indicate how good that policy is for that state.
• As you train your agent, your position on this landscape moves, hopefully, towards the optimal action probabilities for that state.

Now, repeating this visualization for every state in FrozenLake gives you an idea of how the entire learning process progresses. You’re adjusting your policy (flashlight) for each state based on the outcomes of your actions (journeys). Over time, with enough exploration and correct adjustments, you hope to find the best paths (policies) for most, if not all, states.

You:

yes, i’m asking if you can provide sample (plausible) data, which will show how actual computation is happening (which was expressed by mathematical formular), so that I can relate the concept (of climbing) to the mathematical expression more firmly

ChatGPT:

Sure! I’ll walk you through a simplified scenario for one step, providing mock data that’s plausible. This won’t be as complex as a real neural network, but it’ll help conceptualize the process.

Scenario: Let’s say you’re at a particular state in the FrozenLake, and you have a very simple policy network. For the sake of this example, let’s assume our policy network is just a single-layer network (no hidden layers), mapping the state directly to action probabilities.

1. Forward Step: #

Network’s Weights and Biases:

W = [[0.2, 0.1, 0.3, 0.5],
     [0.3, 0.2, 0.1, 0.4],
     [0.4, 0.5, 0.3, 0.2],
     [0.5, 0.4, 0.2, 0.1]]

b = [0.1, 0.1, 0.1, 0.1]

Where each row in W represents the weights for one state, and b is the bias.

Given State (for simplicity, assume it’s one-hot encoded):

state = [0, 1, 0, 0]  # Represents the second state

Compute Action Probabilities:

z = state x W + b
  = [0.3, 0.3, 0.2, 0.5]  # Raw outputs (logits)

action_probs = Softmax(z)
  = [0.23, 0.23, 0.21, 0.33]  # Roughly computed for simplicity

You decide to take the action with the highest probability, i.e., action = 3 (UP).

2. Backward Step: #

Let’s say after taking the action, you fall into a hole and the episode ends with a total reward of -1.

Loss Computation: Using the REINFORCE formula, the loss for this step is:

loss = -log(action_prob of taken action) x reward
     = -log(0.33) x (-1)
     = 0.41  # Simplified

Gradient Computation: For this simple network, the gradient can be thought of as how much each weight contributed to the error (loss). For a real neural network, this involves chain rule calculus, but for our purposes:

Let’s say the computed gradient for our weights related to state 2 is:

gradient_W = [0.01, 0.02, -0.01, -0.02]
gradient_b = 0.02  # For all actions

These gradients indicate how the loss would change if we adjusted the weights and biases by a tiny amount.

3. Optimizer Step: #

We’ll adjust our weights using these gradients. Let’s use a simple learning rate of 0.1.

W[1] = W[1] - 0.1 x gradient_W
     = [0.29, 0.28, 0.21, 0.42]  # Updated weights for state 2

b = [0.08, 0.08, 0.09, 0.08]  # Updated biases

After this update, the next time you’re in state 2, your policy network will produce slightly different action probabilities, hopefully guiding you towards a better action.

This process, when repeated over many episodes and for all states, will ideally guide you to an optimal or near-optimal policy for navigating the FrozenLake.

You:

can you expand on gradient calculation? because, i’m used to thinking of calculating a slope when given a function, but I havent’ seen the function for this example. so i’m not sure how we calucate the slope.

ChatGPT:

Certainly! Gradients (or derivatives) capture the sensitivity of one quantity to changes in another. In neural networks, the gradient tells us how much the loss will change if we adjust the network’s parameters (weights and biases) by a tiny amount. The computation of this gradient involves the chain rule of calculus and is performed through a process called backpropagation.

Let’s delve into the gradient computation for our example.

The Function: #

The function in the context of neural networks is the composition of all operations involved in calculating the loss from the input. For our simple example, this function can be represented as:

$L(\theta) = \text{LossFunction}(\text{Softmax}(s \times W + b))$L(θ)=LossFunction(Softmax(s×W+b))

Where:

• L ( θ ) L(\theta) L( θ) is the loss given the parameters θ \theta θ (weights W W W and biases b b b in our case).
• s s s is our state.
• The Softmax and matrix multiplication are part of the forward step.
• The LossFunction is, in our context, the negative log likelihood of taking the selected action, weighted by the return.

Gradient Computation: #

To compute the gradient, we need to see how a small change in our weights $W$W and biases $b$b would affect the overall loss $L$L.

This is computed using the chain rule:

$\frac{\partial L}{\partial W} = \frac{\partial L}{\partial z} \times \frac{\partial z}{\partial W}$∂W∂L​=∂z∂L​×∂W∂z​

Where:

• z z z is the raw output (logits) from the network before the Softmax, i.e., z = s × W

b z = s \times W + b z #

s × W + b.

• ∂ L ∂ z \frac{\partial L}{\partial z} ∂ z ∂L​ captures how the loss changes with a small change in the logits.
• ∂ z ∂ W \frac{\partial z}{\partial W} ∂ W ∂ z​ captures how the logits change with a small change in the weights.

This is the essence of backpropagation: we’re computing how the loss changes with respect to each intermediate step, all the way back to the weights and biases.

In practice:

• Softmax Gradient: We first compute the gradient of the loss with respect to the logits (raw outputs). This involves the derivative of the Softmax function and the specific loss we’re using (negative log likelihood in our case).
• Weight Gradient: With the gradient of the loss with respect to the logits in hand, we can compute the gradient of the loss with respect to the weights by multiplying it by the gradient of the logits with respect to the weights. This latter gradient is simply the input state s s s for the specific weight connections used in that step, due to the linearity of the matrix multiplication operation.

In PyTorch: #

This gradient computation is abstracted away and performed automatically. When we call loss.backward(), PyTorch uses automatic differentiation to compute the gradients of the loss with respect to all tensors (weights, biases) that have requires_grad=True. Under the hood, it’s doing the chain rule computations described above.

Summing Up: #

The gradient tells us the slope of our loss function concerning our weights and biases. By adjusting our weights in the direction opposite to this slope, we aim to decrease the loss and, consequently, improve our policy network’s predictions (action probabilities). The actual function we’re computing the slope (gradient) of is the combination of all operations (matrix multiplications, activations, loss computation) from input to loss.

You:

ok, whereas Q learning focused on [state/action] pair to update policy, here, we don’t even need to know which action or state happened for learning to take place, it’s like, we assume we know how mountain looks like, (but probably will not be correct) , and we assume our mountain can be modeled with this much weight and functions (softmax and the connection of weights) in each step, we update weight so that we can go top of hill faster, and somehow we reach top of mountiain..

so it’s like .. we assumed , we hoped, and it works.. (as opposed to there’s a certain undisputable steps that will derived the desired output (like mathematical proof)) here, it’s like heuristic search.. (to me) unless there’s some deep mathematical reasoning why this process (gussing, assuming) produces the result..

ChatGPT:

Your understanding aligns well with the intuitions and sentiments of many in the field. Let’s break down a few things:

The Nature of Deep Learning: #

• Heuristic Nature: Yes, many parts of deep learning can be viewed as heuristic. It’s about making an educated guess, based on the data, using a flexible function approximator (the neural network), and then iteratively refining that guess.
• Lack of Complete Theoretical Understanding: While there are many theoretical results related to neural networks and deep learning, we don’t yet fully understand why they work as well as they do. Especially in complex environments, it’s still a topic of active research. However, despite the lack of complete theoretical grounding, empirical results across various domains have been astonishingly good.

Reinforcement Learning vs. Supervised Learning: #

• Guess and Check Nature of RL: In supervised learning, we have clear target labels for each input, and the learning process involves minimizing the difference between the network’s predictions and these labels. However, in reinforcement learning, especially policy-based methods, we don’t have such clear labels. The agent “tries” things out, observes results, and adjusts. It’s much more about trial and error.
• Exploration vs. Exploitation: In RL, agents also have to balance exploration (trying new actions to discover their effects) and exploitation (using what it has learned to get the best possible reward). This adds another layer of complexity.

Why Does It Work?: #

• Gradient Descent and Backpropagation: While the model (neural network) and the environment (like FrozenLake) might be complex, the core idea of adjusting weights based on gradients is grounded in calculus. The idea is simple: find the direction that decreases the error and move a bit in that direction. Over time, this iterative process tends to find good (if not always the best) solutions.
• Function Approximation: Neural networks, especially deep ones, have been shown to be universal function approximators. They can, in theory, approximate any function to a high degree of accuracy. Given enough data and computation, they can capture highly complex relationships. This capability makes them suitable for RL, where the relationship between states, actions, and rewards can be intricate.
• Empirical Success: Even if we don’t have a complete theoretical understanding, the empirical success of these methods across a wide range of tasks (from playing games to controlling robots) gives confidence in their efficacy.

In conclusion, while the learning process in neural networks, especially in the RL context, might seem like heuristic search, the underlying principles are grounded in mathematics. The “guessing” and “assuming” you mentioned are guided by these mathematical principles and refined over many iterations. The beauty and challenge of RL lie in its explorative nature, where agents learn through interaction rather than being explicitly told the right action for every situation.

You:

i’ll be more specific, when i say guess..

i can see NN is grounded in math, so if i’m sure i have the correct function , i can use the “finding steepest” would lead me to a hill (it could be local maxima but still, it’s guranteed to work to find a hill)

but here we are not sure if we have the correct function, although the process of using gradient is sound, the given fact are different , it’s heuristic, not mathematicaly deterministic..

it’s like saying, i’ll give you a funciton of. mountain, but it may not be correct, but if you use gradient decent, you will reach hill.. i’m like.. ok? will i?

ChatGPT:

I see your point and it’s a very valid one. Let’s break down what you’re highlighting.

When we talk about neural networks and reinforcement learning, there are two distinct assumptions being made:

• Function Approximation with Neural Networks: This is the assumption that given a network architecture (which can be thought of as a class of functions), the function that maps our inputs to our outputs can be approximated closely enough to be useful. For supervised tasks, we’ve seen that this is generally true — with large enough networks and enough data, we can approximate the function very well. This is backed up by the universal approximation theorem, which says that a feedforward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of R^n, under mild assumptions on the activation function. However, the theorem doesn’t tell us how many neurons we might need, and it doesn’t say anything about learnability or generalization to unseen data.
• Learning Optimal Policies with Reinforcement Learning: This is where your “guessing” concern comes more into play. In RL, especially with complex environments and models, we are not always sure if the policy we’re approximating is the optimal one, or if it’s even close. This is compounded by the fact that in RL, the data distribution changes as the policy changes (because the policy’s actions influence the next states and rewards). This is unlike supervised learning, where the data distribution is fixed.

This is why reinforcement learning is particularly challenging. The landscape (or the function) we’re trying to optimize is non-stationary — it changes as the agent learns. This is fundamentally different from standard optimization problems where the landscape is fixed, and we’re just trying to find the lowest point.

So yes, while the process of gradient descent is mathematically sound, and while neural networks are powerful function approximators, combining these in the context of RL does introduce a level of uncertainty.

To handle this, researchers employ various techniques:

• Exploration: Trying different actions to learn more about the environment.
• Regularization: Penalizing complex models to prevent overfitting to the data.
• Experience Replay: Using past experiences to stabilize learning.
• Reward Shaping: Modifying rewards to guide learning.
• And many more…

But even with these techniques, there’s no guarantee that the learned policy is optimal. That’s why RL agents are often evaluated based on empirical performance over many episodes rather than theoretical guarantees.

In essence, the practice of deep reinforcement learning can be thought of as an interplay between theory (the math behind neural networks and gradient descent) and heuristics (techniques and tricks to stabilize learning and improve performance in practice).