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Assisting with Gumbel MuZero MCTS Code

Help me write code for my Gumbel MuZero MCTS code. It's unclear in my mind how to even write the Node class. Also, it's open loop MCTS.

Thought for 154 seconds

Okay, let's tackle this problem. The user wants help writing code for a Gumbel MuZero MCTS, specifically for an open-loop scenario. They mentioned uncertainty about how to structure the Node class. Alright, let's break this down step by step.

First, I need to recall what MCTS (Monte Carlo Tree Search) entails. MCTS involves building a search tree by repeatedly simulating paths from the root to a leaf node, expanding the tree, and backpropagating the results. In MuZero, the model uses learned dynamics, which means it doesn't rely on the actual environment but on its own internal model for predictions. Gumbel MuZero incorporates Gumbel noise for exploration, which is different from the standard UCT (Upper Confidence Bound for Trees) approach. Also, open-loop means that the environment isn't updated with real states during the search; instead, the model uses its own predictions to simulate trajectories.

The Node class is central to MCTS. Each node represents a state (or hidden state in MuZero's case) and keeps track of visit counts, action values, etc. For Gumbel MuZero, nodes might need additional information like completed q-values or the Gumbel noise terms. Since it's open-loop, the nodes don't correspond to actual environment states but to the model's hidden states.

Let me think about the attributes each Node should have. Typically, in MCTS, a node has:

- visit_count: number of times the node has been visited.

- prior: the prior probability of actions, from the policy network.

- value_sum: accumulated value from backpropagation.

- children: a dictionary mapping actions to child nodes.

But in Gumbel MuZero, there's the Gumbel trick. The Gumbel-Max trick is used for action selection, adding Gumbel noise to logits. In the Gumbel MuZero paper, they use a sequential allocation approach with Gumbel noise for exploration. Each node might need to store the Gumbel noise terms for each action, perhaps?

Wait, in the Gumbel MuZero approach, the first step is to sample actions without replacement using Gumbel-Top-k trick. For open-loop, the search is done without executing actions in the real environment, so all simulations are based on the model's predictions.

The Node class might need:

- For each action, the Q-value, visit count, prior, and possibly the Gumbel noise added. But since Gumbel MuZero's action selection is based on the Gumbel variables sampled at the root, maybe the noise is only added at the root? Or is it per node?

Hmm. Let me refer to the Gumbel MuZero paper. From what I remember, in Gumbel MuZero, the Gumbel noise is added to the logits (log probabilities plus Q-values) at the root node for the initial expansion. Then, actions are selected in a deterministic way during the simulation steps. Wait, but I might be mixing up details. Alternatively, perhaps each node stores the necessary information for the sequential allocation, like the completed Q-values with Gumbel noise.

Alternatively, in the standard MuZero, during expansion, child nodes are created with prior probabilities from the policy network. In Gumbel MuZero, the selection of actions during the search uses a combination of Gumbel noise and the Q-values. The key idea is to improve exploration efficiency by using the Gumbel statistics.

But maybe for each node, we need to track the action logits (prior probabilities), the Q-values, and the Gumbel noise terms. However, in open-loop MCTS, the tree doesn't update the state based on real environment steps, so all the transitions are model-based.

Wait, perhaps the main difference in the Node class for Gumbel MuZero is the way actions are selected. Instead of using the UCB formula, it uses the Gumbel-based selection. So each node's action selection would involve the Gumbel noise added to the logits. But how is this implemented?

Let me think again. The Gumbel MuZero approach, as per the paper, uses a two-stage process: first, sample actions at the root with Gumbel noise, then perform simulations. The key is that the action selection is based on the improved policy with the Gumbel noise, leading to better exploration.

In the code structure, the Node class would need to store for each action:

- The prior probability (from the policy network)

- The Q-value (accumulated from backpropagation)

- The visit count

- The Gumbel noise term (sampled once per action at the root?)

Wait, perhaps the Gumbel noise is only added at the root node for the initial action selection. Then, during simulations, the actions are selected based on some deterministic criteria. But I'm not entirely sure. The original Gumbel MuZero paper (if I recall correctly) applies the Gumbel noise to the root node's logits (prior probabilities plus some term), and then uses a sequential allocation strategy to select actions during the search.

So for the Node class, maybe the root node has Gumbel noise terms for each action, while other nodes do not. But how does this affect the structure?

Alternatively, the Gumbel noise is part of the action selection process during the tree search. For each node, when expanding, you add Gumbel noise to the logits (prior + Q), and then select actions based on that.

But perhaps in the code, each node has for each action:

- prior: prior probability from the policy network

- visit_count: number of times the action was taken

- value_sum: total value accumulated

- gumbel: the Gumbel noise term for that action (maybe only at the root?)

Wait, perhaps the Gumbel noise is sampled once per root node expansion. That is, when the MCTS starts, for each action in the root node's available actions, a Gumbel noise is sampled and added to the log prior. Then, during the search, the actions are selected based on these perturbed values.

In that case, the root node would have a dictionary of actions, each with a gumbel noise term. The other nodes (non-root) might not need this, as the noise is only added at the root. But I need to confirm.

Looking at the Gumbel MuZero algorithm steps:

1. The root node receives prior logits from the policy network.

2. Sample Gumbel noise for each action, add to the logits to get Gumbel scores.

3. Select actions based on the Gumbel scores in a sequential manner (without replacement), possibly using a simulation-based approach to find the top actions.

4. Expand nodes as necessary and simulate trajectories using the model's dynamics.

5. Backpropagate the values to update Q-values.

In this case, the Gumbel noise is only added at the root node's actions. Then, during the tree traversal, the action selection might follow a deterministic approach based on the Q-values and prior, but not adding more noise. Therefore, the Node class for the root would have Gumbel noise stored per action, while other nodes do not.

But how is the Node class structured then? Let's outline possible attributes for the Node:

Each Node could have:

- state: the hidden state (for MuZero, this is the model's latent state)

- prior: a dictionary of action probabilities (from policy network)

- children: a dictionary mapping actions to child Nodes

- visit_count: total visits to this node

- value_sum: sum of values backpropagated through this node

- reward: the predicted reward (if applicable, for MuZero's model)

- For Gumbel-specific attributes:

- gumbel_scores: for root node, the Gumbel noise added to log_prior + Q?

Wait, maybe not. Let's think again. The Gumbel MuZero process for the root node involves sampling Gumbel noise for each action, then combining with the log prior and the Q-values (or something similar) to select actions. But for open-loop, once the actions are chosen in the search, the model's dynamics function is used to predict the next state, reward, etc., without interacting with the environment.

So, the Node class for open-loop MCTS in Gumbel MuZero would need:

- The prior probabilities for each action (from the policy network)

- The visit counts for each action

- The Q-values (action values) accumulated through backpropagation

- For the root node, the Gumbel noise added to the logits

- The children nodes for each action

- The hidden state associated with this node (if applicable)

Wait, but in open-loop MCTS, the nodes don't necessarily correspond to actual states. Instead, during the simulation, the state is updated using the model's dynamics function. So each node might not need to store the hidden state; instead, the state is maintained during the simulation steps. Hmm, that complicates things.

Alternatively, in closed-loop MCTS, each node represents a specific state, and the tree is built by taking actions and transitioning to child nodes. But in open-loop, the search is conducted without branching on the state; instead, the search is over action sequences, and the state is updated internally by the model. Wait, maybe open-loop MCTS is different in structure. Let me clarify.

In open-loop planning (like in AlphaZero), the MCTS does not consider the state transitions; instead, it builds a tree of actions, assuming that the state is updated deterministically or according to the model. However, in MuZero, the model's dynamics function is used to predict the next state given the previous hidden state and action. So, even in open-loop, each node would correspond to a hidden state predicted by the model, and the tree is built by applying actions and using the dynamics to get the next hidden state. Therefore, each node would need to store the hidden state, which is used by the model to predict the next step.

But storing the hidden state for each node might be memory-intensive, especially for large trees. However, for the sake of correctness, perhaps each node must contain the hidden state. Alternatively, during simulations, the hidden state is generated on the fly by applying the dynamics function from the root's hidden state and the action sequence taken. But that's only possible if the actions are known and the dynamics are deterministic. If the dynamics are stochastic, then each node's hidden state is necessary.

Assuming the dynamics function is deterministic (as in MuZero's usual setup, where the dynamics function is a deterministic function of the previous hidden state and action), then the hidden state for each node can be computed by applying the dynamics function along the path from the root. Therefore, we don't need to store the hidden state in each node. Instead, during the simulation, when we select an action, we can compute the next hidden state by applying the dynamics function to the current hidden state and the action. This would save memory, but requires recomputing the hidden state each time a simulation is run. However, in practice, during MCTS, each simulation is a separate traversal from the root to a leaf, so the hidden states can be computed incrementally as we traverse the tree.

Wait, but in open-loop MCTS, the key is that the planning is done without updating the real environment state. So the model's dynamics are used to simulate the state transitions. Therefore, during the MCTS, each node represents a hypothetical hidden state (as predicted by the model), and each edge represents an action taken from that state. To compute the hidden state for a child node, you apply the dynamics function to the parent's hidden state and the action. Therefore, each node must store the hidden state (or perhaps not, if we can recompute it when needed by replaying the action sequence from the root). However, recomputing the hidden state every time would be computationally expensive, especially for deep trees. Therefore, it's more efficient to cache the hidden state in each node.

So, the Node class would have:

- hidden_state: the latent state associated with this node

- prior: action probabilities from the policy network

- children: dictionary of actions to child nodes

- visit_count: number of times this node has been visited

- value_sum: sum of the values backpropagated through this node

- reward: the predicted reward for transitioning to this node (if applicable)

For Gumbel MuZero, perhaps the root node needs additional information. For example, when the root is expanded, Gumbel noise is added to the logits of the prior probabilities. Then, during the selection phase, actions are chosen based on these perturbed logits plus the Q-values. So, the root node's actions would have their Gumbel noise stored, whereas other nodes do not. Therefore, the Node class might have a 'gumbel' attribute, which is a dictionary of Gumbel noise values per action, but only for the root node.

But how to handle this in code? Maybe the root node is a special case. Alternatively, the Node class can have a 'gumbel' dictionary that is only populated for the root node.

Putting this together, here's a possible structure for the Node class:

class Node:

def __init__(self, hidden_state, prior, reward=0.0):

self.hidden_state = hidden_state # latent state from model

self.prior = prior # dictionary of action: prior_probability

self.reward = reward # reward predicted for arriving here

self.children = {} # action: Node

self.visit_count = 0

self.value_sum = 0.0

# For Gumbel MuZero, root node has Gumbel noise per action

self.gumbel = None # dictionary of action: gumbel_noise (only for root)

def value(self):

if self.visit_count == 0:

return 0.0

return self.value_sum / self.visit_count

# Other methods as needed...

But wait, the Gumbel noise is added to the logits. So for the root node, during the initial expansion, we sample Gumbel noise for each action. Then, the action selection uses these Gumbel values. The non-root nodes would not have Gumbel noise, so their action selection is based on the prior and Q-values.

In the code, when creating the root node, we would generate Gumbel noise for each possible action and store it in the 'gumbel' attribute. Then, when selecting actions during the search, the root node uses the Gumbel noise, while other nodes use standard selection criteria (like in MuZero).

But how exactly is the Gumbel noise used in the action selection? The Gumbel MuZero approach combines the Gumbel noise with the log prior and the Q-values. The action selection at the root uses the sum of the log prior, Gumbel noise, and some function of the Q-values. For example, the score for each action could be:

score = gumbel_noise + log_prior + c * Q_value

Then, the action with the highest score is chosen. However, the exact formula might vary based on the paper's specifics. Alternatively, the Gumbel noise is added to the logits (log prior) to form the Gumbel scores, and then actions are selected based on these scores along with the Q-values.

In the Gumbel MuZero paper, the action selection at the root is done using Sequential Halving, which is a best-arm identification algorithm. The Gumbel noise is added to the logits (log prior + Q), and the actions are sampled in a way that ensures exploration. But I'm not entirely clear on the exact implementation details here.

Alternatively, in the Gumbel MuZero algorithm, the root node's actions are scored using the Gumbel-Max trick. For each action a, the score is:

score(a) = log_prior(a) + Gumbel(0,1) + c * Q(a)

Then, the action with the highest score is selected. But this might be simplified. In practice, the Gumbel noise is added once at the root, and then during the simulations, actions are selected based on some other criteria (like the Q-values plus prior).

But perhaps during the tree search, for the root node's children, the Gumbel noise is added to the prior, and then the action selection is based on that. For other nodes, the action selection uses the standard PUCT formula (as in MuZero) or another selection method.

Alternatively, the Gumbel MuZero modifies the action selection across the entire tree, but that complicates things. Let's assume that the Gumbel noise is only added at the root node's actions, and the rest of the tree uses a different selection method.

In code, the Node class would need to handle whether it's a root node or not. The root node would have the 'gumbel' attribute, while others do not. Then, when expanding the root node, we generate Gumbel noise for each action. For example:

def expand_root(self, actions, gumbel_noises):

for action in actions:

self.children[action] = Node(...)

self.gumbel[action] = gumbel_noises[action]

But how to generate the Gumbel noises? The Gumbel distribution can be sampled using the inverse transform method: sample u ~ Uniform(0,1), then g = -log(-log(u)).

In Python, this can be done with:

import numpy as np

def sample_gumbel(shape):

u = np.random.uniform(0, 1, shape)

return -np.log(-np.log(u))

So, for the root node, we would sample a Gumbel noise for each possible action, then store them in the node's 'gumbel' dictionary.

Then, during the selection phase, for the root node, the action score is computed as:

score = prior[a] + gumbel[a] + c * Q[a]

Wait, but prior[a] is a probability, so perhaps we take the log prior. The Gumbel-Max trick uses log probabilities plus Gumbel noise. So, the correct formula would be:

score[a] = log(prior[a]) + gumbel_noise[a] + Q_contribution

But the Q_contribution could be scaled appropriately. Alternatively, in the Gumbel MuZero paper, the action selection uses logits (log prior) plus the Q-value divided by a parameter, plus Gumbel noise. For example:

score[a] = log_prior[a] + gumbel_noise[a] + (Q[a] / tau)

where tau is a temperature parameter.

But I'm not certain about the exact formula. The user is implementing Gumbel MuZero, so they might have a specific approach in mind. Since the user is unsure about the Node class structure, the example code should focus on the essential elements.

So, putting this together, the Node class would have:

- hidden_state: the model's latent state at this node

- prior: a dictionary or array of prior probabilities for each action

- children: dictionary mapping actions to child nodes

- visit_count: total visits to this node

- value_sum: accumulated value for backpropagation

- reward: the predicted reward for transitioning to this node (from parent)

- gumbel: dictionary of Gumbel noise values per action (only for root)

The root node would be initialized with Gumbel noise for each action. When expanding the root node, each action's child is created with the corresponding hidden state (computed via the model's dynamics function), prior probabilities (from the policy network applied to the new hidden state), and reward.

Now, the MCTS process for Gumbel MuZero in open-loop would involve the following steps:

1. Initialize the root node with the current hidden state, prior from the policy network, and sample Gumbel noise for each action.

2. Perform multiple simulations (e.g., num_simulations times):

a. Traverse the tree from the root to a leaf node by selecting actions with the highest scores (using Gumbel noise at the root and possibly other criteria elsewhere).

b. Expand the leaf node by querying the model's dynamics, reward, and policy networks to get the next hidden state, reward, and action priors.

c. Backpropagate the value (from the leaf node's value prediction) through the path, updating Q-values and visit counts.

3. After all simulations, select the action with the highest visit count (or other criteria) as the chosen action.

In code, the selection step for the root node would use the Gumbel scores, while for other nodes, it might use the standard PUCT formula or another selection method.

But since this is Gumbel MuZero, maybe the selection is different. For example, the Gumbel MuZero uses Sequential Halving, which is a method for best arm identification in multi-armed bandits. This involves multiple rounds where actions are progressively eliminated based on their performance. However, integrating this into MCTS is non-trivial.

Alternatively, the Gumbel MuZero might use the Gumbel noise to select the best action at the root, and then proceed with standard MCTS for the rest of the tree.

But given the complexity, perhaps the example code should focus on the basic structure, highlighting the Node class with Gumbel noise at the root and the MCTS steps.

Another point to consider is that in open-loop MCTS, the same node can be reached through different action sequences, but since the dynamics are deterministic, each action sequence leads to a unique state. Therefore, the tree structure would have each node representing a unique hidden state, and each path from root to node represents a unique action sequence.

Therefore, the Node class must store the hidden state, and when expanding, apply the dynamics function to get the next hidden state for each action.

Now, let's outline the code structure.

First, the Node class:

class Node:

def __init__(self, hidden_state, prior, reward=0.0, is_root=False):

self.hidden_state = hidden_state

self.prior = prior # dictionary or array-like, action indices to probabilities

self.reward = reward

self.children = {} # action: Node

self.visit_count = 0

self.value_sum = 0.0

self.is_root = is_root

self.gumbel = {} if is_root else None # only root has Gumbel noise

def expanded(self):

return len(self.children) > 0

def value(self):

return self.value_sum / self.visit_count if self.visit_count else 0.0

Then, during the MCTS:

The root node is initialized with is_root=True. The prior would be obtained from the policy network. Then, for each action in the root's prior, we sample a Gumbel noise and store it in self.gumbel.

For the simulation step:

def select_action(node):

if node.is_root:

# Compute scores using Gumbel noise, log prior, and Q

scores = {}

for action in node.prior:

log_prior = np.log(node.prior[action])

gumbel_noise = node.gumbel[action]

q_value = node.children[action].value() if action in node.children else 0.0

# Assuming some scaling factor for Q, like temperature

score = log_prior + gumbel_noise + q_value / temperature

scores[action] = score

# Select action with maximum score

selected_action = max(scores, key=lambda a: scores[a])

else:

# Use PUCT or another strategy for non-root nodes

total_visit = sum(child.visit_count for child in node.children.values())

puct_scores = {}

for action, child in node.children.items():

puct_score = child.value() + c_puct * node.prior[action] * np.sqrt(total_visit) / (1 + child.visit_count)

puct_scores[action] = puct_score

selected_action = max(puct_scores, key=lambda a: puct_scores[a])

return selected_action

But this is a simplification. The exact formula for the action selection in Gumbel MuZero might differ. Also, during the simulation, when a leaf node is encountered, it needs to be expanded by querying the model's dynamics, reward, and policy networks.

The expansion step would involve:

def expand(node, model):

# Get prior, reward, and value from model

# model is a function that takes hidden state and returns prior, reward, value, next_hidden_state?

# For MuZero, the model has three functions: representation, dynamics, prediction

# Let's assume node.hidden_state is the current hidden state

# To expand, we need to get the prior for all actions and possible next states

# But in open-loop, we may need to generate the next hidden state for each action

# Wait, in MuZero, the dynamics function takes (hidden_state, action) and returns next_hidden_state and reward

# The prediction function takes hidden_state and returns (policy_logits, value)

# So, for each possible action, we need to compute the next hidden state and reward using dynamics

# and get the prior for the next hidden state using prediction

# But expanding a node would require considering all possible actions, which could be expensive

# Alternatively, during expansion, the node's children are created for each action, using the dynamics function to compute next hidden state and reward, and then the prediction network to get the prior for the next state

# However, in practice, this is done during the expansion when the node is a leaf node

# For the current node (leaf), we need to evaluate it using the model

# The model's prediction function gives the policy (prior) and value

# Then, for each action, apply dynamics to get next hidden state and reward

# Create child nodes for each action

# Let's say model has a function that, given hidden_state and action, returns next_hidden_state, reward

# And the prediction function, given hidden_state, returns policy (prior) and value

# So, for the current node (leaf), first get the policy and value

policy, value = model.predict(node.hidden_state)

# Store the value for backpropagation

node.value = value # Not sure if this is part of the node structure

# Then, for each action in the action space

for action in possible_actions:

# Compute next hidden state and reward using dynamics

next_hidden_state, reward = model.dynamics(node.hidden_state, action)

# Get prior for the next hidden state

next_policy, _ = model.predict(next_hidden_state)

# Create child node

child_node = Node(

hidden_state=next_hidden_state,

prior=next_policy,

reward=reward,

is_root=False # only root is True

)

node.children[action] = child_node

return value

But this is a very simplified version. The exact steps depend on how the model is structured. Also, in open-loop, the model's dynamics are used to generate the next hidden states, and the search doesn't interact with the real environment.

Backpropagation would update the value_sum and visit_count for each node along the path from leaf to root. The value from the leaf is propagated back, adding the reward (discounted?) and the value.

For example:

def backpropagate(path, value, gamma=0.99):

for node in reversed(path):

node.visit_count += 1

node.value_sum += value

value = node.reward + gamma * value

But the exact backpropagation would depend on MuZero's specifics, which uses a multi-step return with the value function.

Putting all this together, the MCTS code would involve:

- Initializing the root node with Gumbel noise.

- Running multiple simulations, each consisting of selection, expansion (if leaf), evaluation, and backpropagation.

- After all simulations, selecting the best action based on visit counts.

However, the user's main confusion is about the Node class structure. So, the example code should focus on that, with comments indicating where the Gumbel noise is used, and how the MCTS proceeds.

Now, considering all this, here's a draft code structure:

import numpy as np

class Node:

def __init__(self, hidden_state, prior, reward=0.0, is_root=False):

self.hidden_state = hidden_state

self.prior = prior # dict or array of action probabilities

self.reward = reward

self.children = {} # maps action to child nodes

self.visit_count = 0

self.value_sum = 0.0

self.is_root = is_root

self.gumbel = {} # only relevant for root node

def value(self):

if self.visit_count == 0:

return 0.0

return self.value_sum / self.visit_count

def expanded(self):

return len(self.children) > 0

class GumbelMCTS:

def __init__(self, model, num_simulations, c_puct=1.0, temperature=1.0):

self.model = model

self.num_simulations = num_simulations

self.c_puct = c_puct

self.temperature = temperature

def run(self, initial_hidden_state):

# Initialize root node

root_policy, _ = self.model.predict(initial_hidden_state)

root = Node(initial_hidden_state, root_policy, is_root=True)

# Sample Gumbel noise for root node's actions

for action in root.prior:

root.gumbel[action] = self._sample_gumbel()

# Perform simulations

for _ in range(self.num_simulations):

path = [root]

node = root

# Selection

while node.expanded():

action = self._select_action(node)

path.append(action)

node = node.children[action]

# node is a leaf, expand it

value = self._expand(node)

# Backpropagation

self._backpropagate(path, value)

# Return action probabilities based on visit counts

return self._action_probs(root)

def _select_action(self, node):

if node.is_root:

scores = {}

for action in node.prior.keys():

log_prior = np.log(node.prior[action])

gumbel = node.gumbel[action]

q_value = node.children[action].value() if action in node.children else 0.0

score = log_prior + gumbel + q_value / self.temperature

scores[action] = score

return max(scores, key=lambda a: scores[a])

else:

# Use PUCT for non-root nodes

total_visit = sum(child.visit_count for child in node.children.values())

scores = {}

for action, child in node.children.items():

puct_score = child.value() + self.c_puct * node.prior[action] * np.sqrt(total_visit) / (1 + child.visit_count)

scores[action] = puct_score

return max(scores, key=lambda a: scores[a])

def _expand(self, node):

# Get policy and value from model

policy, value = self.model.predict(node.hidden_state)

# For each action, create a child node using dynamics

for action in policy.keys():

next_hidden_state, reward = self.model.dynamics(node.hidden_state, action)

node.children[action] = Node(next_hidden_state, policy[action], reward)

return value

def _backpropagate(self, path, value):

gamma = 0.99 # discount factor

for node in reversed(path):

node.visit_count += 1

node.value_sum += value

value = node.reward + gamma * value

def _sample_gumbel(self):

u = np.random.uniform(0, 1)

return -np.log(-np.log(u))

def _action_probs(self, root):

visits = np.array([root.children[a].visit_count for a in root.children])

probs = visits / np.sum(visits)

return {a: probs[i] for i, a in enumerate(root.children.keys())}

But this code has several issues. For example, the path in the simulation is supposed to be a list of nodes, but in the current code, path appends 'action' which is not a node. Also, the expansion logic may not be correctly implemented. Let's see:

In the simulation loop:

path = [root]

node = root

while node.expanded():

action = self._select_action(node)

path.append(action) # This is wrong, should append the child node

node = node.children[action]

Wait, no. The path should track the nodes visited. So, after selecting an action, the next node is node.children[action], which is added to the path.

So the selection loop should be:

path = [root]

current_node = root

while current_node.expanded():

action = self._select_action(current_node)

next_node = current_node.children.get(action)

if next_node is None:

break # leaf node

path.append(next_node)

current_node = next_node

But wait, in the code above, the loop continues while the node is expanded. So if the node is expanded, it has children. Therefore, when selecting an action that leads to a child node that is not expanded (i.e., not in children?), no, if the node is expanded, all possible actions should have children.

Alternatively, the code may have a mistake here. The selection phase should traverse until a leaf node (a node that hasn't been expanded yet). So, the loop should continue while the current node is expanded (i.e., has children). Once we reach a node that hasn't been expanded (no children), then we expand it.

But the code as written in the draft would loop while node.expanded(), which is True if the node has children. So for each step, select an action, move to the child node (which is in the children dict), and add to the path. Once a node is encountered that hasn't been expanded (children is empty), then the loop exits, and we proceed to expand.

But in the code above, the expansion is done by calling self._expand(node), which creates children for all possible actions.

So the code would:

- Start at root (expanded? No, unless children are already present. Wait, initially, root node's children is empty, so node.expanded() returns False. So the loop doesn't run. Then, the expansion step is called on the root node. But in the first simulation, the root node hasn't been expanded yet.

Wait, there's a mistake here. The root node is initialized with children empty. So in the first simulation, node.expanded() is False, so the loop doesn't run. Then, the code proceeds to expand the root node. But after expansion, the root node's children are populated, so in subsequent simulations, the loop will select actions and traverse through the tree.

But in the initial code draft, the root node's children are empty. So the selection phase would immediately exit, and expansion is done. But during expansion, for each action, a child node is created. However, the model's dynamics function requires the current hidden state and action to produce the next hidden state and reward.

But the code in _expand assumes that the model can generate all possible next hidden states for all actions. This might be computationally expensive if the action space is large. However, for the sake of example, this code outlines the structure.

Another issue is that the path variable in the simulation is supposed to track the nodes visited during the traversal. However, in the code draft, after selecting an action, the code appends the action to the path, which is incorrect. The path should be a list of nodes.

Let's correct that part:

In the run method:

for _ in range(self.num_simulations):

path = [root]

current_node = root

# Selection phase: traverse until leaf node

while current_node.expanded():

action = self._select_action(current_node)

next_node = current_node.children[action]

path.append(next_node)

current_node = next_node

# Now, current_node is a leaf node (not expanded)

# Expand the leaf node

value = self._expand(current_node)

# Backpropagate the value through the path

self._backpropagate(path, value)

But then, the leaf node expansion involves creating children for all actions. After expansion, the leaf node is no longer a leaf.

But in the code draft, the _expand function creates children for all actions, so after expansion, the node is expanded.

Now, the backpropagation function needs to update all nodes in the path with the value.

The value returned by _expand is the predicted value from the model for the leaf node. Then, during backpropagation, we update each node's value_sum and visit_count, considering the reward and discount factor.

The _backpropagate function in the draft adds the value and then computes the new value as reward + gamma * value. This assumes that the value is the discounted cumulative reward from the leaf node. However, MuZero's backup uses the value estimate plus the discounted rewards along the path. The exact backpropagation would depend on the model's setup.

For example, in MuZero, the value network predicts the value of the current state, which is used as the bootstrap for the return. The backpropagation would then compute the discounted return as reward_1 + gamma * reward_2 + gamma^2 * value.

But in the code draft, the backpropagate function starts with the leaf's value and then multiplies by gamma at each step, adding the node's reward. This might be correct if each node's reward is the immediate reward received when transitioning to that node. So the path is root -> node1 (reward r1) -> node2 (reward r2) -> leaf (value v). Then, the return is v + gamma r2 + gamma^2 r1 + ... ?

No, the usual backup is to start from the leaf's value and apply the rewards in reverse order. Let's say the leaf node has value v. The parent of the leaf has reward r. Then, the parent's value is updated with r + gamma * v. The grandparent's value is updated with its reward + gamma * (r + gamma * v), etc.

In the _backpropagate function:

def _backpropagate(self, path, value):

gamma = 0.99

for node in reversed(path):

node.visit_count += 1

node.value_sum += value

value = node.reward + gamma * value

This correctly accumulates the discounted rewards. For example, starting from the leaf node (which has value v), then its parent node's contribution is its reward plus gamma * v, and so on.

This should be correct.

Now, the code has the Node class and the GumbelMCTS class. The model is expected to have predict and dynamics methods. The predict method takes a hidden state and returns a policy (prior) and value. The dynamics method takes a hidden state and action, and returns the next hidden state and reward.

The Gumbel noise is only applied at the root node during action selection. For non-root nodes, the PUCT formula is used.

Potential issues with the code:

1. The root node's Gumbel noise is sampled once during initialization. However, in some implementations, the Gumbel noise might need to be resampled for each simulation or each MCTS run. But according to the Gumbel MuZero paper, the Gumbel noise is sampled once per root node per MCTS run.

2. The expansion function creates child nodes for all actions, which could be computationally intensive for large action spaces. In practice, this is handled by only expanding nodes when they are selected during simulation, but the code here expands all actions at once. However, for open-loop MCTS, this might not be feasible. Alternatively, the code could generate child nodes on-demand as actions are selected, but that would require modifying the expansion step.

3. The selection of actions in non-root nodes uses the PUCT formula, but Gumbel MuZero might require a different approach. However, without specific details, using PUCT is a reasonable placeholder.

4. The handling of hidden states: in this code, each node stores the hidden state, which is generated by the dynamics function when expanding. This assumes that the dynamics function is deterministic, which is the case in MuZero. If the dynamics were stochastic, each child node would need to store a different hidden state for the same action, which complicates the tree structure.

5. The _expand function currently returns the value predicted by the model for the leaf node. This value is then backpropagated. However, in MuZero, the value is part of the prediction network's output, so this is correct.

6. The action selection for the root node uses the Gumbel noise, log prior, and Q-value. The Q-value is the child node's value (average of backpropagated values). The division by temperature is a way to scale the Q-value's contribution, but the exact implementation might vary.

Overall, this code provides a basic structure for Gumbel MuZero's MCTS with open-loop planning. The Node class includes the necessary attributes, and the MCTS class outlines the simulation steps. The user would need to adapt this code to their specific model and environment, especially the model's predict and dynamics functions.

To implement a Gumbel MuZero MCTS with an open-loop approach, you'll need a Node class and a search algorithm that handles Gumbel noise for exploration. Here's a structured solution:

Node Class

python

Copy

import numpy as np

class Node:
def __init__(self, hidden_state, prior, reward=0.0, is_root=False):
self.hidden_state = hidden_state # Model's latent state
self.prior = prior # Prior probabilities for actions (dict or array)
self.reward = reward # Predicted reward for transitioning here
self.children = {} # Maps actions to child nodes
self.visit_count = 0 # Number of visits
self.value_sum = 0.0 # Accumulated value for backpropagation
self.is_root = is_root # True for root node
self.gumbel = {} if is_root else None # Gumbel noise (root only)

@property
def value(self):
return self.value_sum / self.visit_count if self.visit_count else 0.0

def expanded(self):
return len(self.children) > 0

Gumbel MuZero MCTS Implementation

python

Copy

class GumbelMuZeroMCTS:
def __init__(self, model, num_simulations, c_puct=1.0, temperature=1.0, gamma=0.99):
self.model = model # Neural network model with dynamics and prediction
self.num_simulations = num_simulations # Number of MCTS simulations
self.c_puct = c_puct # Exploration constant for PUCT
self.temperature = temperature # Temperature for Q-value scaling
self.gamma = gamma # Discount factor

def run(self, initial_state):
"""Run MCTS from initial hidden state and return action probabilities."""
# Initialize root node
root_policy, _ = self.model.predict(initial_state)
root = Node(initial_state, root_policy, is_root=True)

# Sample Gumbel noise for root actions
for action in root.prior:
root.gumbel[action] = self._sample_gumbel()

# Perform simulations
for _ in range(self.num_simulations):
self._simulate(root)

# Return action probabilities based on visit counts
return self._action_probs(root)

def _simulate(self, root):
"""Perform one MCTS simulation."""
path = [root]
current = root

# Selection phase
while current.expanded():
action = self._select_action(current)
current = current.children[action]
path.append(current)

# Expansion and evaluation
value = self._expand(current)

# Backpropagation
self._backpropagate(path, value)

def _select_action(self, node):
"""Select action using Gumbel noise (root) or PUCT (non-root)."""
if node.is_root:
scores = {
a: np.log(node.prior[a]) + node.gumbel[a] + (self.model.q_value(node.hidden_state, a) / self.temperature)
for a in node.prior
}
else:
total_visits = sum(c.visit_count for c in node.children.values())
scores = {
a: child.value + self.c_puct * node.prior[a] * np.sqrt(total_visits) / (1 + child.visit_count)
for a, child in node.children.items()
}
return max(scores, key=lambda k: scores[k])

def _expand(self, node):
"""Expand leaf node using model dynamics."""
policy, value = self.model.predict(node.hidden_state)
for action in policy:
next_state, reward = self.model.dynamics(node.hidden_state, action)
node.children[action] = Node(next_state, policy[action], reward)
return value

def _backpropagate(self, path, value):
"""Update node values along the path."""
for node in reversed(path):
node.visit_count += 1
node.value_sum += value
value = node.reward + self.gamma * value

def _action_probs(self, node):
"""Normalize visit counts to get action probabilities."""
total = sum(c.visit_count for c in node.children.values())
return {a: c.visit_count / total for a, c in node.children.items()}

@staticmethod
def _sample_gumbel():
"""Sample from Gumbel(0,1) distribution."""
u = np.random.uniform(0, 1)
return -np.log(-np.log(u))

Key Features:

Gumbel Noise: Added only to the root node's actions during selection.

Open-Loop Planning: Uses the model's dynamics to simulate future states without environment interaction.

Backpropagation: Discounted reward accumulation using γ (gamma).

PUCT: Used for non-root nodes to balance exploration/exploitation.

Usage:

Implement your neural network model with:

predict(hidden_state): Returns policy logits and value.

dynamics(hidden_state, action): Returns next hidden state and reward.

q_value(hidden_state, action): Optional Q-value estimator.

Initialize MCTS with your model and run with the initial state.

This implementation provides a foundation for Gumbel MuZero's open-loop planning. Adapt the model integration and hyperparameters based on your specific problem.

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