Initial commit

LICENCE

+MIT License
+
+Copyright (c) 2020 Werner Duvaud
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
\ No newline at end of file

README.md

+<p>
+<img src="https://img.shields.io/badge/licence-MIT-green">
+<img src="https://img.shields.io/badge/dependencies-up%20to%20date-brightgreen">
+<a href="https://github.com/psf/black"><img alt="Code style: black" src="https://img.shields.io/badge/code%20style-black-000000.svg"></a>
+</p>
+
+# MuZero General
+
+A flexible, commented and [documented](https://github.com/werner-duvaud/muzero-general/wiki/MuZero-Documentation) implementation of MuZero based on the Google DeepMind [paper](https://arxiv.org/abs/1911.08265) and the associated [pseudocode](https://arxiv.org/src/1911.08265v1/anc/pseudocode.py).
+It is designed to be easily adaptable for every games or reinforcement learning environnements (like [gym](https://github.com/openai/gym)). You only need to edit the game file with the parameters and the game class. Please refer to the documentation and the tutorial.
+
+MuZero is a model based reinforcement learning algorithm, successor of AlphaZero. It learns to master games whithout knowing the rules. It only know actions and then learn to play and master the game. It is at least more efficient than similar algorithms like [AlphaZero](https://arxiv.org/abs/1712.01815), [SimPLe](https://arxiv.org/abs/1903.00374) and [World Models](https://arxiv.org/abs/1803.10122).
+
+It uses [PyTorch](https://github.com/pytorch/pytorch) and [Ray](https://github.com/ray-project/ray) for self-playing on multiple threads. A synchronous mode (easier for debug) will be released. There is a complete GPU support.
+The code has three parts, muzero.py with the entry class, self-play.py with the replay-buffer and the MCTS classes, and network.py with the neural networks and the shared storage classes.
+
+## Games already implemented with pretrained network available
+* Lunar Lander
+* Cartpole
+
+## Getting started
+### Installation
+```bash
+cd muzero-general
+pip install -r requirements.txt
+```
+
+### Training
+Edit the end of muzero.py :
+```python
+muzero = Muzero("cartpole")
+muzero.train()
+```
+Then run :
+```bash
+python muzero.py
+```
+
+### Testing
+Edit the end of muzero.py :
+```python
+muzero = Muzero("cartpole")
+muzero.load_model()
+muzero.test()
+```
+Then run :
+```bash
+python muzero.py
+```
+
+## Coming soon
+* [ ] Convolutionnal / Atari mode
+* [ ] Performance tracking
+* [ ] Synchronous mode
+* [ ] [Open spiel](https://github.com/deepmind/open_spiel) integration
+* [ ] Checkers game
+* [ ] TensorFlow mode
+
+## Authors
+* Werner Duvaud
+* Aurèle Hainaut
+* Paul Lenoir

games/cartpole.py

+import gym
+import numpy
+import torch
+
+
+class MuZeroConfig:
+    def __init__(self):
+        self.seed = 0  # Seed for numpy, torch and the game
+
+        ### Game
+        self.observation_shape = 4  # Dimensions of the game observation
+        self.action_space = [i for i in range(2)]  # Fixed list of all possible actions
+
+        ### Self-Play
+        self.num_actors = 10  # Number of simultaneous threads self-playing to feed the replay buffer
+        self.max_moves = 500  # Maximum number of moves if game is not finished before
+        self.num_simulations = 50  # Number of futur moves self-simulated
+        self.discount = 0.997  # Chronological discount of the reward
+
+        # Root prior exploration noise
+        self.root_dirichlet_alpha = 0.25
+        self.root_exploration_fraction = 0.25
+
+        # UCB formula
+        self.pb_c_base = 19652
+        self.pb_c_init = 1.25
+
+        # If we already have some information about which values occur in the environment, we can use them to initialize the rescaling
+        # This is not strictly necessary, but establishes identical behaviour to AlphaZero in board games
+        self.min_known_bound = None
+        self.max_known_bound = None
+
+        ### Network
+        self.encoding_size = 32
+        self.hidden_size = 64
+
+        # Training
+        self.results_path = "./pretrained"  # Path to store the model weights
+        self.device = torch.device("cuda" if torch.cuda.is_available() else "cpu")  # Automatically use GPU instead of CPU if available
+        self.training_steps = 400  # Total number of training steps (ie weights update according to a batch)
+        self.batch_size = 128  # Number of parts of games to train on at each training step
+        self.num_unroll_steps = 5  # Number of game moves to keep for every batch element
+        self.test_interval = 20  # Number of training steps before evaluating the network on the game to track the performance
+        self.test_episodes = 2  # Number of game played to evaluate the network
+        self.checkpoint_interval = 10  # Number of training steps before using the model for sef-playing
+        self.window_size = 1000  # Number of self-play games to keep in memory (in the replay buffer)
+        self.td_steps = 10 # Number of steps in the futur to take into account for calculating the target value
+
+        self.weight_decay = 1e-4  # L2 weights regularization
+        self.momentum = 0.9
+
+        # Exponential learning rate schedule
+        self.lr_init = 0.005  # Initial learning rate
+        self.lr_decay_rate = 0.1
+        self.lr_decay_steps = 3500
+
+    def visit_softmax_temperature_fn(self, num_moves, trained_steps):
+        """
+        Parameter to alter the visit count distribution to ensure that the action selection becomes greedier as training progresses.
+        The smaller it is, the more likely the best action (ie with the highest visit count) is chosen.
+
+        Returns:
+            Positive float.
+        """
+        if trained_steps < 0.5 * self.training_steps:
+            return 1.0
+        elif trained_steps < 0.75 * self.training_steps:
+            return 0.5
+        else:
+            return 0.25
+
+
+class Game:
+    """Game wrapper.
+    """
+
+    def __init__(self, seed=None):
+        self.env = gym.make("CartPole-v1")
+        if seed is not None:
+            self.env.seed(seed)
+
+    def step(self, action):
+        """Apply action to the game.
+        
+        Args:
+            action : action of the action_space to take.
+
+        Returns:
+            The new observation, the reward and a boolean if the game has ended.
+        """
+        observation, reward, done, _ = self.env.step(action)
+        return numpy.array(observation).flatten(), reward, done
+
+    def reset(self):
+        """Reset the game for a new game.
+        
+        Returns:
+            Initial observation of the game.
+        """
+        return self.env.reset()
+
+    def close(self):
+        """Properly close the game.
+        """
+        self.env.close()
+
+    def render(self):
+        """Display the game observation.
+        """
+        self.env.render()
+        input("Press enter to take a step ")

games/lunarlander.py

+import gym
+import numpy
+import torch
+
+
+class MuZeroConfig:
+    def __init__(self):
+        self.seed = 0  # Seed for numpy, torch and the game
+
+        ### Game
+        self.observation_shape = 8  # Dimensions of the game observation
+        self.action_space = [i for i in range(4)]  # Fixed list of all possible actions
+
+        ### Self-Play
+        self.num_actors = 10  # Number of simultaneous threads self-playing to feed the replay buffer
+        self.max_moves = 100  # Maximum number of moves if game is not finished before
+        self.num_simulations = 50  # Number of futur moves self-simulated
+        self.discount = 0.997  # Chronological discount of the reward
+
+        # Root prior exploration noise
+        self.root_dirichlet_alpha = 0.25
+        self.root_exploration_fraction = 0.25
+
+        # UCB formula
+        self.pb_c_base = 19652
+        self.pb_c_init = 1.25
+
+        # If we already have some information about which values occur in the environment, we can use them to initialize the rescaling
+        # This is not strictly necessary, but establishes identical behaviour to AlphaZero in board games
+        self.min_known_bound = None
+        self.max_known_bound = None
+
+        ### Network
+        self.encoding_size = 16
+        self.hidden_size = 8
+
+        # Training
+        self.results_path = "./pretrained"  # Path to store the model weights
+        self.device = torch.device("cuda" if torch.cuda.is_available() else "cpu")  # Automatically use GPU instead of CPU if available
+        self.training_steps = 700  # Total number of training steps (ie weights update according to a batch)
+        self.batch_size = 128  # Number of parts of games to train on at each training step
+        self.num_unroll_steps = 50  # Number of game moves to keep for every batch element
+        self.test_interval = 20  # Number of training steps before evaluating the network on the game to track the performance
+        self.test_episodes = 2  # Number of game played to evaluate the network
+        self.checkpoint_interval = 20  # Number of training steps before using the model for sef-playing
+        self.window_size = 1000  # Number of self-play games to keep in memory (in the replay buffer)
+        self.td_steps = 50 # Number of steps in the futur to take into account for calculating the target value
+
+        self.weight_decay = 1e-4  # L2 weights regularization
+        self.momentum = 0.9
+
+        # Exponential learning rate schedule
+        self.lr_init = 0.005  # Initial learning rate
+        self.lr_decay_rate = 0.01
+        self.lr_decay_steps = 3500
+
+    def visit_softmax_temperature_fn(self, num_moves, trained_steps):
+        """
+        Parameter to alter the visit count distribution to ensure that the action selection becomes greedier as training progresses.
+        The smaller it is, the more likely the best action (ie with the highest visit count) is chosen.
+
+        Returns:
+            Positive float.
+        """
+        if trained_steps < 0.25 * self.training_steps:
+            return 1000
+        elif trained_steps < 0.5 * self.training_steps:
+            return 1
+        elif trained_steps < 0.75 * self.training_steps:
+            return 0.5
+        else:
+            return 0.1
+
+
+class Game:
+    """Game wrapper.
+    """
+
+    def __init__(self, seed=None):
+        self.env = gym.make("LunarLander-v2")
+        if seed is not None:
+            self.env.seed(seed)
+
+    def step(self, action):
+        """Apply action to the game.
+        
+        Args:
+            action : action of the action_space to take.
+
+        Returns:
+            The new observation, the reward and a boolean if the game has ended.
+        """
+        observation, reward, done, _ = self.env.step(action)
+        return numpy.array(observation).flatten(), reward, done
+
+    def reset(self):
+        """Reset the game for a new game.
+        
+        Returns:
+            Initial observation of the game.
+        """
+        return self.env.reset()
+
+    def close(self):
+        """Properly close the game.
+        """
+        self.env.close()
+
+    def render(self):
+        """Display the game observation.
+        """
+        self.env.render()
+        input("Press enter to take a step ")

muzero.py

+import importlib
+import os
+import time
+
+import numpy
+import ray
+import torch
+
+import network
+import self_play
+
+
+class MuZero:
+    """
+    Main class to manage MuZero.
+
+    Args:
+        game_name (str): Name of the game module, it should match the name of a .py file in the "./games" directory.
+
+    Example:
+        >>> muzero = MuZero("cartpole")
+        >>> muzero.train()
+        >>> muzero.test()
+    """
+
+    def __init__(self, game_name):
+        self.game_name = game_name
+
+        # Load the game and the config from the module with the game name
+        try:
+            game_module = importlib.import_module("games." + self.game_name)
+            self.config = game_module.MuZeroConfig()
+            self.Game = game_module.Game
+        except Exception as err:
+            print(
+                '{} is not a supported game name, try "cartpole" or refer to the documentation for adding a new game.'.format(
+                    self.game_name
+                )
+            )
+            raise err
+
+        # Fix random generator seed for reproductibility
+        # TODO: check if results do not change from one run to another
+        numpy.random.seed(self.config.seed)
+        torch.manual_seed(self.config.seed)
+
+        self.best_model = network.Network(
+            self.config.observation_shape,
+            len(self.config.action_space),
+            self.config.encoding_size,
+            self.config.hidden_size,
+        )
+
+    def train(self):
+        # Initialize and launch components that work simultaneously
+        ray.init()
+        model = self.best_model
+        model.train()
+        storage = network.SharedStorage.remote(model)
+        replay_buffer = self_play.ReplayBuffer.remote(self.config)
+        for seed in range(self.config.num_actors):
+            self_play.run_selfplay.remote(
+                self.Game,
+                self.config,
+                storage,
+                replay_buffer,
+                model,
+                self.config.seed + seed,
+            )
+
+        # Initialize network for training
+        model = model.to(self.config.device)
+        optimizer = torch.optim.SGD(
+            model.parameters(),
+            lr=self.config.lr_init,
+            momentum=self.config.momentum,
+            weight_decay=self.config.weight_decay,
+        )
+
+        # Wait for replay buffer to be non-empty
+        while ray.get(replay_buffer.length.remote()) == 0:
+            time.sleep(0.1)
+
+        # Training loop
+        best_test_rewards = None
+        for training_step in range(self.config.training_steps):
+            model.train()
+            storage.set_training_step.remote(training_step)
+
+            # Make the model available for self-play
+            if training_step % self.config.checkpoint_interval == 0:
+                storage.set_weights.remote(model.state_dict())
+
+            # Update learning rate
+            lr = self.config.lr_init * self.config.lr_decay_rate ** (
+                training_step / self.config.lr_decay_steps
+            )
+            for param_group in optimizer.param_groups:
+                param_group["lr"] = lr
+
+            # Train on a batch.
+            batch = ray.get(
+                replay_buffer.sample_batch.remote(
+                    self.config.num_unroll_steps, self.config.td_steps
+                )
+            )
+            loss = network.update_weights(optimizer, model, batch, self.config)
+
+            # Test the current model and save it on disk if it is the best
+            if training_step % self.config.test_interval == 0:
+                total_test_rewards = self.test(model=model, render=False)
+                if best_test_rewards is None or sum(total_test_rewards) >= sum(
+                    best_test_rewards
+                ):
+                    self.best_model = model
+                    best_test_rewards = total_test_rewards
+                    self.save_model()
+
+            print(
+                "Training step: {}\nBuffer Size: {}\nLearning rate: {}\nLoss: {}\nLast test score: {}\nBest sest score: {}\n".format(
+                    training_step,
+                    ray.get(replay_buffer.length.remote()),
+                    lr,
+                    loss,
+                    str(total_test_rewards),
+                    str(best_test_rewards),
+                )
+            )
+
+        # Finally, save the latest network in the shared storage and end the self-play
+        storage.set_weights.remote(model.state_dict())
+        ray.shutdown()
+
+    def test(self, model=None, render=True):
+        if not model:
+            model = self.best_model
+
+        model.to(self.config.device)
+        test_rewards = []
+        game = self.Game()
+
+        model.eval()
+        with torch.no_grad():
+            for _ in range(self.config.test_episodes):
+                observation = game.reset()
+                done = False
+                total_reward = 0
+                while not done:
+                    if render:
+                        game.render()
+                    root = self_play.MCTS(self.config).run(model, observation, False)
+                    action = self_play.select_action(root, temperature=0)
+                    observation, reward, done = game.step(action)
+                    total_reward += reward
+                test_rewards.append(total_reward)
+
+        return test_rewards
+
+    def save_model(self, model=None, path=None):
+        if not model:
+            model = self.best_model
+        if not path:
+            path = os.path.join(self.config.results_path, self.game_name)
+
+        torch.save(model.state_dict(), path)
+
+    def load_model(self, path=None):
+        if not path:
+            path = os.path.join(self.config.results_path, self.game_name)
+        self.best_model = network.Network(
+            self.config.observation_shape,
+            len(self.config.action_space),
+            self.config.encoding_size,
+            self.config.hidden_size,
+        )
+        try:
+            self.best_model.load_state_dict(torch.load(path))
+        except FileNotFoundError:
+            print("There is no model saved in {}.".format(path))
+
+
+if __name__ == "__main__":
+    # Load the game and the parameters from ./games/file_name.py
+    muzero = MuZero("cartpole")
+    muzero.load_model()
+    muzero.train()
+    muzero.test()

network.py

+import ray
+import torch
+
+
+class Network(torch.nn.Module):
+    def __init__(self, input_size, action_space_n, encoding_size, hidden_size):
+        super().__init__()
+        self.action_space_n = action_space_n
+
+        self.representation_network = FullyConnectedNetwork(
+            input_size, [], encoding_size
+        )
+
+        self.dynamics_state_network = FullyConnectedNetwork(
+            encoding_size + self.action_space_n, [hidden_size], encoding_size
+        )
+        self.dynamics_reward_network = FullyConnectedNetwork(
+            encoding_size + self.action_space_n, [hidden_size], 1
+        )
+
+        self.prediction_actor_network = FullyConnectedNetwork(
+            encoding_size, [], self.action_space_n, activation=None
+        )
+        self.prediction_value_network = FullyConnectedNetwork(
+            encoding_size, [], 1, activation=None
+        )
+
+    def prediction(self, state):
+        actor_logit = self.prediction_actor_network(state)
+        value = self.prediction_value_network(state)
+        return actor_logit, value
+
+    def representation(self, observation):
+        return self.representation_network(observation)
+
+    def dynamics(self, state, action):
+        action_one_hot = (
+            torch.zeros((action.shape[0], self.action_space_n))
+            .to(action.device)
+            .float()
+        )
+        action_one_hot.scatter_(1, action.long(), 1.0)
+
+        x = torch.cat((state, action_one_hot), dim=1)
+        next_state = self.dynamics_state_network(x)
+        reward = self.dynamics_reward_network(x)
+        return next_state, reward
+
+    def initial_inference(self, observation):
+        state = self.representation(observation)
+        actor_logit, value = self.prediction(state)
+        return (
+            value,
+            torch.zeros(len(observation)).to(observation.device),
+            actor_logit,
+            state,
+        )
+
+    def recurrent_inference(self, hidden_state, action):
+        state, reward = self.dynamics(hidden_state, action)
+        actor_logit, value = self.prediction(state)
+        return value, reward, actor_logit, state
+
+
+def update_weights(optimizer, model, batch, config):
+    observation_batch, action_batch, target_reward, target_value, target_policy = batch
+
+    observation_batch = torch.tensor(observation_batch).float().to(config.device)
+    action_batch = torch.tensor(action_batch).float().to(config.device).unsqueeze(-1)
+    target_value = torch.tensor(target_value).float().to(config.device)
+    target_reward = torch.tensor(target_reward).float().to(config.device)
+    target_policy = torch.tensor(target_policy).float().to(config.device)
+
+    value, reward, policy_logits, hidden_state = model.initial_inference(
+        observation_batch
+    )
+    predictions = [(value, reward, policy_logits)]
+    for action_i in range(config.num_unroll_steps):
+        value, reward, policy_logits, hidden_state = model.recurrent_inference(
+            hidden_state, action_batch[:, action_i]
+        )
+        predictions.append((value, reward, policy_logits))
+
+    loss = 0
+    for i, prediction in enumerate(predictions):
+        value, reward, policy_logits = prediction
+        loss += loss_function(
+            value.squeeze(-1),
+            reward.squeeze(-1),
+            policy_logits,
+            target_value[:, i],
+            target_reward[:, i],
+            target_policy[:, i, :],
+        )
+
+    # Scale gradient by number of unroll steps (See paper Training appendix)
+    loss = loss.mean() / config.num_unroll_steps
+
+    # Optimize
+    optimizer.zero_grad()
+    loss.backward()
+    optimizer.step()
+
+    return loss.item()
+
+
+def loss_function(
+    value, reward, policy_logits, target_value, target_reward, target_policy
+):
+    # TODO: paper promotes cross entropy instead of MSE
+    value_loss = torch.nn.MSELoss(reduction="none")(value, target_value)
+    reward_loss = torch.nn.MSELoss(reduction="none")(reward, target_reward)
+    policy_loss = -(torch.log_softmax(policy_logits, dim=1) * target_policy).sum(1)
+    return value_loss + reward_loss + policy_loss
+
+
+class FullyConnectedNetwork(torch.nn.Module):
+    def __init__(
+        self, input_size, layers_sizes, output_size, activation=torch.nn.Tanh()
+    ):
+        super(FullyConnectedNetwork, self).__init__()
+        layers_sizes.insert(0, input_size)
+        layers = []
+        if 1 < len(layers_sizes):
+            for i in range(len(layers_sizes) - 1):
+                layers.extend(
+                    [
+                        torch.nn.Linear(layers_sizes[i], layers_sizes[i + 1]),
+                        torch.nn.ReLU(),
+                    ]
+                )
+        layers.append(torch.nn.Linear(layers_sizes[-1], output_size))
+        if activation:
+            layers.append(activation)
+        self.layers = torch.nn.ModuleList(layers)
+
+    def forward(self, x):
+        for layer in self.layers:
+            x = layer(x)
+        return x
+
+
+@ray.remote
+class SharedStorage:
+    def __init__(self, model):
+        self.training_step = 0
+        self.model = model
+
+    def get_weights(self):
+        return self.model.state_dict()
+
+    def set_weights(self, weights):
+        return self.model.load_state_dict(weights)
+
+    def set_training_step(self, training_step):
+        self.training_step = training_step
+
+    def get_training_step(self):
+        return self.training_step

requirements.txt

+numpy
+torch
+ray
+gym[all]
+psutil
+setproctitle

self_play.py

+import math
+
+import numpy
+import ray
+import torch
+
+
+@ray.remote  # (num_gpus=1) #  Uncoomment num_gpus and model.to(config.device) to self-play on GPU
+def run_selfplay(Game, config, shared_storage, replay_buffer, model, seed):
+    """
+    Function which run simultaneously in multiple threads and is continuously playing games and saving them to the replay-buffer.
+    """
+    # model.to(config.device) # Uncoomment this line and num_gpus from the decorator to self-play on GPU
+    with torch.no_grad():
+        while True:
+            # Initialize a self-play
+            model.load_state_dict(ray.get(shared_storage.get_weights.remote()))
+            game = Game(seed)
+            done = False
+            game_history = GameHistory(config.discount)
+
+            # Self-play with actions based on the Monte Carlo tree search at each moves
+            observation = game.reset()
+            game_history.observation_history.append(observation)
+            while not done and len(game_history.history) < config.max_moves:
+                root = MCTS(config).run(model, observation, True)
+
+                temperature = config.visit_softmax_temperature_fn(
+                    num_moves=len(game_history.history),
+                    trained_steps=ray.get(shared_storage.get_training_step.remote()),
+                )
+                action = select_action(root, temperature)
+
+                observation, reward, done = game.step(action)
+
+                game_history.observation_history.append(observation)
+                game_history.rewards.append(reward)
+                game_history.history.append(action)
+                game_history.store_search_statistics(root, config.action_space)
+
+            game.close()
+            # Save the game history
+            replay_buffer.save_game.remote(game_history)
+
+
+# Game independant
+class MCTS:
+    """
+    Core Monte Carlo Tree Search algorithm.
+    To decide on an action, we run N simulations, always starting at the root of
+    the search tree and traversing the tree according to the UCB formula until we
+    reach a leaf node.
+    """
+
+    def __init__(self, config):
+        self.config = config
+
+    def run(self, model, observation, add_exploration_noise):
+        """
+        At the root of the search tree we use the representation function to obtain a hidden state given the current observation.
+        We then run a Monte Carlo Tree Search using only action sequences and the model learned by the network.
+        """
+        root = Node(0)
+        observation = (
+            torch.from_numpy(observation).to(self.config.device).float().unsqueeze(0)
+        )
+        _, expected_reward, policy_logits, hidden_state = model.initial_inference(
+            observation
+        )
+        root.expand(
+            self.config.action_space, expected_reward, policy_logits, hidden_state
+        )
+        if add_exploration_noise:
+            root.add_exploration_noise(
+                dirichlet_alpha=self.config.root_dirichlet_alpha,
+                exploration_fraction=self.config.root_exploration_fraction,
+            )
+
+        min_max_stats = MinMaxStats(
+            self.config.min_known_bound, self.config.max_known_bound
+        )
+
+        for _ in range(self.config.num_simulations):
+            node = root
+            search_path = [node]
+
+            while node.expanded():
+                action, node = self.select_child(node, min_max_stats)
+                last_action = action
+                search_path.append(node)
+
+            # Inside the search tree we use the dynamics function to obtain the next hidden state given an action and the previous hidden state
+            parent = search_path[-2]
+            value, reward, policy_logits, hidden_state = model.recurrent_inference(
+                parent.hidden_state,
+                torch.tensor([[last_action]]).to(parent.hidden_state.device),
+            )
+            node.expand(self.config.action_space, reward, policy_logits, hidden_state)
+
+            self.backpropagate(search_path, value.item(), min_max_stats)
+
+        return root
+
+    def select_child(self, node, min_max_stats):
+        """
+        Select the child with the highest UCB score.
+        """
+        _, action, child = max(
+            (self.ucb_score(node, child, min_max_stats), action, child)
+            for action, child in node.children.items()
+        )
+        return action, child
+
+    def ucb_score(self, parent, child, min_max_stats):
+        """
+        The score for a node is based on its value, plus an exploration bonus based on the prior.
+        """
+        pb_c = (
+            math.log(
+                (parent.visit_count + self.config.pb_c_base + 1) / self.config.pb_c_base
+            )
+            + self.config.pb_c_init
+        )
+        pb_c *= math.sqrt(parent.visit_count) / (child.visit_count + 1)
+
+        prior_score = pb_c * child.prior
+        value_score = min_max_stats.normalize(child.value())
+
+        return prior_score + value_score
+
+    def backpropagate(self, search_path, value, min_max_stats):
+        """
+        At the end of a simulation, we propagate the evaluation all the way up the tree to the root.
+        """
+        for node in search_path:
+            # Always the same player, the other players minds should be modeled in network because environment do not act always in the best way to make you lose
+            node.value_sum += value  # if node.to_play == to_play else -value
+            node.visit_count += 1
+            min_max_stats.update(node.value())
+
+            value = node.reward + self.config.discount * value
+
+
+def select_action(node, temperature, random=False):
+    """
+    Select action according to the vivist count distribution and the temperature.
+    The temperature is changed dynamically with the visit_softmax_temperature function in the config.
+    """
+    visit_counts = numpy.array(
+        [[child.visit_count, action] for action, child in node.children.items()]
+    ).T
+    if temperature == 0:
+        action_pos = numpy.argmax(visit_counts[0])
+    else:
+        # See paper Data Generation appendix
+        visit_count_distribution = visit_counts[0] ** (1 / temperature)
+        visit_count_distribution = visit_count_distribution / sum(
+            visit_count_distribution
+        )
+        action_pos = numpy.random.choice(
+            len(visit_counts[1]), p=visit_count_distribution
+        )
+
+    if random:
+        action_pos = numpy.random.choice(len(visit_counts[1]))
+
+    return visit_counts[1][action_pos]
+
+
+class Node:
+    def __init__(self, prior):
+        self.visit_count = 0
+        self.to_play = -1
+        self.prior = prior
+        self.value_sum = 0
+        self.children = {}
+        self.hidden_state = None
+        self.reward = 0
+
+    def expanded(self):
+        return len(self.children) > 0
+
+    def value(self):
+        if self.visit_count == 0:
+            return 0
+        return self.value_sum / self.visit_count
+
+    def expand(self, actions, reward, policy_logits, hidden_state):
+        """
+        We expand a node using the value, reward and policy prediction obtained from the neural network.
+        """
+        self.reward = reward
+        self.hidden_state = hidden_state
+        policy = {a: math.exp(policy_logits[0][a]) for a in actions}
+        policy_sum = sum(policy.values())
+        for action, p in policy.items():
+            self.children[action] = Node(p / policy_sum)
+
+    def add_exploration_noise(self, dirichlet_alpha, exploration_fraction):
+        """
+        At the start of each search, we add dirichlet noise to the prior of the root to encourage the search to explore new actions.
+        """
+        actions = list(self.children.keys())
+        noise = numpy.random.dirichlet([dirichlet_alpha] * len(actions))
+        frac = exploration_fraction
+        for a, n in zip(actions, noise):
+            self.children[a].prior = self.children[a].prior * (1 - frac) + n * frac
+
+
+class GameHistory:
+    """
+    Store only usefull information of a self-play game.
+    """
+
+    def __init__(self, discount):
+        self.observation_history = []
+        self.history = []
+        self.rewards = []
+        self.child_visits = []
+        self.root_values = []
+        self.discount = discount
+
+    def store_search_statistics(self, root, action_space):
+        sum_visits = sum(child.visit_count for child in root.children.values())
+        self.child_visits.append(
+            [
+                root.children[a].visit_count / sum_visits if a in root.children else 0
+                for a in action_space
+            ]
+        )
+        self.root_values.append(root.value())
+
+
+@ray.remote
+class ReplayBuffer:
+    # Store list of game history and generate batch
+    def __init__(self, config):
+        self.window_size = config.window_size
+        self.batch_size = config.batch_size
+        self.buffer = []
+
+    def save_game(self, game_history):
+        if len(self.buffer) > self.window_size:
+            self.buffer.pop(0)
+        self.buffer.append(game_history)
+
+    def sample_batch(self, num_unroll_steps, td_steps):
+        observation_batch, action_batch, reward_batch, value_batch, policy_batch = (
+            [],
+            [],
+            [],
+            [],
+            [],
+        )
+        for _ in range(self.batch_size):
+            game_history = self.sample_game()
+            game_pos = self.sample_position(game_history)
+            actions = game_history.history[game_pos : game_pos + num_unroll_steps]
+            # Repeat precedent action to make "actions" of length "num_unroll_steps"
+            actions.extend(
+                [actions[-1] for _ in range(num_unroll_steps - len(actions) + 1)]
+            )
+            observation_batch.append(game_history.observation_history[game_pos])
+            action_batch.append(actions)
+            value, reward, policy = self.make_target(
+                game_history, game_pos, num_unroll_steps, td_steps
+            )
+            reward_batch.append(reward)
+            value_batch.append(value)
+            policy_batch.append(policy)
+
+        return observation_batch, action_batch, reward_batch, value_batch, policy_batch
+
+    def sample_game(self):
+        """
+        Sample game from buffer either uniformly or according to some priority.
+        """
+        # TODO: sample with probability link to the highest difference between real and predicted value (see paper appendix Training)
+        return self.buffer[numpy.random.choice(range(len(self.buffer)))]
+
+    def sample_position(self, game):
+        """
+        Sample position from game either uniformly or according to some priority.
+        """
+        # TODO: according to some priority
+        return numpy.random.choice(range(len(game.rewards)))
+
+    def make_target(self, game, state_index, num_unroll_steps, td_steps):
+        """
+        The value target is the discounted root value of the search tree td_steps into the future, plus the discounted sum of all rewards until then.
+        """
+        target_values, target_rewards, target_policies = [], [], []
+        for current_index in range(state_index, state_index + num_unroll_steps + 1):
+            bootstrap_index = current_index + td_steps
+            if bootstrap_index < len(game.root_values):
+                value = game.root_values[bootstrap_index] * game.discount ** td_steps
+            else:
+                value = 0
+
+            for i, reward in enumerate(game.rewards[current_index:bootstrap_index]):
+                value += reward * game.discount ** i
+
+            if current_index < len(game.root_values):
+                target_values.append(value)
+                target_rewards.append(game.rewards[current_index])
+                target_policies.append(game.child_visits[current_index])
+            else:
+                # States past the end of games are treated as absorbing states
+                target_values.append(0)
+                target_rewards.append(0)
+                # Uniform policy to give the tensor a valid dimension
+                target_policies.append(
+                    [
+                        1 / len(game.child_visits[0])
+                        for _ in range(len(game.child_visits[0]))
+                    ]
+                )
+
+        return target_values, target_rewards, target_policies
+
+    def length(self):
+        return len(self.buffer)
+
+
+class MinMaxStats:
+    """
+    A class that holds the min-max values of the tree.
+    """
+
+    def __init__(self, min_value_bound, max_value_bound):
+        self.maximum = min_value_bound if min_value_bound else -float("inf")
+        self.minimum = max_value_bound if max_value_bound else float("inf")
+
+    def update(self, value):
+        self.maximum = max(self.maximum, value)
+        self.minimum = min(self.minimum, value)
+
+    def normalize(self, value):
+        if self.maximum > self.minimum:
+            # We normalize only when we have set the maximum and minimum values
+            return (value - self.minimum) / (self.maximum - self.minimum)
+        return value