Overview of Nested Monte Carlo Search (NMC)
Nested Monte Carlo Search (NMC) is a type of Monte Carlo Tree Search (MCTS) which is described in “Overview of Monte Carlo Tree Search and Examples of Algorithms and Implementations., which is a method for efficiently exploring a search space NMC combines multiple levels of search to achieve high search efficiency.
An overview of NMC is as follows.
1. Hierarchical search: NMC performs search at multiple levels. At each level, search is conducted at different depths and with different numbers of samples. In general, shallow search is conducted at the upper levels and deep search is conducted at the lower levels.
2. information propagation: Information obtained at the upper levels is propagated to the lower levels. This makes the search at the lower levels more efficient, and heuristic information and optimal move priorities obtained at the higher levels are used in the search at the lower levels.
3. Aggregation of search: Search results at lower levels are aggregated at higher levels. This makes it possible to comprehensively evaluate the search results at multiple levels and select the optimal move. Weighted averages and statistical methods are used to aggregate search results.
4. efficient search: NMC is a method for efficient search of the search space. Shallow search at the upper levels covers the entire search space widely, while deep search at the lower levels searches for locally optimal solutions. This results in high search efficiency.
NMC is a powerful method for efficient search in complex search problems and games, and by combining search at different levels, it is an approach that allows both extensive search and local optimization.
Application of Nested Monte Carlo Search (NMC)
Nested Monte Carlo Search (NMC) has been used in a variety of areas, including game play and search problems. They are described below.
1. Shogi and Go: NMC is used as an effective search method in board games such as Shogi and Go. In particular, NMC is used in professional Shogi and Go programs because it enables more efficient search by combining multiple layers of search in complex phases of the game.
2. real-time strategy games (RTS): RTS games require real-time decision making; NMC helps to determine the optimal action in real-time by combining multiple layers of search; in RTS games, NMC is used to optimize player strategies and predict enemy actions NMC is used for prediction.
3. Combinatorial Optimization: In combinatorial optimization problems, the search space is complex and finding the optimal solution is difficult. NMC is used for
4. business strategy optimization: NMC can also be useful in business strategy optimization. In business problems such as ad placement, inventory management, and pricing strategies, NMC is used to help find the optimal strategy by combining multiple layers of search, and NMC is used to assist in business strategy decision making.
Example implementation of Nested Monte Carlo Search (NMC)
An example implementation of Nested Monte Carlo Search (NMC) is presented. The following Python code is a simple implementation of the NMC algorithm for the Tic-Tac-Toe game.
import random
class Node:
def __init__(self, state, parent=None):
self.state = state # Game State
self.parent = parent # parent node
self.children = {} # Child Node (Action: Node)
self.visits = 0 # Number of visits to node
self.score = 0 # Total node score
def is_fully_expanded(self):
return len(self.children) == len(self.state.get_legal_moves())
def select_child(self, exploration_factor=0.7):
return max(self.children.values(), key=lambda child: child.get_score(exploration_factor))
def get_score(self, exploration_factor):
if self.visits == 0:
return float('inf')
exploitation = self.score / self.visits
exploration = exploration_factor * (self.parent.visits / (1 + self.visits))
return exploitation + exploration
def add_child(self, action, child_state):
child = Node(child_state, self)
self.children[action] = child
return child
class TicTacToeState:
def __init__(self):
self.board = [[' ' for _ in range(3)] for _ in range(3)]
self.current_player = 'X'
self.last_move = None
def clone(self):
clone_state = TicTacToeState()
clone_state.board = [row[:] for row in self.board]
clone_state.current_player = self.current_player
clone_state.last_move = self.last_move
return clone_state
def get_legal_moves(self):
legal_moves = []
for i in range(3):
for j in range(3):
if self.board[i][j] == ' ':
legal_moves.append((i, j))
return legal_moves
def move(self, move):
i, j = move
new_state = self.clone()
new_state.board[i][j] = self.current_player
new_state.current_player = 'O' if self.current_player == 'X' else 'X'
new_state.last_move = move
return new_state
def is_terminal(self):
for i in range(3):
if self.board[i][0] == self.board[i][1] == self.board[i][2] != ' ':
return True
if self.board[0][i] == self.board[1][i] == self.board[2][i] != ' ':
return True
if self.board[0][0] == self.board[1][1] == self.board[2][2] != ' ':
return True
if self.board[0][2] == self.board[1][1] == self.board[2][0] != ' ':
return True
if all(self.board[i][j] != ' ' for i in range(3) for j in range(3)):
return True
return False
def get_winner(self):
for i in range(3):
if self.board[i][0] == self.board[i][1] == self.board[i][2] != ' ':
return 1 if self.board[i][0] == 'X' else -1
if self.board[0][i] == self.board[1][i] == self.board[2][i] != ' ':
return 1 if self.board[0][i] == 'X' else -1
if self.board[0][0] == self.board[1][1] == self.board[2][2] != ' ':
return 1 if self.board[0][0] == 'X' else -1
if self.board[0][2] == self.board[1][1] == self.board[2][0] != ' ':
return 1 if self.board[0][2] == 'X' else -1
return 0
def nmc_search(root_state, num_iterations):
root_node = Node(root_state)
for _ in range(num_iterations):
node = root_node
state = root_state.clone()
# selection
while not state.is_terminal() and node.is_fully_expanded():
node = node.select_child()
state = state.move(node.state.last_move)
# development
if not state.is_terminal():
unexplored_moves = [move for move in state.get_legal_moves() if move not in node.children]
chosen_move = random.choice(unexplored_moves)
state = state.move(chosen_move)
node = node.add_child(chosen_move, state)
# Simulation
winner = state.get_winner()
while winner == 0:
move = random.choice(state.get_legal_moves())
state = state.move(move)
winner = state.get_winner()
# back-propagation
while node is not None:
node.visits += 1
node.score += winner
node = node.parent
return max(root_node.children, key=lambda c: c.visits).state.last_move
def main():
state = TicTacToeState()
while not state.is_terminal():
if state.current_player == 'X':
move = nmc_search(state, 10000)
else:
print("O's turn. Enter move (row,col): ")
move = tuple(map(int, input().split(',')))
state = state.move(move)
print(state.board[0])
print(state.board[1])
print(state.board[2])
winner = state.get_winner()
if winner == 1:
print("X wins!")
elif winner == -1:
print("O wins!")
else:
print("Draw!")
if __name__ == "__main__":
main()This code is a simple implementation of the NMC algorithm for the Tic-Tac-Toe (Tric-Tac-Toe) game, which combines multiple layers of search for efficiency.
Challenges of Nested Monte Carlo Search (NMC) and how to address them
Nested Monte Carlo Search (NMC) is an effective search method, but there are some challenges. These issues and their countermeasures are described below.
1. increased computational load:
Challenge: NMC tends to increase the computational load because it performs multi-layered search. The computation time increases especially when the search space is large or the hierarchy is deep.
Solution: To reduce the computational load, efficient search algorithms and parallel processing techniques should be used. Techniques to limit the search space and tuning parameters to control the search depth can also be effective.
2. over-computation:
Challenge: NMC is computationally expensive because it performs search at multiple levels. In particular, unnecessary computations may occur because the search results in the upper layers are reflected in the lower layers.
Solution: To avoid unnecessary computation, it is important to set appropriate search strategies and termination conditions. In addition, methods to control propagation of search results and methods to promote reuse of search results are effective.
3. convergence to a locally optimal solution:
Challenge: NMC may converge to a locally optimal solution because it performs search at multiple levels of hierarchy. In particular, if the search at lower levels is biased toward locally optimal solutions, the overall search result will often converge to a locally optimal solution.
Solution: To prevent convergence to the locally optimal solution, search strategies to provide diversity and search methods that encourage departure from the locally optimal solution are necessary, and methods that introduce randomness are also effective to maintain search diversity.
4. delay in information propagation:
Challenge: In NMC, information propagation delays occur because search results in the upper hierarchies are propagated to the lower hierarchies. In particular, as the depth of search and the number of hierarchies increase, the delay in information propagation may become more pronounced.
Solution: To reduce information propagation delays, it is important to use efficient data structures and algorithms. Another effective approach is to predict information propagation delays and apply appropriate search strategies.
Reference Information and Reference Books
For general machine learning algorithms including search algorithms, see “Algorithms and Data Structures” or “General Machine Learning and Data Analysis.
“Algorithms” and other reference books are also available.
Nested Monte-Carlo Search” by Tristan Cazenave dke.maastrichtuniversity.nl+3lamsade.dauphine.fr+3ijcai.org+3
This paper proposes the NMC algorithm and shows its application and effectiveness in games such as Morpion Solitaire, SameGame, and 16×16 Sudoku.
Nested Monte-Carlo Tree Search for Online Planning in Large MDPs” by Hendrik Baier and Mark H. M. Winands, dke.maastrichtuniversity.nl
This paper proposes Nested Monte-Carlo Tree Search (NMCTS), an extension of NMC, and examines its application to online planning in large Markov decision processes (MDPs).
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