Reinforcement learning for tic-tac-toe

At the study session I was attending, there was a story about strengthening tic-tac-toe, so I wrote a code.

References

• [How to make a stronger robotic game player-Reinforcement learning to learn in practice-](http://www.amazon.co.jp/gp/product/B00CXEXQ7U/ref=as_li_qf_sp_asin_tl?ie=UTF8&camp=247&creative=1211&creativeASIN= B00CXEXQ7U & linkCode = as2 & tag = shimashimao06-22)
• Reinforcement Learning <img src = "http" //ir-jp.amazon-adsystem.com/e/ir?t=shimashimao06-22&l=as2&o=9&a=4627826613" width = "1" height = "1" border = "0" alt = "" style = "border: none! important; margin: 0px! important;" />

Reinforcement learning super overview by Monte Carlo method

Since I used the Monte Carlo method (policy on type) this time, I will write only around the Monte Carlo method. (I'm not confident in the content because I just read a little book when I heard the story at the study session.) To write about the Monte Carlo method in one word, it is a method (apparently) of policy evaluation and policy improvement while repeating policies by learning the number of values and optimal measures from the experience of sample episode format. (The code can also be roughly divided into blocks of policy iteration, policy evaluation, and policy improvement.) The advantages and disadvantages are listed below.

Advantages of the Monte Carlo method

• Does not require prior knowledge (model) of environmental dynamics
• Still, you can achieve the optimum behavior
• Does not require a complete probability distribution of all transitions as in dynamic programming (DP)

Problems with the Monte Carlo method

• Cannot be implemented in a fully progressive procedure because you don't know your earnings until the end of the episode (If it is the TD method, it can be implemented in a completely progressive procedure)
• Even if the behavior in the middle of the episode is evaluated, a discount is required
• Sufficient episodes required for learning and high sampling time cost (The above reference book also states that the convergence is not good compared to the TD method)

However, looking at the winning percentage of tic-tac-toe described in "How to make a stronger robotic game player", the Monte Carlo method was the highest, so I am writing the code for the Monte Carlo method.

Reference code

Since it is a tic-tac-toe, it is a code when learning in a discrete space and the final time step T exists. The code below is almost the same as the one described in "How to make a stronger robotic game player" puck (ry </ del>). Then, I will write the code below.

python

# coding: utf-8

import numpy as np
from math import floor
import matplotlib.pyplot as plt

class MonteCarloPolycyIteration:
    n_states = 3**9 #Number of states
    n_actions = 9   #Number of actions
    T = 5           #Maximum number of steps

    visites = None
    states = None
    actions = None
    rewards = None
    drewards = None
    results = None
    rate = []
    
    #It will be in the same state when the board is rotated
    #Perform that conversion and reduce the number of states
    convert = [[0,1,2,3,4,5,6,7,8], #Original state
               [2,1,0,5,4,3,8,7,6], #conversion(2)
               [6,3,0,7,4,1,8,5,2], #conversion(3)
               [0,3,8,1,4,7,2,5,8], #conversion(4)
               [8,7,6,5,4,3,2,1,0], #conversion(5)
               [6,7,8,3,4,5,0,1,2], #conversion(6)
               [2,5,8,1,4,7,0,3,6], #conversion(7)
               [8,5,2,7,4,1,6,3,0]  #conversion(8)
               ]
    #Vector for conversion from decimal number to decimal number
    power = np.array([ 3**i for i in xrange(8,-1,-1) ], dtype=np.float64)
    
    def __init__(self,L,M,options):
        self.L = L #Number of policy iterations
        self.M = M #Number of episodes
        self.options = options
        self.Q =  self.init_Q()
        np.random.seed(555)
    
    def init_Q(self):
        """Initialization of Q function(initialize look up table)  """
        return np.zeros((self.n_states,self.n_actions))
    
    def train(self):
        # policy iteration
        for l in xrange(self.L):
            self.visites = self.init_visites()
            self.states = self.init_matrix()
            self.actions = self.init_matrix()
            self.rewards = self.init_matrix()
            self.drewards = self.init_matrix()
            self.results = self.init_results()
            
            # episode
            for m in xrange(self.M):
                state3 = self.init_state3()
                
                # step
                for t in xrange(self.T):
                    state = self.encode(state3)
                    policy = self.generate_policy()
                    policy = self.policy_improvement(policy,state)
                    
                    #Action selection and execution
                    action, reward, state3, fin = self.action_train(policy, t, state3)
                    
                    self.update(m, t, state, action, reward)
                    
                    if self.isfinished(fin):
                        #print 'fin_state',state3
                        self.results[m] = fin
                        
                        #Calculation of discount reward sum
                        self.calculate_discounted_rewards(m, t)
                        break
            self.Q = self.calculate_state_action_value_function()
            self.output_results(l)
            self.rate.append( float(len(self.results[self.results==2]))/self.M )
            
    def init_visites(self):
        return np.ones((self.n_states, self.n_actions))
    
    def init_matrix(self):
        return np.ones((self.M, self.T))
    
    def init_results(self):
        return np.zeros(self.M)
    
    def init_state3(self):
        return np.zeros(self.n_actions)

    def generate_policy(self):
        return np.zeros(self.n_actions)

    def policy_improvement(self,policy,state):
        if self.options['pmode']==0:
            q = self.Q[state] #State value in current policy
            v = max(q) #Value in optimal behavior in the current state
            a = np.where(q==v)[0][0] #Optimal behavior
            policy[a] = 1
        elif self.options['pmode']==1:
            q = self.Q[state] #State value in current policy
            v = max(q) #Value in optimal behavior in the current state
            a = np.where(q==v)[0][0] #Optimal behavior
            policy = np.ones(self.n_actions)*self.options['epsilon']/self.n_actions
            policy[a] = 1-self.options['epsilon']+self.options['epsilon']/self.n_actions
        elif self.options['pmode']==2:
            policy = np.exp(self.Q[state]/self.options['tau'])/ \
                        sum(np.exp(self.Q[state]/self.options['tau']))
        return policy
    
    def action_train(self, policy, t, state3):
        npc_action = self.select_npc_action(t, state3, policy)
        state3[npc_action] = 2
        fin = self.check_game(state3)
        reward = self.calculate_reward(fin)
        if reward is not None:
            return npc_action, reward, state3, fin
        
        enemy_action = self.select_enemy_action(t, state3)
        state3[enemy_action] = 1
        fin = self.check_game(state3)
        reward = self.calculate_reward(fin)
        return npc_action, reward, state3, fin
            
    def select_npc_action(self, step, state3, policy):
        a = None
        if step==0:
            a = 0
        else:
            while 1:
                random = np.random.rand()
                cprob = 0
                for a in xrange(self.n_actions):
                    cprob += policy[a]
                    if random<cprob: break  
                #Check if the square is already filled
                if state3[a]==0:break
        return a

    def select_enemy_action(self, step, state3):
        reach = 0
        pos = [[0,1,2],[3,4,5],[6,7,8],[0,3,6],[1,4,7],[1,5,8],[0,4,8],[2,4,6]]
        a = None
        for i in xrange(len(pos)):
            state_i = state3[pos[i]]
            val = sum(state_i)
            num = len(state_i[state_i==0])
            if val==2 and num==1:
                idx = int( state_i[state_i==0][0] )
                a = pos[i][idx]
                reach = 1
                break
        if reach==0:
            while 1:
                a = floor(np.random.rand()*8)+1
                if state3[a]==0: break
        return a        
        
    def check_game(self, state3):
        pos = [[0,1,2],[3,4,5],[6,7,8],[0,3,6],[1,4,7],[1,5,8],[0,4,8],[2,4,6]]
        for i in xrange(len(pos)):
            state_i = state3[pos[i]]
            val_npc = sum(state_i==1)
            val_enemy = sum(state_i==2)
            if val_npc==3:
                return 1 #win
            elif val_enemy==3:
                return 2 #Lose
        if sum(state3==0)==0:
            return 3 #draw
        return 0 #Continue game
    
    def calculate_reward(self, fin):
        if fin==2:
            return 10
        elif fin==1:
            return -10
        elif fin==3:
            return 0
        elif fin==0:
            return None        
    
    def update(self,m,t,state,action,reward):
        """Update status, actions, rewards, and appearance count"""
        self.states[m,t] = state
        self.actions[m,t] = action
        self.rewards[m,t] = reward
        self.visites[state, action] += 1
        
    def isfinished(self,fin):
        if fin>0: return True
        else: return False
    
    def calculate_discounted_rewards(self, m, t):
        for pstep in xrange(t-1,-1,-1):
            self.drewards[m,t] = self.rewards[m,t]
            self.drewards[m,pstep] = \
                    self.options['gamma']*self.drewards[m,pstep+1]
    
    def calculate_state_action_value_function(self):
        Q = self.init_Q()
        for m in xrange(self.M):
            for t in xrange(self.T):
                s = self.states[m,t]
                if s==0: break #If it is finished by the 9th move, the process is finished
                a =self.actions[m,t]
                Q[s,a] += self.drewards[m,t]
        return Q/self.visites
    
    def output_results(self,l):
        print 'l=%d: Win=%d/%d, Draw=%d/%d, Lose=%d/%d\n' % (l, \
                        len(self.results[self.results==2]),self.M, \
                        len(self.results[self.results==3]),self.M, \
                        len(self.results[self.results==1]),self.M)        
    
    def encode(self, state3):        
        #to state(2)～(8)After adding 8 types of conversion, convert to decimal number
        cands = [ sum(state3[self.convert[i]]*self.power) #Swap indexes and convert to decimal
                    for i in xrange(len(self.convert))]
        #Choose the smallest of the 8 candidates
        return min(cands)+1
    
if __name__=='__main__':
    options = {'pmode':1,'epsilon':0.05,'tau':2,'gamma':0.9}
    mcpi= MonteCarloPolycyIteration(100,100,options)
    mcpi.train()
    
    plt.plot(range(len(mcpi.rate)), mcpi.rate)
    plt.show()

Experimental result

I experimented using the ε-greedy method with 100 steps as one set. Below, we will post the winning percentage for 100 sets during learning.

Summary

Learning seems to be working, but it's slow to calculate. .. It's reckless to implement the Monte Carlo method in python. .. By the way, the same result was obtained with softmax.

We apologize for the inconvenience, but if you make a mistake, we would appreciate it if you could point it out.