2.1 马尔科夫决策过程理论讲解

马尔科夫决策过程由元组（$S,A,P,R,\gamma$）描述，其中：
$S$ 为有限的状态集
$A$ 为有限的动作集
$P$ 为状态转移概率
$R$ 为回报函数
$\gamma$ 为折扣因子，用来计算累积回报。

强化学习的目标是给定一个马尔科夫决策过程，寻找最优策略。

所谓策略是指状态到动作的映射，策略常用符号 $\pi$ 表示，它是指给定状态 $s$ 时，动作集上的一个分布，即
$$\pi (a|s)=p[A_t=a|S_t=s]\text{\hspace{1em}(2.1)}$$

公式（2.1）的含义是：策略 $\pi$ 在每个状态 $s$ 指定一个动作概率。如果给出的策略 $\pi$ 是确定性的，那么策略 $\pi$ 在每个状态 $s$ 指定一个确定的动作。

当给定一个策略 $\pi$ 时，就可以计算累积回报了。

定义累积回报：
$$G_t=R_{t+1}+\gamma R_{t+2}+{\gamma }^2{R}_{t+3}+\cdots =\sum _{k=0}^{\infty }{\gamma }^k{R}_{t+k+1}\text{\hspace{1em}(2.2)}$$

累积回报 $G_1$ 是个随机变量，不是一个确定值，因此无法描述，但其期望是个确定值，可以作为状态值函数的定义。

当智能体采用策略 $\pi$ 时，累积回报服从一个分布，累积回报在状态 $s$ 处的期望值定义为状态-值函数：

$${\nu }_{\pi }(s)=E_{\pi }[\sum _{k=0}^{\infty }{\gamma }^k{R}_{t+k+1}|S_t=s]\text{\hspace{1em}(2.3)}$$

注意：状态值函数是与策略 $\pi$ 相对应的，这是因为策略 $\pi$ 决定了累积回报 $G$ 的状态分布。

状态-行为值函数为
$$q_{\pi }(s,a)=E_{\pi }[\sum _{k=0}^{\infty }{\gamma }^k{R}_{t+k+1}|S_t=s,A_t=a]\text{\hspace{1em}(2.4)}$$

2.3 基于 gym 的 MDP 实例讲解

Appendix

grid_mdp.py 文件代码

import logging
import numpy
import random
from gym import spaces
import gym

logger = logging.getLogger(__name__)

class GridEnv(gym.Env):
    metadata = {
    
        'render.modes': ['human', 'rgb_array'],
        'video.frames_per_second': 2
    }

    def __init__(self):

        self.states = [1,2,3,4,5,6,7,8] #状态空间
        self.x=[140,220,300,380,460,140,300,460]
        self.y=[250,250,250,250,250,150,150,150]
        self.terminate_states = dict()  #终止状态为字典格式
        self.terminate_states[6] = 1
        self.terminate_states[7] = 1
        self.terminate_states[8] = 1

        self.actions = ['n','e','s','w']

        self.rewards = dict();        #回报的数据结构为字典
        self.rewards['1_s'] = -1.0
        self.rewards['3_s'] = 1.0
        self.rewards['5_s'] = -1.0

        self.t = dict();             #状态转移的数据格式为字典
        self.t['1_s'] = 6
        self.t['1_e'] = 2
        self.t['2_w'] = 1
        self.t['2_e'] = 3
        self.t['3_s'] = 7
        self.t['3_w'] = 2
        self.t['3_e'] = 4
        self.t['4_w'] = 3
        self.t['4_e'] = 5
        self.t['5_s'] = 8
        self.t['5_w'] = 4

        self.gamma = 0.8         #折扣因子
        self.viewer = None
        self.state = None

    def getTerminal(self):
        return self.terminate_states

    def getGamma(self):
        return self.gamma

    def getStates(self):
        return self.states

    def getAction(self):
        return self.actions
    def getTerminate_states(self):
        return self.terminate_states
    def setAction(self,s):
        self.state=s

    def _step(self, action):
        #系统当前状态
        state = self.state
        if state in self.terminate_states:
            return state, 0, True, {
    }
        key = "%d_%s"%(state, action)   #将状态和动作组成字典的键值

        #状态转移
        if key in self.t:
            next_state = self.t[key]
        else:
            next_state = state
        self.state = next_state

        is_terminal = False

        if next_state in self.terminate_states:
            is_terminal = True

        if key not in self.rewards:
            r = 0.0
        else:
            r = self.rewards[key]


        return next_state, r,is_terminal,{
    }
    def _reset(self):
        self.state = self.states[int(random.random() * len(self.states))]
        return self.state
    def render(self, mode='human', close=False):
        if close:
            if self.viewer is not None:
                self.viewer.close()
                self.viewer = None
            return
        screen_width = 600
        screen_height = 400

        if self.viewer is None:
            from gym.envs.classic_control import rendering
            self.viewer = rendering.Viewer(screen_width, screen_height)
            #创建网格世界
            self.line1 = rendering.Line((100,300),(500,300))
            self.line2 = rendering.Line((100, 200), (500, 200))
            self.line3 = rendering.Line((100, 300), (100, 100))
            self.line4 = rendering.Line((180, 300), (180, 100))
            self.line5 = rendering.Line((260, 300), (260, 100))
            self.line6 = rendering.Line((340, 300), (340, 100))
            self.line7 = rendering.Line((420, 300), (420, 100))
            self.line8 = rendering.Line((500, 300), (500, 100))
            self.line9 = rendering.Line((100, 100), (180, 100))
            self.line10 = rendering.Line((260, 100), (340, 100))
            self.line11 = rendering.Line((420, 100), (500, 100))
            #创建第一个骷髅
            self.kulo1 = rendering.make_circle(40)
            self.circletrans = rendering.Transform(translation=(140,150))
            self.kulo1.add_attr(self.circletrans)
            self.kulo1.set_color(0,0,0)
            #创建第二个骷髅
            self.kulo2 = rendering.make_circle(40)
            self.circletrans = rendering.Transform(translation=(460, 150))
            self.kulo2.add_attr(self.circletrans)
            self.kulo2.set_color(0, 0, 0)
            #创建金条
            self.gold = rendering.make_circle(40)
            self.circletrans = rendering.Transform(translation=(300, 150))
            self.gold.add_attr(self.circletrans)
            self.gold.set_color(1, 0.9, 0)
            #创建机器人
            self.robot= rendering.make_circle(30)
            self.robotrans = rendering.Transform()
            self.robot.add_attr(self.robotrans)
            self.robot.set_color(0.8, 0.6, 0.4)

            self.line1.set_color(0, 0, 0)
            self.line2.set_color(0, 0, 0)
            self.line3.set_color(0, 0, 0)
            self.line4.set_color(0, 0, 0)
            self.line5.set_color(0, 0, 0)
            self.line6.set_color(0, 0, 0)
            self.line7.set_color(0, 0, 0)
            self.line8.set_color(0, 0, 0)
            self.line9.set_color(0, 0, 0)
            self.line10.set_color(0, 0, 0)
            self.line11.set_color(0, 0, 0)

            self.viewer.add_geom(self.line1)
            self.viewer.add_geom(self.line2)
            self.viewer.add_geom(self.line3)
            self.viewer.add_geom(self.line4)
            self.viewer.add_geom(self.line5)
            self.viewer.add_geom(self.line6)
            self.viewer.add_geom(self.line7)
            self.viewer.add_geom(self.line8)
            self.viewer.add_geom(self.line9)
            self.viewer.add_geom(self.line10)
            self.viewer.add_geom(self.line11)
            self.viewer.add_geom(self.kulo1)
            self.viewer.add_geom(self.kulo2)
            self.viewer.add_geom(self.gold)
            self.viewer.add_geom(self.robot)

        if self.state is None: return None
        #self.robotrans.set_translation(self.x[self.state-1],self.y[self.state-1])
        self.robotrans.set_translation(self.x[self.state-1], self.y[self.state- 1])



        return self.viewer.render(return_rgb_array=mode == 'rgb_array')