文章目录

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()描述,其中:

为有限的状态集

为有限的动作集

为状态转移概率

为回报函数

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 表示,它是指给定状态 时,动作集上的一个分布,即

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


当给定一个策略 时,就可以计算累积回报了。

定义累积回报:

累积回报 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 时,累积回报服从一个分布,累积回报在状态 处的期望值定义为状态-值函数

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


状态-行为值函数为


2.2 MDP 中的概率学基础讲解

(1)随机变量

(2)概率分布

(3)条件概率

(4)期望和方差

(5)方差

最常用的概率分布也就是最常用的随机策略。

(1)贪婪策略

(2)-greedy 策略

(3)高斯策略

(4)玻尔兹曼分布

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')