【强化学习】Q-Learning算法详解

2022-11-09 16:33:11 浏览数 (1)

【强化学习】Q-Learning详解

https://morvanzhou.github.io/tutorials/machine-learning/reinforcement-learning/2-1-general-rl/ 莫凡大神的有趣的强化学习视频通俗易懂

发现了很多RL资料搬砖过来，刚入门的可以用得上

David Silver 博士的 UCL 公开课：http://www0.cs.ucl.ac.uk/staff/d.silver/web/Teaching.html DeepMind 和 UCL 的DL、RL课程：https://www.youtube.com/playlist?list=PLqYmG7hTraZDNJre23vqCGIVpfZ_K2RZs Sergey Levine 的DRL课程：http://rail.eecs.berkeley.edu/deeprlcourse/ OpenAI 的 Spinning Up in Deep RL：https://blog.openai.com/spinning-up-in-deep-rl/ 关于深度强化学习良心paper：https://arxiv.org/abs/1810.06339

1、算法思想

QLearning是强化学习算法中value-based的算法，Q即为Q（s,a）就是在某一时刻的 s 状态下(s∈S)，采取 动作a (a∈A)动作能够获得收益的期望，环境会根据agent的动作反馈相应的回报reward r，所以算法的主要思想就是将State与Action构建成一张Q-table来存储Q值，然后根据Q值来选取能够获得最大的收益的动作。

Q-Table

a1

a2

s1

q(s1,a1)

q(s1,a2)

s2

q(s2,a1)

q(s2,a2)

s3

q(s3,a1)

q(s3,a2)

2、公式推导

举个例子如图有一个GridWorld的游戏从起点出发到达终点为胜利掉进陷阱为失败。智能体（Agent）、环境状态（environment）、奖励（reward）、动作（action）可以将问题抽象成一个马尔科夫决策过程，我们在每个格子都算是一个状态 s t s_t st​ , π(a|s)在s状态下采取动作a策略 。 P(s’|s,a)也可以写成 P s s ′ a P_{ss’}^a Pss′a​为在s状态下选择a动作转换到下一个状态s’的概率。R(s’|s,a)表示在s状态下采取a动作转移到s’的奖励reward，我们的目的很明确就是找到一条能够到达终点获得最大奖赏的策略。

所以目标就是求出累计奖励最大的策略的期望：

Goal： max ⁡ π E [ ∑ t = 0 H γ t R ( S t , A t , S t 1 ) ∣ π ] max_πE[sum_{t=0}^{H}γ^tR(S_t,A_t,S_{t 1}) | π] maxπ​E[∑t=0H​γtR(St​,At​,St 1​)∣π]

Qlearning的主要优势就是使用了时间差分法TD（融合了蒙特卡洛和动态规划）能够进行离线学习, 使用bellman方程可以对马尔科夫过程求解最优策略

贝尔曼方程

通过bellman方程求解马尔科夫决策过程的最佳决策序列，状态值函数 V π ( s ) V_pi(s) Vπ​(s)可以评价当前状态的好坏，每个状态的值不仅由当前状态决定还要由后面的状态决定，所以状态的累计奖励求期望就可得出当前s的状态值函数V(s)。bellman方程如下

V π ( s ) = E ( U t ∣ S t = s ) V_π(s) = E(U_t|S_t = s) Vπ​(s)=E(Ut​∣St​=s) V π ( s ) = E π [ R t 1 γ [ R t 2 γ [ . . . . . . . ] ] ∣ S t = s ] V_π(s) = E_π[R_{t 1} γ[R_{t 2} γ[…….]]|S_t = s] Vπ​(s)=Eπ​[Rt 1​ γ[Rt 2​ γ[.......]]∣St​=s] V π ( s ) = E π [ R t 1 γ V ( s ′ ) ∣ S t = s ] V_π(s) = E_π[R_{t 1} γV(s’)|S_t = s] Vπ​(s)=Eπ​[Rt 1​ γV(s′)∣St​=s]

最优累计期望可用 V ∗ ( s ) V^*(s) V∗(s)表示，可知最优值函数就是 V ∗ ( s ) = m a x π V π ( s ) V^*(s)=max_πV_pi(s) V∗(s)=maxπ​Vπ​(s) V ∗ ( s ) = max ⁡ π E [ ∑ t = 0 H γ t R ( S t , A t , S t 1 ) ∣ π , s 0 = s ] V^*(s)=max_πE[sum_{t=0}^{H}γ^tR(S_t,A_t,S_{t 1}) | π,s_0=s] V∗(s)=maxπ​E[∑t=0H​γtR(St​,At​,St 1​)∣π,s0​=s]

Q(s,a)状态动作值函数 q π ( s , a ) = E π [ r t 1 γ r t 2 γ 2 r t 3 . . . . ∣ A t = a , S t = s ] q_π(s,a) = E_π[r_{t 1} γr_{t 2} γ^2r_{t 3} ….|A_t=a,S_t=s] qπ​(s,a)=Eπ​[rt 1​ γrt 2​ γ2rt 3​ ....∣At​=a,St​=s] q π ( s , a ) = E π [ G t ∣ A t = a , S t = s ] q_π(s,a) = E_π[G_t|A_t=a,S_t=s] qπ​(s,a)=Eπ​[Gt​∣At​=a,St​=s] 其中 G t G_t Gt​是t时刻开始的总折扣奖励，从这里我们能看出来 γ衰变值对Q函数的影响，γ越接近于1代表它越有远见会着重考虑后续状态的的价值，当γ接近0的时候就会变得近视只考虑当前的利益的影响。所以从0到1，算法就会越来越会考虑后续回报的影响。 q π ( s , a ) = E π [ R t 1 γ q π ( S t 1 , A t 1 ) ∣ A t = a , S t = s ] q_π(s,a) = E_π[R_{t 1} γq_π(S_{t 1},A_{t 1})|A_t=a,S_t=s] qπ​(s,a)=Eπ​[Rt 1​ γqπ​(St 1​,At 1​)∣At​=a,St​=s]

最优价值动作函数 Q ∗ ( s , a ) = m a x π Q ∗ ( s , a ) Q^*(s,a)=max_pi Q^*(s,a) Q∗(s,a)=maxπ​Q∗(s,a),打开期望如下 Q ∗ ( s , a ) = ∑ s ′ P ( s ′ ∣ s , a ) ( R ( s , a , s ′ ) γ max ⁡ a ′ Q ∗ ( s ′ , a ′ ) ) Q^*(s,a)=sum_{s’} P(s’|s,a)(R(s,a,s’) γmax_{a’}Q^*(s’,a’)) Q∗(s,a)=∑s′​P(s′∣s,a)(R(s,a,s′) γmaxa′​Q∗(s′,a′))

Bellman方程实际上就是价值动作函数的转换关系

V π ( s ) = ∑ a ∈ A π ( a ∣ s ) q π ( s , a ) V_π(s) = sum_{a∈A}π(a|s)q_π(s,a) Vπ​(s)=∑a∈A​π(a∣s)qπ​(s,a) q π ( s , a ) = R s a γ ∑ s ′ ∈ S P s s ′ a V π ( s ′ ) q_π(s,a) = R_s^a γsum_{s’∈S}P_{ss’}^aV_π(s’) qπ​(s,a)=Rsa​ γ∑s′∈S​Pss′a​Vπ​(s′) V π ( s ) = ∑ a ′ ∈ A π ( a ∣ s ) [ R s a γ ∑ s ′ P s s ′ a V π ( s ′ ) ] V_π(s)=sum_{a’∈A}π(a|s)[R_s^a γsum_{s’}P_{ss’}^aV_π(s’)] Vπ​(s)=∑a′∈A​π(a∣s)[Rsa​ γ∑s′​Pss′a​Vπ​(s′)]

根据下图更直观的了解V(s)与Q(s,a)的关系

时间差分法 https://blog.csdn.net/qq_30615903/article/details/80821061

时间差分方法结合了蒙特卡罗的采样方法和动态规划方法的bootstrapping(利用后继状态的值函数估计当前值函数)使得他可以适用于model-free的算法并且是单步更新，速度更快。值函数计算方式如下

V ( s ) ← V ( s ) α ( R t 1 γ V ( s ′ ) − V ( s ) ) V(s)←V(s) alpha (R_{t 1} gamma V(s’)-V(s)) V(s)←V(s) α(Rt 1​ γV(s′)−V(s))

其中 R t 1 γ V ( s ′ ) R_{t 1} gamma V(s’) Rt 1​ γV(s′)被称为TD目标， δ t = R t 1 γ V ( s ′ ) − V ( s ) delta_t=R_{t 1} gamma V(s’)-V(s) δt​=Rt 1​ γV(s′)−V(s) 称为TD偏差。

3、更新公式

根据以上推导可以对Q值进行计算，所以有了Q值我们就可以进行学习，也就是Q-table的更新过程，其中α为学习率γ为奖励性衰变系数，采用时间差分法的方法进行更新。

Q ( s , a ) ← Q ( s , a ) α [ r γ m a x a ′ Q ( s ′ , a ′ ) − Q ( s , a ) ] Q(s,a) ← Q(s,a) α[r γmax_{a’}Q(s’,a’)-Q(s,a)] Q(s,a)←Q(s,a) α[r γmaxa′​Q(s′,a′)−Q(s,a)]

上式就是Q-learning更新的公式，根据下一个状态s’中选取最大的 Q ( s ′ , a ′ ) Q(s’,a’) Q(s′,a′)值乘以衰变γ加上真实回报值最为Q现实，而根据过往Q表里面的Q(s,a)作为Q估计。

4、实现代码

代码来自网上各路大神的源码，非原创，据反映没图片跑不通，所以建了个github，https://github.com/xshura/reinforcement_learning Q-Learning agent

代码语言：javascript复制
# -*- coding: utf-8 -*-
import random
from environment import Env
from collections import defaultdict


class QLearningAgent:
    def __init__(self, actions):
        # actions = [0, 1, 2, 3]
        self.actions = actions
        self.learning_rate = 0.01
        self.discount_factor = 0.9
        self.epsilon = 0.1
        self.q_table = defaultdict(lambda: [0.0, 0.0, 0.0, 0.0])

    # 采样 <s, a, r, s'>
    def learn(self, state, action, reward, next_state):
        current_q = self.q_table[state][action]
        # 贝尔曼方程更新
        new_q = reward   self.discount_factor * max(self.q_table[next_state])
        self.q_table[state][action]  = self.learning_rate * (new_q - current_q)

    # 从Q-table中选取动作
    def get_action(self, state):
        if np.random.rand() < self.epsilon:
            # 贪婪策略随机探索动作
            action = np.random.choice(self.actions)
        else:
            # 从q表中选择
            state_action = self.q_table[state]
            action = self.arg_max(state_action)
        return action

    @staticmethod
    def arg_max(state_action):
        max_index_list = []
        max_value = state_action[0]
        for index, value in enumerate(state_action):
            if value > max_value:
                max_index_list.clear()
                max_value = value
                max_index_list.append(index)
            elif value == max_value:
                max_index_list.append(index)
        return random.choice(max_index_list)


if __name__ == "__main__":
    env = Env()
    agent = QLearningAgent(actions=list(range(env.n_actions)))
    for episode in range(1000):
        state = env.reset()
        while True:
            env.render()
            # agent产生动作
            action = agent.get_action(str(state))
            next_state, reward, done = env.step(action)
            # 更新Q表
            agent.learn(str(state), action, reward, str(next_state))
            state = next_state
            env.print_value_all(agent.q_table)
            # 当到达终点就终止游戏开始新一轮训练
            if done:
                break

环境部分

代码语言：javascript复制
import time
import numpy as np
import tkinter as tk
from PIL import ImageTk, Image
np.random.seed(1)
PhotoImage = ImageTk.PhotoImage
UNIT = 100
HEIGHT = 5
WIDTH = 5
class Env(tk.Tk):
def __init__(self):
super(Env, self).__init__()
self.action_space = ['u', 'd', 'l', 'r']
self.n_actions = len(self.action_space)
self.title('Q Learning')
self.geometry('{0}x{1}'.format(HEIGHT * UNIT, HEIGHT * UNIT))
self.shapes = self.load_images()
self.canvas = self._build_canvas()
self.texts = []
def _build_canvas(self):
canvas = tk.Canvas(self, bg='white',
height=HEIGHT * UNIT,
width=WIDTH * UNIT)
# create grids
for c in range(0, WIDTH * UNIT, UNIT):  # 0~400 by 80
x0, y0, x1, y1 = c, 0, c, HEIGHT * UNIT
canvas.create_line(x0, y0, x1, y1)
for r in range(0, HEIGHT * UNIT, UNIT):  # 0~400 by 80
x0, y0, x1, y1 = 0, r, HEIGHT * UNIT, r
canvas.create_line(x0, y0, x1, y1)
# add img to canvas
self.rectangle = canvas.create_image(50, 50, image=self.shapes[0])
self.triangle1 = canvas.create_image(250, 150, image=self.shapes[1])
self.triangle2 = canvas.create_image(150, 250, image=self.shapes[1])
self.circle = canvas.create_image(250, 250, image=self.shapes[2])
# pack all
canvas.pack()
return canvas
def load_images(self):
rectangle = PhotoImage(
Image.open("../img/rectangle.png").resize((65, 65)))
triangle = PhotoImage(
Image.open("../img/triangle.png").resize((65, 65)))
circle = PhotoImage(
Image.open("../img/circle.png").resize((65, 65)))
return rectangle, triangle, circle
def text_value(self, row, col, contents, action, font='Helvetica', size=10,
style='normal', anchor="nw"):
if action == 0:
origin_x, origin_y = 7, 42
elif action == 1:
origin_x, origin_y = 85, 42
elif action == 2:
origin_x, origin_y = 42, 5
else:
origin_x, origin_y = 42, 77
x, y = origin_y   (UNIT * col), origin_x   (UNIT * row)
font = (font, str(size), style)
text = self.canvas.create_text(x, y, fill="black", text=contents,
font=font, anchor=anchor)
return self.texts.append(text)
def print_value_all(self, q_table):
for i in self.texts:
self.canvas.delete(i)
self.texts.clear()
for i in range(HEIGHT):
for j in range(WIDTH):
for action in range(0, 4):
state = [i, j]
if str(state) in q_table.keys():
temp = q_table[str(state)][action]
self.text_value(j, i, round(temp, 2), action)
def coords_to_state(self, coords):
x = int((coords[0] - 50) / 100)
y = int((coords[1] - 50) / 100)
return [x, y]
def state_to_coords(self, state):
x = int(state[0] * 100   50)
y = int(state[1] * 100   50)
return [x, y]
def reset(self):
self.update()
time.sleep(0.5)
x, y = self.canvas.coords(self.rectangle)
self.canvas.move(self.rectangle, UNIT / 2 - x, UNIT / 2 - y)
self.render()
# return observation
return self.coords_to_state(self.canvas.coords(self.rectangle))
def step(self, action):
state = self.canvas.coords(self.rectangle)
base_action = np.array([0, 0])
self.render()
if action == 0:  # up
if state[1] > UNIT:
base_action[1] -= UNIT
elif action == 1:  # down
if state[1] < (HEIGHT - 1) * UNIT:
base_action[1]  = UNIT
elif action == 2:  # left
if state[0] > UNIT:
base_action[0] -= UNIT
elif action == 3:  # right
if state[0] < (WIDTH - 1) * UNIT:
base_action[0]  = UNIT
# 移动
self.canvas.move(self.rectangle, base_action[0], base_action[1])
self.canvas.tag_raise(self.rectangle)
next_state = self.canvas.coords(self.rectangle)
# 判断得分条件
if next_state == self.canvas.coords(self.circle):
reward = 100
done = True
elif next_state in [self.canvas.coords(self.triangle1),
self.canvas.coords(self.triangle2)]:
reward = -100
done = True
else:
reward = 0
done = False
next_state = self.coords_to_state(next_state)
return next_state, reward, done
# 渲染环境
def render(self):
time.sleep(0.03)
self.update()

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