-
An episode consists of an alternating sequence of states and state-action pairs:
-
epsilon-greedy policy for one step, then bootstrap the action value function:
$$Q\left(S_{t}, A_{t}\right) \leftarrow Q\left(S_{t}, A_{t}\right)+\alpha\left[R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right)-Q\left(S_{t}, A_{t}\right)\right]$$
We allow both behavior and target policies to improve
The target policy
和SARSA不同的是，Q-learning采用不应的policy，这样收集数据的policy就可以有更多的探索。
其实我觉得on-policy和off-policy之间差别就是看你优化的策略和你采数据的策略是不是同一个策略，如果是，那就是on-policy，如果不是，那就是off-policy。也就说off-policy有两个策略，一个是target policy，另一个是behavior policy。
off-policy 相对于on-policy会有一些好处：
- Learn about optimal policy while following exploratory policy
- Learn from observing humans or other agents
- Re-use experience generated from old policies
$\pi_{1}, \pi_{2}, \ldots, \pi_{t-1}$
重要性采样（Importance Sampling ）是统计中的一种采样方法。在强化学习中经常用到这种采样方法，特别是off-policy方法当中，我们会用通过与环境交互的策略采集到的数据来去优化我们的策略。它主要用在一些难以直接采样的数据分布上。我们虽然无法从这个分布函数采样，但我们还有其他常见的、可以采样的分布，我们能不能对上面的公式进行一些变换，使用常见的分布采样呢？我们令待采样的分布为p(x) ，另一个简单可采样且定义域与p(x) 相同的概率密度函数为p(x)，我们可以得到
$$\begin{aligned} \mathbb{E}{T \sim \pi}[g(T)] &=\int P(T) g(T) d T \ &=\int Q(T) \frac{P(T)}{Q(T)} g(T) d T \ &=\mathbb{E}{T \sim \mu}\left[\frac{P(T)}{Q(T)} g(T)\right] \ & \approx \frac{1}{n} \sum_{i} \frac{P\left(T_{i}\right)}{Q\left(T_{i}\right)} g\left(T_{i}\right) \end{aligned}$$
此时我们发现，公式变成了类似上一个方法的形式，而且我们只需要从这个简单 分布
Why don't use importance sampling on Q-Learning?
Q-learning is over the transition distribution, not over policy distribution thus no need to correct different policy distributions
Short answer: Because Q-learning does not make expected value estimates over the policy distribution. For the full answer click here
这里我们主要是用Q-learning和Sarsa去解决Cliffwalk问题
# 作者：Yunhui
# 创建时间：2020/9/27 10:27
# IDE：PyCharm
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.colors import hsv_to_rgb
def change_range(values, vmin=0, vmax=1):
"""
这个函数的作用是把values的值缩放到(vmin, vmax)之间，除数里面的1e-7是为了防止values里面都是0的时候造成分母为0
"""
start_zero = values - np.min(values)
return (start_zero / (np.max(start_zero) + 1e-7)) * (vmax - vmin) + vmin
class GridWorld:
# 这一块其实没懂地形的颜色是怎么用的
terrain_color = dict(normal=[127 / 360, 0, 96 / 100],
objective=[26 / 360, 100 / 100, 100 / 100],
cliff=[247 / 360, 92 / 100, 70 / 100],
player=[344 / 360, 93 / 100, 100 / 100])
def __init__(self):
self.player = None
self._create_grid()
self._draw_grid()
self.num_steps = 0
'''
(1)、以单下划线开头，表示这是一个保护成员，只有类对象和子类对象自己能访问到这些变量。以单下划线开头的变量和函数被默认当作是内部函数，
使用from module improt *时不会被获取，但是使用import module可以获取
(2)、以单下划线结尾仅仅是为了区别该名称与关键词
(3)、双下划线开头，表示为私有成员，只允许类本身访问，子类也不行。在文本上被替换为_class__method
(4)、双下划线开头，双下划线结尾。一种约定，Python内部的名字，用来区别其他用户自定义的命名,以防冲突。是一些 Python 的“魔术”对象
，表示这是一个特殊成员，例如：定义类的时候，若是添加__init__方法，那么在创建类的实例的时候，实例会自动调用这个方法，一般用来对实
例的属性进行初使化，Python不建议将自己命名的方法写为这种形式。
'''
def _create_grid(self, initial_grid=None):
self.grid = self.terrain_color['normal'] * np.ones((4, 12, 3)) # 这里的（4,12）应该表示的是格子数，3表示的是RGB三个通道？？？
self._add_objectives(self.grid)
def _add_objectives(self, grid):
grid[-1, 1:11] = self.terrain_color['cliff'] # 整个Clifwalk的左上角是坐标（0,0），最下面一行的第1个到第10个格子是cliff陷阱
grid[-1, -1] = self.terrain_color['objective'] # 最下面一行的第11个格子是目标
def _draw_grid(self):
self.fig, self.ax = plt.subplots(figsize=(12, 4))
self.ax.grid(which='minor')
self.q_texts = [self.ax.text(*self._id_to_position(i)[::-1], '0',
fontsize=11, verticalalignment='center',
horizontalalignment='center') for i in range(12 * 4)]
self.im = self.ax.imshow(hsv_to_rgb(self.grid), cmap='terrain',
interpolation='nearest', vmin=0, vmax=1)
self.ax.set_xticks(np.arange(12))
self.ax.set_xticks(np.arange(12) - 0.5, minor=True)
self.ax.set_yticks(np.arange(4))
self.ax.set_yticks(np.arange(4) - 0.5, minor=True)
def reset(self): # 位置复原到坐标（3,0）
self.player = (3, 0)
self.num_steps = 0
return self._position_to_id(self.player)
# 下面的这两个函数是将二维坐标与一维数字进行相互转换，从左上角的0（0,0）到右下角的47（3,11），//表示除之后取整
def _position_to_id(self, pos):
""" Maps a position in x,y coordinates to a unique ID """
return pos[0] * 12 + pos[1]
def _id_to_position(self, idx):
return (idx // 12), (idx % 12)
# 动作0,1,2,3分别表示上、下、右、左，掉入cliff奖励为-100，结束；普通的格子奖励为-1，不结束；objective奖励为0，结束；
def step(self, action):
# Possible actions
if action == 0 and self.player[0] > 0:
self.player = (self.player[0] - 1, self.player[1])
if action == 1 and self.player[0] < 3:
self.player = (self.player[0] + 1, self.player[1])
if action == 2 and self.player[1] < 11:
self.player = (self.player[0], self.player[1] + 1)
if action == 3 and self.player[1] > 0:
self.player = (self.player[0], self.player[1] - 1)
self.num_steps = self.num_steps + 1
# Rules
if all(self.grid[self.player] == self.terrain_color['cliff']):
reward = -100
done = True
elif all(self.grid[self.player] == self.terrain_color['objective']):
reward = 0
done = True
else:
reward = -1
done = False
return self._position_to_id(self.player), reward, done
def render(self, q_values=None, action=None, max_q=False, colorize_q=False):
assert self.player is not None, 'You first need to call .reset()'
if colorize_q:
assert q_values is not None, 'q_values must not be None for using colorize_q'
grid = self.terrain_color['normal'] * np.ones((4, 12, 3))
values = change_range(np.max(q_values, -1)).reshape(4, 12)
grid[:, :, 1] = values
self._add_objectives(grid)
else:
grid = self.grid.copy()
grid[self.player] = self.terrain_color['player']
self.im.set_data(hsv_to_rgb(grid))
if q_values is not None:
xs = np.repeat(np.arange(12), 4)
ys = np.tile(np.arange(4), 12)
for i, text in enumerate(self.q_texts):
if max_q:
q = max(q_values[i])
txt = '{:.2f}'.format(q)
text.set_text(txt)
else:
actions = ['U', 'D', 'R', 'L']
txt = '\n'.join(['{}: {:.2f}'.format(k, q) for k, q in zip(actions, q_values[i])])
text.set_text(txt)
if action is not None:
self.ax.set_title(action, color='r', weight='bold', fontsize=32)
plt.pause(0.01)
def egreedy_policy(q_values, state, epsilon=0.1):
"""
Choose an action based on a epsilon greedy policy.
A random action is selected with epsilon probability, else select the best action.
"""
if np.random.random() < epsilon:
return np.random.choice(4)
else:
return np.argmax(q_values[state])
def q_learning(env, num_episodes=500, render=True, exploration=0.1, learning_rate=0.5, gamma=0.9):
q_values = np.zeros((num_states, num_actions))
ep_rewards = []
for i in range(num_episodes):
state = env.reset()
done = False
reward_sum = 0
while not done:
action = egreedy_policy(q_values, state, exploration)
next_state, reward, done = env.step(action)
reward_sum += reward
# 更新q表
td_target = reward + gamma * np.max(q_values[next_state])
td_error = td_target - q_values[state][action]
q_values[state][action] += learning_rate * td_error
state = next_state
if render:
env.render(q_values, action=actions[action], colorize_q=True)
if done:
print("第%d个epsiode已经结束" % i)
ep_rewards.append(reward_sum)
return ep_rewards, q_values
def sarsa(env, num_episodes=500, render=True, exploration_rate=0.1, learning_rate=0.5, gamma=0.9):
q_values_sarsa = np.zeros((num_states, num_actions))
ep_rewards = []
for _ in range(num_episodes):
state = env.reset()
done = False
reward_sum = 0
action = egreedy_policy(q_values_sarsa, state, exploration_rate) # 这里是和q_learning不一样的地方
while not done:
next_state, reward, done = env.step(action)
reward_sum += reward
# 选择动作
next_action = egreedy_policy(q_values_sarsa, next_state, exploration_rate)
td_target = reward + gamma * (q_values[next_state][next_action])
td_error = td_target - q_values_sarsa[state][action]
q_values_sarsa[state][action] += learning_rate * td_error
state = next_state
action = next_action
if render:
env.render(q_values, action=action[action], colorize=True)
ep_rewards.append(reward_sum)
return ep_rewards, q_values_sarsa
def play(q_values):
env = GridWorld()
state, done = env.reset()
while not done:
action = egreedy_policy(q_values, state, 0.0)
next_state, reward, done = env.step(action)
state = next_state
env.render(q_values=q_values, action=actions[action], colorize_q=True)
UP = 0
DOWN = 1
RIGHT = 2
LEFT = 3
actions = ['UP', 'DOWN', 'RIGHT', 'LEFT']
env = GridWorld()
num_states = 4 * 12 # The number of states in simply the number of "squares" in our grid world, in this case 4 * 12
num_actions = 4 # We have 4 possible actions, up, down, right and left
q_learning_rewards, q_values = q_learning(env, gamma=0.9, learning_rate=1, render=False)
print("q_learning_rewards:", q_learning_rewards)
print("q_values:", q_values)
env.render(q_values, colorize_q=True)
# 下面的这个zip其实相当于是要把10次试验的每个epsiode的reward取平均然后去画图，先得到的是一个10*5000的矩阵
q_learning_rewards, _ = zip(*[q_learning(env, render=True, exploration=0.1,
learning_rate=1) for _ in range(10)])
# 得到avg_reward其实就可以画图，就mean_reward主要是为了画平均reward的参考线以及打印出平均reward
avg_rewards = np.mean(q_learning_rewards, axis=0)
mean_reward = [np.mean(avg_rewards)] * len(avg_rewards)
fig, ax = plt.subplots()
ax.set_xlabel('Episodes using Q-learning')
ax.set_ylabel('Rewards')
ax.plot(avg_rewards)
ax.plot(mean_reward, 'g--')
print('Mean Reward using Q-Learning: {}'.format(mean_reward[0]))
# Sarsa learning for cliff walk
sarsa_rewards, q_values_sarsa = sarsa(env, render=False, learning_rate=0.5, gamma=0.99)
sarsa_rewards, _ = zip(*[sarsa(env, render=False, exploration_rate=0.2) for _ in range(10)])
avg_rewards = np.mean(sarsa_rewards, axis=0)
mean_reward = [np.mean(avg_rewards)] * len(avg_rewards)
fig, ax = plt.subplots()
ax.set_xlabel('Episodes using Sarsa')
ax.set_ylabel('Rewards')
ax.plot(avg_rewards)
ax.plot(mean_reward, 'g--')
print('Mean Reward using Sarsa: {}'.format(mean_reward[0]))
# visualize the episode in inference for Q-learing and Sarsa-learning
play(q_values)
play(q_values_sarsa)Qlearning解决Frozenlake和Mountaincar的Python代码放在github上了。
- Previous lectures on small RL problems:
- Cliff walk:
$4 \times 16$ states - Mountain car: 1600 states
- Tic-Tac-Toe:
$10^{3}$ states
- Cliff walk:
- Large-scale problems:
- Backgammon:
$10^{20}$ states - Chess:
$10^{47}$ states - Game of Go:
$10^{170}$ states - Q Robot Arm and Helicopter have continuous state space
- (5 Number of atomics in universe:
$10^{80}$
- Backgammon:
即使智能体具有完整和准确的环境模型，智能体也通常没有足够的计算能力在每一时刻都全面利用它。 而可用的存储资源也是一个重要的限制。精确的价值函数、策略和模型都需要占用储存资源。在大多数实际问题中，环境状态远远不是一个表格可以装下的。所以就想着用函数近似去解决这个问题，一来可以解决q表太大的问题，二来可以对于没有见过的状态有泛化性。
当我们去对V函数做价值优化的时候，我们有两种情况
- 一种是我们已经知道真值（这里的意思其实是知道部分真值，然后去对V函数做近似就可以有泛化能力）：
We assumed that true value function
-
还有一种情况也是更现实的情况就是我们不知道V函数，我们只知道reward，那么我们就可以用前面的方法去估算V：
- For
$\mathrm{M} \mathrm{C}$ , the target is the actual return$G_{t}$
$$ \Delta \mathbf{w}=\alpha\left(G_{t}-\hat{v}\left(s_{t}, \mathbf{w}\right)\right) \nabla_{\mathbf{w}} \hat{v}\left(s_{t}, \mathbf{w}\right) $$
$\mathrm{Return} G_{t}$ is an unbiased, but noisy sample of true value$v^{\pi}\left(s_{t}\right)$ - For
$T D(0),$ the target is the$T D$ target$R_{t+1}+\gamma \hat{v}\left(s_{t+1}, w\right)$
$$ \Delta \mathbf{w}=\alpha\left(R_{t+1}+\gamma \hat{v}\left(s_{t+1}, \mathbf{w}\right)-\hat{v}\left(s_{t}, \mathbf{w}\right)\right) \nabla_{\mathbf{w}} \hat{v}\left(s_{t}, \mathbf{w}\right) $$
- For
TD target
Why biased?
It is drawn from our previous estimate, rather than the true value:
Using linear
- Semi-gradient Sarsa for VFA Control:
具体我们可以用一下几种模型去进行拟合：
-
Linear combinations of features 特征的线性组合
-
Neural networks 神经网络
-
Decision trees 决策树
-
Nearest neighbors
上面的选择其实是对label的选择，对于模型来说，我们可以表示为状态提取出的特征的线性变换：
Represent value function by a linear combination of features
$$
\hat{v}(s, \mathbf{w})=\mathbf{x}(s)^{T} \mathbf{w}=\sum_{j=1}^{n} x_{j}(s) w_{j}
$$
The objective function is quadratic in parameter
-
Function approximation
-
Bootstrapping
-
off-policy
如果包含这三个要素，很有可能不稳定性就难以避免，如果只出现两个要素，那么不稳定性就是有可能避免的。
在这三个要素中，FA是最不可能舍弃的，状态聚合或者非参数化的方法的复杂性随数据的增大而增大，都是效果太差或价格太昂贵。
不使用Boot strapping是有可能的，付出的代价是计算和数据上的效率。
很多强化学习算法都是在解决这个不稳定性的问题
