在强化学习（十）Double DQN (DDQN)中，我们讲到了DDQN使用两个Q网络，用当前Q网络计算最大Q值对应的动作，用目标Q网络计算这个最大动作对应的目标Q值，进而消除贪婪法带来的偏差。今天我们在DDQN的基础上，对经验回放部分的逻辑做优化。对应的算法是Prioritized Replay DQN。
本章内容主要参考了ICML 2016的deep RL tutorial和Prioritized Replay DQN的论文<Prioritized Experience Replay>(ICLR 2016)。
1. Prioritized Replay DQN之前算法的问题
在Prioritized Replay DQN之前，我们已经讨论了很多种DQN，比如Nature DQN， DDQN等，他们都是通过经验回放来采样，进而做目标Q值的计算的。在采样的时候，我们是一视同仁，在经验回放池里面的所有的样本都有相同的被采样到的概率。
但是注意到在经验回放池里面的不同的样本由于TD误差的不同，对我们反向传播的作用是不一样的。TD误差越大，那么对我们反向传播的作用越大。而TD误差小的样本，由于TD误差小，对反向梯度的计算影响不大。在Q网络中，TD误差就是目标Q网络计算的目标Q值和当前Q网络计算的Q值之间的差距。
这样如果TD误差的绝对值
|δ(t)|较大的样本更容易被采样，则我们的算法会比较容易收敛。下面我们看看Prioritized Replay DQN的算法思路。
2. Prioritized Replay DQN算法的建模
Prioritized Replay DQN根据每个样本的TD误差绝对值
|δ(t)|，给定该样本的优先级正比于
|δ(t)|，将这个优先级的值存入经验回放池。回忆下之前的DQN算法，我们仅仅只保存和环境交互得到的样本状态，动作，奖励等数据，没有优先级这个说法。
由于引入了经验回放的优先级，那么Prioritized Replay DQN的经验回放池和之前的其他DQN算法的经验回放池就不一样了。因为这个优先级大小会影响它被采样的概率。在实际使用中，我们通常使用SumTree这样的二叉树结构来做我们的带优先级的经验回放池样本的存储。
具体的SumTree树结构如下图：
所有的经验回放样本只保存在最下面的叶子节点上面，一个节点一个样本。内部节点不保存样本数据。而叶子节点除了保存数据以外，还要保存该样本的优先级，就是图中的显示的数字。对于内部节点每个节点只保存自己的儿子节点的优先级值之和，如图中内部节点上显示的数字。
这样保存有什么好处呢？主要是方便采样。以上面的树结构为例，根节点是42，如果要采样一个样本，那么我们可以在[0,42]之间做均匀采样，采样到哪个区间，就是哪个样本。比如我们采样到了26， 在（25-29）这个区间，那么就是第四个叶子节点被采样到。而注意到第三个叶子节点优先级最高，是12，它的区间13-25也是最长的，会比其他节点更容易被采样到。
如果要采样两个样本，我们可以在[0,21],[21,42]两个区间做均匀采样，方法和上面采样一个样本类似。
类似的采样算法思想我们在word2vec原理(三) 基于Negative Sampling的模型第四节中也有讲到。
除了经验回放池，现在我们的Q网络的算法损失函数也有优化，之前我们的损失函数是：
现在我们新的考虑了样本优先级的损失函数是
第三个要注意的点就是当我们对Q网络参数进行了梯度更新后，需要重新计算TD误差，并将TD误差更新到SunTree上面。
除了以上三个部分，Prioritized Replay DQN和DDQN的算法流程相同。
3. Prioritized Replay DQN算法流程
下面我们总结下Prioritized Replay DQN的算法流程，基于上一节的DDQN，因此这个算法我们应该叫做Prioritized Replay DDQN。主流程参考论文<Prioritized Experience Replay>(ICLR 2016)。
注意，上述第二步的f步和g步的Q值计算也都需要通过Q网络计算得到。另外，实际应用中，为了算法较好的收敛，探索率εϵ需要随着迭代的进行而变小。
4. Prioritized Replay DDQN算法流程
下面我们给出Prioritized Replay DDQN算法的实例代码。仍然使用了OpenAI Gym中的CartPole-v0游戏来作为我们算法应用。CartPole-v0游戏的介绍参见这里。它比较简单，基本要求就是控制下面的cart移动使连接在上面的pole保持垂直不倒。这个任务只有两个离散动作，要么向左用力，要么向右用力。而state状态就是这个cart的位置和速度， pole的角度和角速度，4维的特征。坚持到200分的奖励则为过关。
完整的代码参见我的github: https://github.com/ljpzzz/machinelearning/blob/master/reinforcement-learning/ddqn_prioritised_replay.py， 代码中的SumTree的结构和经验回放池的结构参考了morvanzhou的github代码。
def sample(self, n):
b_idx, b_memory, ISWeights = np.empty((n,), dtype=np.int32), np.empty((n, self.tree.data[0].size)), np.empty((n, 1))
pri_seg = self.tree.total_p / n # priority segment
self.beta = np.min([1., self.beta + self.beta_increment_per_sampling]) # max = 1
min_prob = np.min(self.tree.tree[-self.tree.capacity:]) / self.tree.total_p # for later calculate ISweight
if min_prob == 0:
min_prob = 0.00001
for i in range(n):
a, b = pri_seg * i, pri_seg * (i + 1)
v = np.random.uniform(a, b)
idx, p, data = self.tree.get_leaf(v)
prob = p / self.tree.total_p
ISWeights[i, 0] = np.power(prob/min_prob, -self.beta)
b_idx[i], b_memory[i, :] = idx, data
return b_idx, b_memory, ISWeights
上述代码的采样在第二节已经讲到。根据树的优先级的和total_p和采样数n，将要采样的区间划分为n段，每段来进行均匀采样，根据采样到的值落到的区间，决定被采样到的叶子节点。当我们拿到第i段的均匀采样值v以后，就可以去SumTree中找对应的叶子节点拿样本数据，样本叶子节点序号以及样本优先级了。代码如下：
def get_leaf(self, v):
"""
Tree structure and array storage:
Tree index:
-> storing priority sum
/ \
2
/ \ / \
4 5 6 -> storing priority for transitions
Array type for storing:
[0,1,2,3,4,5,6]
"""
parent_idx = 0
while True: # the while loop is faster than the method in the reference code
cl_idx = 2 * parent_idx + 1 # this leaf's left and right kids
cr_idx = cl_idx + 1
if cl_idx >= len(self.tree): # reach bottom, end search
leaf_idx = parent_idx
break
else: # downward search, always search for a higher priority node
if v <= self.tree[cl_idx]:
parent_idx = cl_idx
else:
v -= self.tree[cl_idx]
parent_idx = cr_idx
data_idx = leaf_idx - self.capacity + 1
return leaf_idx, self.tree[leaf_idx], self.data[data_idx]
除了采样部分，要注意的就是当梯度更新完毕后，我们要去更新SumTree的权重，代码如下，注意叶子节点的权重更新后，要向上回溯，更新所有祖先节点的权重。
self.memory.batch_update(tree_idx, abs_errors) # update priority
def batch_update(self, tree_idx, abs_errors):
abs_errors += self.epsilon # convert to abs and avoid 0
clipped_errors = np.minimum(abs_errors, self.abs_err_upper)
ps = np.power(clipped_errors, self.alpha)
for ti, p in zip(tree_idx, ps):
self.tree.update(ti, p)
def update(self, tree_idx, p):
change = p - self.tree[tree_idx]
self.tree[tree_idx] = p
# then propagate the change through tree
while tree_idx != 0: # this method is faster than the recursive loop in the reference code
tree_idx = (tree_idx - 1) // 2
self.tree[tree_idx] += change
除了上面这部分的区别，和DDQN比，TensorFlow的网络结构流程中多了一个TD误差的计算节点，以及损失函数多了一个ISWeights系数。此外，区别不大。
5. Prioritized Replay DQN小结
Prioritized Replay DQN和DDQN相比，收敛速度有了很大的提高，避免了一些没有价值的迭代，因此是一个不错的优化点。同时它也可以直接集成DDQN算法，所以是一个比较常用的DQN算法。
下一篇我们讨论DQN家族的另一个优化算法Duel DQN，它将价值Q分解为两部分，第一部分是仅仅受状态但不受动作影响的部分，第二部分才是同时受状态和动作影响的部分，算法的效果也很好。
----------------------------------------------------------------------------------------------------------------
#######################################################################
# Copyright (C) #
# 2016 - 2019 Pinard Liu() #
# #
# Permission given to modify the code as long as you keep this #
# declaration at the top #
#######################################################################
# SumTree and Memory class are referred from https://github.com/MorvanZhou #
## javascript:void(0) ##
## 强化学习(十一) Prioritized Replay DQN ##
import gym
import tensorflow as tf
import numpy as np
import random
from collections import deque
# Hyper Parameters for DQN
GAMMA = 0.9 # discount factor for target Q
INITIAL_EPSILON = 0.5 # starting value of epsilon
FINAL_EPSILON = 0.01 # final value of epsilon
REPLAY_SIZE = 10000 # experience replay buffer size
BATCH_SIZE = 128 # size of minibatch
REPLACE_TARGET_FREQ = 10 # frequency to update target Q network
class SumTree(object):
"""
This SumTree code is a modified version and the original code is from:
https://github.com/jaara/AI-blog/blob/master/SumTree.py
Story data with its priority in the tree.
"""
data_pointer = 0
def __init__(self, capacity):
self.capacity = capacity # for all priority values
self.tree = np.zeros(2 * capacity - 1)
# [--------------Parent nodes-------------][-------leaves to recode priority-------]
# size: capacity - 1 size: capacity
self.data = np.zeros(capacity, dtype=object) # for all transitions
# [--------------data frame-------------]
# size: capacity
def add(self, p, data):
tree_idx = self.data_pointer + self.capacity - 1
self.data[self.data_pointer] = data # update data_frame
self.update(tree_idx, p) # update tree_frame
self.data_pointer += 1
if self.data_pointer >= self.capacity: # replace when exceed the capacity
self.data_pointer = 0
def update(self, tree_idx, p):
change = p - self.tree[tree_idx]
self.tree[tree_idx] = p
# then propagate the change through tree
while tree_idx != 0: # this method is faster than the recursive loop in the reference code
tree_idx = (tree_idx - 1) // 2
self.tree[tree_idx] += change
def get_leaf(self, v):
"""
Tree structure and array storage:
Tree index:
0 -> storing priority sum
/ \
1 2
/ \ / \
3 4 5 6 -> storing priority for transitions
Array type for storing:
[0,1,2,3,4,5,6]
"""
parent_idx = 0
while True: # the while loop is faster than the method in the reference code
cl_idx = 2 * parent_idx + 1 # this leaf's left and right kids
cr_idx = cl_idx + 1
if cl_idx >= len(self.tree): # reach bottom, end search
leaf_idx = parent_idx
break
else: # downward search, always search for a higher priority node
if v <= self.tree[cl_idx]:
parent_idx = cl_idx
else:
v -= self.tree[cl_idx]
parent_idx = cr_idx
data_idx = leaf_idx - self.capacity + 1
return leaf_idx, self.tree[leaf_idx], self.data[data_idx]
@property
def total_p(self):
return self.tree[0] # the root
class Memory(object): # stored as ( s, a, r, s_ ) in SumTree
"""
This Memory class is modified based on the original code from:
https://github.com/jaara/AI-blog/blob/master/Seaquest-DDQN-PER.py
"""
epsilon = 0.01 # small amount to avoid zero priority
alpha = 0.6 # [0~1] convert the importance of TD error to priority
beta = 0.4 # importance-sampling, from initial value increasing to 1
beta_increment_per_sampling = 0.001
abs_err_upper = 1. # clipped abs error
def __init__(self, capacity):
self.tree = SumTree(capacity)
def store(self, transition):
max_p = np.max(self.tree.tree[-self.tree.capacity:])
if max_p == 0:
max_p = self.abs_err_upper
self.tree.add(max_p, transition) # set the max p for new p
def sample(self, n):
b_idx, b_memory, ISWeights = np.empty((n,), dtype=np.int32), np.empty((n, self.tree.data[0].size)), np.empty((n, 1))
pri_seg = self.tree.total_p / n # priority segment
self.beta = np.min([1., self.beta + self.beta_increment_per_sampling]) # max = 1
min_prob = np.min(self.tree.tree[-self.tree.capacity:]) / self.tree.total_p # for later calculate ISweight
if min_prob == 0:
min_prob = 0.00001
for i in range(n):
a, b = pri_seg * i, pri_seg * (i + 1)
v = np.random.uniform(a, b)
idx, p, data = self.tree.get_leaf(v)
prob = p / self.tree.total_p
ISWeights[i, 0] = np.power(prob/min_prob, -self.beta)
b_idx[i], b_memory[i, :] = idx, data
return b_idx, b_memory, ISWeights
def batch_update(self, tree_idx, abs_errors):
abs_errors += self.epsilon # convert to abs and avoid 0
clipped_errors = np.minimum(abs_errors, self.abs_err_upper)
ps = np.power(clipped_errors, self.alpha)
for ti, p in zip(tree_idx, ps):
self.tree.update(ti, p)
class DQN():
# DQN Agent
def __init__(self, env):
# init experience replay
self.replay_total = 0
# init some parameters
self.time_step = 0
self.epsilon = INITIAL_EPSILON
self.state_dim = env.observation_space.shape[0]
self.action_dim = env.action_space.n
self.memory = Memory(capacity=REPLAY_SIZE)
self.create_Q_network()
self.create_training_method()
# Init session
self.session = tf.InteractiveSession()
self.session.run(tf.global_variables_initializer())
def create_Q_network(self):
# input layer
self.state_input = tf.placeholder("float", [None, self.state_dim])
self.ISWeights = tf.placeholder(tf.float32, [None, 1])
# network weights
with tf.variable_scope('current_net'):
W1 = self.weight_variable([self.state_dim,20])
b1 = self.bias_variable([20])
W2 = self.weight_variable([20,self.action_dim])
b2 = self.bias_variable([self.action_dim])
# hidden layers
h_layer = tf.nn.relu(tf.matmul(self.state_input,W1) + b1)
# Q Value layer
self.Q_value = tf.matmul(h_layer,W2) + b2
with tf.variable_scope('target_net'):
W1t = self.weight_variable([self.state_dim,20])
b1t = self.bias_variable([20])
W2t = self.weight_variable([20,self.action_dim])
b2t = self.bias_variable([self.action_dim])
# hidden layers
h_layer_t = tf.nn.relu(tf.matmul(self.state_input,W1t) + b1t)
# Q Value layer
self.target_Q_value = tf.matmul(h_layer,W2t) + b2t
t_params = tf.get_collection(tf.GraphKeys.GLOBAL_VARIABLES, scope='target_net')
e_params = tf.get_collection(tf.GraphKeys.GLOBAL_VARIABLES, scope='current_net')
with tf.variable_scope('soft_replacement'):
self.target_replace_op = [tf.assign(t, e) for t, e in zip(t_params, e_params)]
def create_training_method(self):
self.action_input = tf.placeholder("float",[None,self.action_dim]) # one hot presentation
self.y_input = tf.placeholder("float",[None])
Q_action = tf.reduce_sum(tf.multiply(self.Q_value,self.action_input),reduction_indices = 1)
self.cost = tf.reduce_mean(self.ISWeights *(tf.square(self.y_input - Q_action)))
self.abs_errors =tf.abs(self.y_input - Q_action)
self.optimizer = tf.train.AdamOptimizer(0.0001).minimize(self.cost)
def store_transition(self, s, a, r, s_, done):
transition = np.hstack((s, a, r, s_, done))
self.memory.store(transition) # have high priority for newly arrived transition
def perceive(self,state,action,reward,next_state,done):
one_hot_action = np.zeros(self.action_dim)
one_hot_action[action] = 1
#print(state,one_hot_action,reward,next_state,done)
self.store_transition(state,one_hot_action,reward,next_state,done)
self.replay_total += 1
if self.replay_total > BATCH_SIZE:
self.train_Q_network()
def train_Q_network(self):
self.time_step += 1
# Step 1: obtain random minibatch from replay memory
tree_idx, minibatch, ISWeights = self.memory.sample(BATCH_SIZE)
state_batch = minibatch[:,0:4]
action_batch = minibatch[:,4:6]
reward_batch = [data[6] for data in minibatch]
next_state_batch = minibatch[:,7:11]
# Step 2: calculate y
y_batch = []
current_Q_batch = self.Q_value.eval（feed_dict={self.state_input: next_state_batch})
max_action_next = np.argmax(current_Q_batch, axis=1)
target_Q_batch = self.target_Q_value.eval（feed_dict={self.state_input: next_state_batch})
for i in range(0,BATCH_SIZE):
done = minibatch[i][11]
if done:
y_batch.append(reward_batch[i])
else :
target_Q_value = target_Q_batch[i, max_action_next[i]]
y_batch.append(reward_batch[i] + GAMMA * target_Q_value)
self.optimizer.run(feed_dict={
self.y_input:y_batch,
self.action_input:action_batch,
self.state_input:state_batch,
self.ISWeights: ISWeights
})
_, abs_errors, _ = self.session.run([self.optimizer, self.abs_errors, self.cost], feed_dict={
self.y_input: y_batch,
self.action_input: action_batch,
self.state_input: state_batch,
self.ISWeights: ISWeights
})
self.memory.batch_update(tree_idx, abs_errors) # update priority
def egreedy_action(self,state):
Q_value = self.Q_value.eval（feed_dict = {
self.state_input:[state]
})[0]
if random.random() <= self.epsilon:
self.epsilon -= (INITIAL_EPSILON - FINAL_EPSILON) / 10000
return random.randint(0,self.action_dim - 1)
else:
self.epsilon -= (INITIAL_EPSILON - FINAL_EPSILON) / 10000
return np.argmax(Q_value)
def action(self,state):
return np.argmax(self.Q_value.eval（feed_dict = {
self.state_input:[state]
})[0])
def update_target_q_network(self, episode):
# update target Q netowrk
if episode % REPLACE_TARGET_FREQ == 0:
self.session.run(self.target_replace_op)
#print('episode '+str(episode) +', target Q network params replaced!')
def weight_variable(self,shape):
initial = tf.truncated_normal(shape)
return tf.Variable(initial)
def bias_variable(self,shape):
initial = tf.constant(0.01, shape = shape)
return tf.Variable(initial)
# ---------------------------------------------------------
# Hyper Parameters
ENV_NAME = 'CartPole-v0'
EPISODE = 3000 # Episode limitation
STEP = 300 # Step limitation in an episode
TEST = 5 # The number of experiment test every 100 episode
def main():
# initialize OpenAI Gym env and dqn agent
env = gym.make(ENV_NAME)
agent = DQN(env)
for episode in range(EPISODE):
# initialize task
state = env.reset()
# Train
for step in range(STEP):
action = agent.egreedy_action(state) # e-greedy action for train
next_state,reward,done,_ = env.step(action)
# Define reward for agent
reward = -1 if done else 0.1
agent.perceive(state,action,reward,next_state,done)
state = next_state
if done:
break
# Test every 100 episodes
if episode % 100 == 0:
total_reward = 0
for i in range(TEST):
state = env.reset()
for j in range(STEP):
env.render()
action = agent.action(state) # direct action for test
state,reward,done,_ = env.step(action)
total_reward += reward
if done:
break
ave_reward = total_reward/TEST
print ('episode: ',episode,'Evaluation Average Reward:',ave_reward)
agent.update_target_q_network(episode)
if __name__ == '__main__':
main()
ps: prioritized replay DQN 运算速度奇慢，大约能有数倍分之前面的DQN算法，但是效果看来却有提升。
