【转载】 强化学习(十一) Prioritized Replay DQN

原文地址：

https://www.cnblogs.com/pinard/p/9797695.html

----------------------------------------------------------------------------------------

        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。

（欢迎转载，转载请注明出处。欢迎沟通交流： liujianping-ok@163.com）

----------------------------------------------------------------------------------------------------------------

#######################################################################
# Copyright (C)                                                       #
# 2016 - 2019 Pinard Liu(liujianping-ok@163.com)                      #
# https://www.cnblogs.com/pinard                                      #
# 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 #

## https://www.cnblogs.com/pinard/p/9797695.html ##
## 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。