强化学习原理python篇08——actor-critic

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本章全篇参考赵世钰老师的教材 Mathmatical-Foundation-of-Reinforcement-Learning Actor-Critic Methods 章节，请各位结合阅读，本合集只专注于数学概念的代码实现。

前置知识

TD Error

如果用$\hat{v}(s,w)$代表状态值函数，则TD Error表示为
$$r_{t+1}+\gamma \hat{v}(s_{t+1},w)-\hat{v}(s_t,w)$$

令损失函数
$$J_w=E[v(s_t)-\hat{v}(s_t,w){]}^2$$

则利用梯度下降法最小化$J_{\theta }$为
$$\begin{array}{rl}{w}_{k+1}=& w_k-\alpha {\nabla }_{w}J(w_k)\\ =& w_k-\alpha [-2E([r_{t+1}+\gamma \hat{v}(s_{t+1},w)-\hat{v}(s_t,w)])]{\nabla }_w\hat{v}(s_t,w))\end{array}$$

用随机梯度来估算，则最小化$J_{\theta }$为
$$\begin{array}{rl}{w}_{k+1}=& w_k-\alpha {\nabla }_{w}J(w_k)\\ =& w_k+\alpha [r_{t+1}+\gamma \hat{v}(s_{t+1},w)-\hat{v}(s_t,w)]{\nabla }_w\hat{v}(s_t,w))\\ =& w_k+\alpha [v(s_t)-\hat{v}(s_t,w)]{\nabla }_w\hat{v}(s_t,w))\end{array}$$

对于q—value来说，
$$\begin{array}{rl}{w}_{k+1}=& w_k-\alpha {\nabla }_{w}J(w_k)\\ =& w_k+\alpha [r_{t+1}+\gamma \hat{q}(s_{t+1},a_{t+1},w)-\hat{q}(s_t,a_t,w)]{\nabla }_w\hat{q}(s_t,a_t,w))\end{array}$$

REINFORCE

参考上一节

$$\begin{array}{rl}{\theta }_{t+1}=& {\theta }_t+{\nabla }_{\theta }J({\theta }_t)\\ =& {\theta }_t+{\nabla }_{\theta }{E}_{S-d,a-\pi (S,\Theta )}[q(s,a){\nabla }_{\theta }ln\pi (a|s,\theta )]\end{array}$$
一般来说，${\nabla }_{\theta }ln\pi (a|s,\theta )$是未知的，可以用随机梯度法来估计，则
$$\begin{array}{rl}{\theta }_{t+1}=& {\theta }_t+{\nabla }_{\theta }J({\theta }_t)\\ =& {\theta }_t+{\nabla }_{\theta }[q(s,a){\nabla }_{\theta }ln\pi (a|s,\theta )]\end{array}$$

QAC

The simplest actor-critic algorithm

• actor：更新策略

  我们采用reinforce的方法来更新策略函数$\pi$，$$\begin{array}{rl}{\theta }_{t+1}=& {\theta }_t+{\nabla }_{\theta }[q(s,a){\nabla }_{\theta }ln\pi (a|s,\theta )]\end{array}$$

• critic：更新值

  我们采用优化td-error的方法来更新行动值$q$，
  $$\begin{array}{rl}{w}_{k+1}=& w_k+\alpha [r_{t+1}+\gamma \hat{q}(s_{t+1},a_{t+1},w)-\hat{q}(s_t,a_t,w)]{\nabla }_w\hat{q}(s_t,a_t,w))\end{array}$$

Advantage actor-critic (A2C)

减小方差的下一步是使基线与状态相关（这是一个好主意，因为不同的状态可能具有非常不同的基线）。确实，要决定某个特定动作在某种状态下的适用性，我们会使用该动作的折扣总奖励。但是，总奖励本身可以表示为状态的价值加上动作的优势值：Q(s,a)=V(s)+A(s,a)（参见DuelingDQN）。

知道每个状态的价值（至少有一个近似值）后，我们就可以用它来计算策略梯度并更新策略网络，以增加具有良好优势值的动作的执行概率，并减少具有劣势优势值的动作的执行概率。策略网络（返回动作的概率分布）被称为行动者（actor），因为它会告诉我们该做什么。另一个网络称为评论家（critic），因为它能使我们了解自己的动作有多好。这种改进有一个众所周知的名称，即advantage actorcritic方法，通常被简称为A2C。
$$E_{S-d,a-\pi (S,\Theta )}[q(s,a){\nabla }_{\theta }ln\pi (a|s,\theta )]=E_{S-d,a-\pi (S,\Theta )}[{\nabla }_{\theta }ln\pi (a|s,\theta )[q(s,a)-v(s)]]$$

• Advantage(TD error)

  ${\delta }_t=r_{t+1}+\gamma v(s_{t+1};w_t)-v(s_t;w_t)$

• actor：更新策略

  我们采用reinforce的方法来更新策略函数$\pi$，

  $$\begin{array}{rl}{\theta }_{t+1}=& {\theta }_t+a{\delta }_t{\nabla }_{\theta }[{\nabla }_{\theta }ln\pi (a|s,\theta )]\end{array}$$

• critic：更新值

  1、我们采用优化td-error的方法来更新状态值$v$，$$\begin{array}{rl}{w}_{k+1}=& w_k-\alpha {\nabla }_w[v(s_t,w)-\hat{v}(s_t,w){]}^2\end{array}$$

  2、在这里，使用实际发生的discount reward来估算$v(s_t,w)$

  3、$$\begin{array}{rl}{w}_{k+1}=& w_k-\alpha {\nabla }_w[R-\hat{v}(s_t,w){]}^2\end{array}$$

torch实现步骤

第一步

1. 初始化A2CNet，使其返回策略函数pi(s, a)，和价值V(s)

import collections
import copy
import math
import random
import time
from collections import defaultdictimport gym
import gym.spaces
import numpy as np
import torch
import torch.nn as nn
import torch.nn.utils as nn_utils
import torch.optim as optim
from gym.envs.toy_text import frozen_lake
from torch.utils.tensorboard import SummaryWriterclass A2CNet(nn.Module):def __init__(self, obs_size, hidden_size, q_table_size):super().__init__()# 策略函数pi(s, a)self.policy_net = nn.Sequential(nn.Linear(obs_size, hidden_size),nn.ReLU(),nn.Linear(hidden_size, q_table_size),nn.Softmax(dim=1),)# 价值V(s)self.v_net = nn.Sequential(nn.Linear(obs_size, hidden_size),nn.ReLU(),nn.Linear(hidden_size, 1),)def forward(self, state):if len(torch.Tensor(state).size()) == 1:state = state.reshape(1, -1)return self.policy_net(state), self.v_net(state)

第二步

2. 使用当前策略πθ在环境中交互N步，并保存状态（st）、动作（at）和奖励（rt）
3. 如果片段到达结尾，则R=0，否则为Vθ(st)，这里采用环境产生的R来近似。

def discount_reward(R, gamma):# r 为历史得分n = len(R)dr = 0for i in range(n):dr += gamma**i * R[i]return drdef generate_episode(env, n_steps, net, gamma, predict=False):episode_history = dict()r_list = []for _ in range(n_steps):episode = []predict_reward = []state, info = env.reset()while True:p, v = net(torch.Tensor(state))p = p.detach().numpy().reshape(-1)action = np.random.choice(list(range(env.action_space.n)), p=p)next_state, reward, terminated, truncted, info = env.step(action)# 如果截断，则展开 v(state) = r + gamma*v(next_state)if truncted and not terminated:reward = reward + gamma * float(net(torch.Tensor(next_state))[1].detach())episode.append([state, action, next_state, reward, terminated])predict_reward.append(reward)state = next_stateif terminated or truncted:episode_history[_] = episoder_list.append(len(episode))episode = []predict_reward = []breakif predict:return np.mean(r_list)return episode_historydef calculate_t_discount_reward(reward_list, gamma):discount_reward = []total_reward = 0for i in reward_list[::-1]:total_reward = total_reward * gamma + idiscount_reward.append(total_reward)return discount_reward[::-1]

第三步

4. 累积策略梯度$$\begin{array}{rl}{\theta }_{t+1}=& {\theta }_t+a{\delta }_t{\nabla }_{\theta }[{\nabla }_{\theta }ln\pi (a|s,\theta )]\end{array}$$

5. 累积价值梯度
   $$\begin{array}{rl}{w}_{k+1}=& w_k-\alpha {\nabla }_w[R-\hat{v}(s_t,w){]}^2\end{array}$$

# actor策略损失函数
def loss(net, batch, gamma, entropy_beta=False):l = 0for episode in batch.values():reward_list = [reward for state, action, next_state, reward, terminated in episode]state = [state for state, action, next_state, reward, terminated in episode]action = [action for state, action, next_state, reward, terminated in episode]# actor策略损失函数## max entropyqt = calculate_t_discount_reward(reward_list, gamma)pi = net(torch.Tensor(state))[0]entropy_loss = -torch.sum((pi * torch.log(pi)), axis=1).mean() * entropy_betapi = pi.gather(dim=1, index=torch.LongTensor(action).reshape(-1, 1))l_policy = -torch.Tensor(qt) @ torch.log(pi)if entropy_beta:l_policy -= entropy_loss# critic损失函数critic_loss = nn.MSELoss()(net(torch.Tensor(state))[1].reshape(-1), torch.Tensor(qt))l += l_policy + critic_lossreturn l / len(batch.values())

训练

## 初始化环境
env = gym.make("CartPole-v1", max_episode_steps=200)
# env = gym.make("CartPole-v1", render_mode = "human")state, info = env.reset()obs_n = env.observation_space.shape[0]
hidden_num = 64
act_n = env.action_space.n
a2c = A2CNet(obs_n, hidden_num, act_n)# 定义优化器
opt = optim.Adam(a2c.parameters(), lr=0.01)# 记录
writer = SummaryWriter(log_dir="logs/PolicyGradient/A2C", comment="test1")epochs = 200
batch_size = 20
gamma = 0.9
entropy_beta = 0.01
# 避免梯度太大
CLIP_GRAD = 0.1for epoch in range(epochs):batch = generate_episode(env, batch_size, a2c, gamma)l = loss(a2c, batch, gamma, entropy_beta)# 反向传播opt.zero_grad()l.backward()# 梯度裁剪nn_utils.clip_grad_norm_(a2c.parameters(), CLIP_GRAD)opt.step()max_steps = generate_episode(env, 10, a2c, gamma, predict=True)writer.add_scalars("Loss",{"loss": l.item(), "max_steps": max_steps},epoch,)print("epoch:{},  Loss: {}, max_steps: {}".format(epoch, l.detach(), max_steps))

结果

可以看到，对比上一节的几种方法，收敛速度和收敛方向都稳定了不少。

Ref

[1] Mathematical Foundations of Reinforcement Learning，Shiyu Zhao
[2] 深度学习强化学习实践（第二版），Maxim Lapan

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