From a3f1f49a1716cfb686eb80849bf146d9f38f8892 Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Fri, 17 Jul 2020 09:07:12 +0800 Subject: [PATCH] fix some errors --- docs/chapter1/chapter1.md | 135 ++++++++++++++++++++++++++++++------- docs/chapter1/img/1.35.png | Bin 89475 -> 64598 bytes docs/chapter1/img/1.6.png | Bin 284872 -> 176819 bytes docs/chapter2/chapter2.md | 96 ++++++++++++++++++++------ docs/chapter5/chapter5.md | 14 +++- 5 files changed, 198 insertions(+), 47 deletions(-) diff --git a/docs/chapter1/chapter1.md b/docs/chapter1/chapter1.md index 0cfab11..9eece4c 100644 --- a/docs/chapter1/chapter1.md +++ b/docs/chapter1/chapter1.md @@ -13,6 +13,7 @@ * 我们告诉了这个 learner 正确的标签是什么,这样它可以通过正确的标签来修正自己的这个预测。 ![](img/1.3.png) + 在强化学习里面,这两点其实都不满足。举一个 Atari Breakout 游戏的例子,这是一个打砖块的游戏,控制木板,然后把这个球反弹到上面来消除这些砖块。在游戏过程中,大家可以发现你这个 agent,你这个 learner 得到的观测其实不是个 i.i.d 的分布。就是你上一帧下一帧其实有非常强的这个连续性。另外一点,你这个玩游戏的过程中,其实并没有立刻获得这个反馈,比如你现在把这个木板往右移,那么只会使得这个球然后往上或者往左上去一点,你并不会得到立刻的反馈。所以这也是为什么强化学习这么困难,就是你没有得到很好的反馈,然后你依然希望这个 agent 在这个环境里面学习。 ![](img/1.4.png) @@ -31,7 +32,7 @@ ![](img/1.6.png)通过跟监督学习比较,我们可以总结出这个强化学习的一些特征。 -* 首先它是有这个 trial-and-error exploration,它需要通过在环境里面探索来获取对这个环境的理解。 +* 首先它是有这个 trial-and-error exploration,它需要通过探索环境来获取对这个环境的理解。 * 第二点是这个强化学习 agent 会从环境里面获得延迟的奖励。 * 第三点是这个强化学习的训练过程中时间非常重要,因为你得到的数据都是有这个时间关联的,而不是这个 i.i.d 分布的。大家如果做过机器学习的话就会发现,如果你的观测数据有非常强的关联,其实会使得这个训练非常不稳定。这也是为什么监督学习我们希望我们的 data 尽量是 i.i.d 了,这样就可以消除数据之间的相关性。 * 第四点是这个 agent 的行为会影响它随后得到的数据,这一点其实也是非常重要的。就是在我们训练这个 agent 的过程中,很多时候我们也是通过正在学习的这个 agent 去跟这个环境交互得到数据。所以如果我们在训练过程中,这个 agent 的模型如果快死掉了,那么就会使得我们采集到的数据也是非常糟糕的。这样整个训练过程就失败了。所以其实在强化学习里面一个非常重要的问题就是怎么使得让这个 agent 的行为一直稳定的提升。 @@ -106,9 +107,9 @@ ## Introduction to Sequential Decision Making ![](img/1.18.png) -接下来我们讲序列决策过程,强化学习研究的问题 是 agent 跟环境交互,这幅图左边画的是一个 agent,agent 一直在跟环境进行交互。这个agent 把它输出的动作给环境,然后环境取得这个动作过后,会进行到下一步,然后会把下一步的观测跟它上一步是否得到奖励返还给 agent。通过这样的交互过程,然后会产生很多观测,agent 的目的就是为了从这些观测之中学到能极大化奖励的策略。 +接下来我们讲`序列决策过程`,强化学习研究的问题 是 agent 跟环境交互,这幅图左边画的是一个 agent,agent 一直在跟环境进行交互。这个agent 把它输出的动作给环境,然后环境取得这个动作过后,会进行到下一步,然后会把下一步的观测跟它上一步是否得到奖励返还给 agent。通过这样的交互过程,然后会产生很多观测,agent 就是为了从这些观测之中学到能极大化奖励的策略。 -### Rewards +### Reward ![](img/1.19.png) 奖励是由环境给的一个反馈信号,这个信号指定了这个 agent 在某一步采取了某个策略是否得到奖励。强化学习的目的就是为了极大化 agent 可以获得的奖励,这个 agent 在这个环境里面存在的目的就是为了极大它的期望积累的奖励。 @@ -118,7 +119,7 @@ 这里我给大家举一些奖励的例子。不同的环境,奖励的也是不同的。 * 比如说一个下象棋的选手,它的目的其实就为了赢棋,奖励是说在最后棋局结束的时候,他知道会得到一个正奖励或者负奖励。 -* 羚羊站立其实也是一个强化学习过程,那它得到的奖励就是它是否可以最后跟它妈妈一块离开或者它被吃掉。 +* 羚羊站立也是一个强化学习过程,那它得到的奖励就是它是否可以最后跟它妈妈一块离开或者它被吃掉。 * 股票管理里面,奖励定义由你的股票获取的收益跟损失决定。 * 在玩雅达利游戏的时候,奖励就是你有没有在增加游戏的分数,奖励本身的稀疏程度其实也决定了这个游戏的难度。 @@ -155,12 +156,18 @@ Policy 就是决定了这个 agent 的行为,它其实是一个函数,把输入的状态变成行为。所以这里有有两种 policy。 -* 一种是 stochastic policy,它就是这个 $\pi$ 函数,当你输入一个状态 s 的时候,它输出其实是一个概率。这概率就是你所有行为的一个概率,然后你可以进一步对这个概率分布进行采样,然后得到真实的你采取的行为。比如说这个概率可能是有70%的概率往左,30%的概率往右,那么你通过采样就可以得到一个 action。 +* 一种是 `stochastic policy(随机性策略)`,它就是 $\pi$ 函数 $\pi(a | s)=P\left[A_{t}=a | S_{t}=s\right]$ ,当你输入一个状态 s 的时候,它输出其实是一个概率。这概率就是你所有行为的一个概率,然后你可以进一步对这个概率分布进行采样,然后得到真实的你采取的行为。比如说这个概率可能是有 70% 的概率往左,30% 的概率往右,那么你通过采样就可以得到一个 action。 -* 另外一种是 deterministic policy,就是说你这里有可能只是采取它的极大化,采取最有可能的概率。所以你现在这个概率就是事先决定好的。 +* 一种是 `deterministic policy(确定性策略)`,就是说你这里有可能只是采取它的极大化,采取最有可能的概率。所以你现在这个概率就是事先决定好的。 从 Atari 游戏的来看的话,policy function 的输入就是一游戏的一帧,然后它的输出决定你是往左走或者是往右走。 +通常情况下,强化学习一般使用`随机性策略`。随机性策略可以有很多优点: + +* 在学习时可以通过引入一定随机性更好地探索环境; + +* 随机性策略的动作具有多样性,这一点在多个智能体博弈时也非常重要。采用确定性策略的智能体总是对同样的环境做出相同的动作,会导致它的策略很容易被对手预测。 + ### Value Function @@ -187,8 +194,6 @@ Policy 就是决定了这个 agent 的行为,它其实是一个函数,把输 这里我们来看一个走迷宫的例子,这个例子是要求这个 agent 从 start 开始,然后到达 goal 的位置。我们这里设定的奖励是每走一步,你就会得到一个负的奖励,然后这里可以采取的动作是往上下左右走。然后当前状态用现在 agent 所在的位置来描述。 - - ![](img/1.31.png) 我们可以用不同的强化学习算法来解这个环境,如果我们这里采取的是 Policy-based RL。当我们学习好了这个环境过后,然后在每一个状态,我们就会得到一个最佳的行为。比如说现在在第一格开始的时候,我们知道它最佳行为是往右走,然后第二格的时候,得到的最佳策略是往上走,第三格是往右走。通过这个最佳的策略,我们就可以最快地到达终点。 @@ -198,22 +203,31 @@ Policy 就是决定了这个 agent 的行为,它其实是一个函数,把输 ### Types of RL Agents ![](img/1.33.png) -所以根据强化学习 agent 的不同,我们可以把 agent 进行归类。 +根据强化学习 agent 的不同,我们可以把 agent 进行归类。 * 基于价值函数的 agent。这一类 agent 显式地学习的就是价值函数,隐式地学习了它的策略。因为这个策略是从我们学到的价值函数里面推算出来的。 -* 另外一种基于策略导向的 agent。它直接去学习 policy,就是说你直接给它一个 state,它就会输出这个动作的概率。然后在这个 policy-based agent 里面并没有去学习它的价值函数。 - +* 基于策略的 agent。它直接去学习 policy,就是说你直接给它一个 state,它就会输出这个动作的概率。然后在这个 policy-based agent 里面并没有去学习它的价值函数。 * 然后另外还有一种 agent 是把这两者结合。把 value-based 和 policy-based 结合起来就有了 `Actor-Critic agent`。这一类 agent 就把它的策略函数和价值函数都学习了,然后通过两者的交互得到一个最佳的行为。 +Q: 基于策略迭代和基于价值迭代的强化学习方法有什么区别? + +A: 对于一个状态转移概率已知的马尔可夫决策过程,我们可以使用动态规划算法来求解;从决策方式来看,强化学习又可以划分为基于策略迭代的方法和基于价值迭代的方法。`决策方式`是智能体在给定状态下从动作集合中选择一个动作的依据,它是静态的,不随状态变化而变化。 + +在`基于策略迭代`的强化学习方法中,智能体会`制定一套动作策略`(确定在给定状态下需要采取何种动作),并根据这个策略进行操作。强化学习算法直接对策略进行优化,使制定的策略能够获得最大的奖励。 + +而在`基于价值迭代`的强化学习方法中,智能体不需要制定显式的策略,它`维护一个价值表格或价值函数`,并通过这个价值表格或价值函数来选取价值最大的动作。基于价值迭代的方法只能应用在不连续的、离散的环境下(如围棋或某些游戏领域),对于行为集合规模庞大、动作连续的场景(如机器人控制领域),其很难学习到较好的结果(此时基于策略迭代的方法能够根据设定的策略来选择连续的动作)。 + +基于价值迭代的强化学习算法有 Q-learning、 Sarsa 等,而基于策略迭代的强化学习算法有策略梯度算法等。此外, Actor-Critic 算法同时使用策略和价值评估来做出决策,其中,智能体会根据策略做出动作,而价值函数会对做出的动作给出价值,这样可以在原有的策略梯度算法的基础上加速学习过程,取得更好的效果。 + ![](img/1.34.png) 另外,我们是可以通过 agent 到底有没有学习这个环境模型来分类。 -* 第一种是 `model-based` RL agent,它通过学习这个状态的转移来采取措施。 -* 另外一种是 ` model-free` RL agent,它没有去直接估计这个状态的转移,也没有得到环境的具体转移变量。它通过学习 value function 和 policy function 进行决策。这种 model-free 的模型里面没有一个环境转移的一个模型。 +* 第一种是 `model-based(有模型)` RL agent,它通过学习这个状态的转移来采取措施。 +* 另外一种是 ` model-free(免模型)` RL agent,它没有去直接估计这个状态的转移,也没有得到环境的具体转移变量。它通过学习 value function 和 policy function 进行决策。这种 model-free 的模型里面没有一个环境转移的一个模型。 Q: 有模型学习和免模型学习有什么区别? -针对是否需要对真实环境建模,强化学习可以分为有模型学习和免模型学习。 +A: 针对是否需要对真实环境建模,强化学习可以分为有模型学习和免模型学习。 * 有模型学习是指根据环境中的经验,构建一个虚拟世界,同时在真实环境和虚拟世界中学习; @@ -233,7 +247,7 @@ Q: 有模型学习和免模型学习有什么区别? 总的来说,有模型学习相比于免模型学习仅仅多出一个步骤,即对真实环境进行建模。因此,一些有模型的强化学习方法,也可以在免模型的强化学习方法中使用。在实际应用中,如果不清楚该用有模型强化学习还是免模型强化学习,可以先思考一下,在智能体执行动作前,是否能对下一步的状态和奖励进行预测,如果可以,就能够对环境进行建模,从而采用有模型学习。 -免模型学习通常属于数据驱动型方法,需要大量的采样来估计状态、动作及奖励函数,从而优化动作策略。例如,在 Atari 平台上的《太空侵略者》(Space Invader)游戏中,免模型的深度强化学习需要大约 2 亿帧游戏画面才能学到比较理想的效果。相比之下,有模型学习可以在一定程度上缓解训练数据匮乏的问题,因为智能体可以在虚拟世界中行训练。 +免模型学习通常属于数据驱动型方法,需要大量的采样来估计状态、动作及奖励函数,从而优化动作策略。例如,在 Atari 平台上的 Space Invader 游戏中,免模型的深度强化学习需要大约 2 亿帧游戏画面才能学到比较理想的效果。相比之下,有模型学习可以在一定程度上缓解训练数据匮乏的问题,因为智能体可以在虚拟世界中行训练。 免模型学习的泛化性要优于有模型学习,原因是有模型学习算需要对真实环境进行建模,并且虚拟世界与真实环境之间可能还有差异,这限制了有模型学习算法的泛化性。 @@ -254,9 +268,9 @@ Q: 有模型学习和免模型学习有什么区别? ![](img/1.37.png) 在强化学习里面,Exploration 和 Exploitation 是两个很核心的问题。 -* Exploration 的意思是说我们怎么去探索这个环境。通过尝试不同的行为,然后可以得到一个最佳的策略,得到最大奖励的策略。 +* Exploration 是说我们怎么去探索这个环境。通过尝试不同的行为,然后可以得到一个最佳的策略,得到最大奖励的策略。 -* Exploitation 的意思是我们现在不去尝试新的东西,就采取我们已知的可以得到很大奖励的行为。 +* Exploitation 是说我们现在不去尝试新的东西,就采取我们已知的可以得到很大奖励的行为。 因为在刚开始的时候这个强化学习 agent 并不知道它采取了某个行为会发生什么,所以它只能通过试错去探索。所以 Exploration 就是在试错来理解采取的这个行为到底可不可以得到好的奖励。Exploitation 是说我们直接采取已知的可以得到很好奖励的行为。所以这里就面临一个 trade-off。怎么通过牺牲一些短期的 reward 来获得行为的理解。 @@ -280,7 +294,7 @@ Q: 有模型学习和免模型学习有什么区别? -### Experiment with Reinforcement Learning +## Experiment with Reinforcement Learning ![](img/1.39.png) 接下来我们会进入一个实践环节。强化学习其实是一个理论跟实践结合的一个机器学习分支,需要去推导很多算法公式。然后去理解它算法背后的一些数学原理。另外一方面,上机实践通过实现算法,然后在很多实验环境里面去真正探索这个算法是不是可以得到预期效果也是一个非常重要的一个过程。所以我希望大家把实践提到一个很高的高度,真正去实践这个强化学习的算法。 @@ -296,13 +310,31 @@ Q: 有模型学习和免模型学习有什么区别? ![](img/1.42.png) -强化学习的话就不得不提 OpenAI 这家公司。OpenAI 是一个非盈利的人工智能研究公司。Open AI 公布了非常多的学习资源以及这个算法资源,他们之所以叫 Open AI,就是他们把他们所有开发的这些算法都 open source 出来。 - +强化学习的话就不得不提 [ OpenAI](https://openai.com/)。OpenAI 是一个非盈利的人工智能研究公司。Open AI 公布了非常多的学习资源以及这个算法资源,他们之所以叫 Open AI,就是他们把他们所有开发的这些算法都 open source 出来。 +### Gym ![](img/1.43.png) -`OpenAI Gym` 里面包含了很多现有的环境,比如说这个 Atari 游戏,然后还有一些强化学习里面比较经典的一些控制的环境。Gym Retro 是这个 gym 环境的进一步扩展,包含了更多的一些游戏。 +[OpenAI Gym](https://gym.openai.com/) 是一个环境仿真库,里面包含了很多现有的环境。针对不同的场景,我们可以选择不同的环境, + +* 离散控制场景(输出的动作是可数的,比如 Pong 游戏中输出的向上或向下动作):一般使用 Atari 环境评估 +* 连续控制场景(输出的动作是不可数的,比如机器人走路时不仅有方向,还要角度,角度就是不可数的,是一个连续的量 ):一般使用 mujoco 环境评估 + +Gym Retro 是对 Gym 环境的进一步扩展,包含了更多的一些游戏。 + +我们可以通过 pip 来安装 Gym: + +```bash +pip install gym +``` + +在 Python 环境中导入Gym,如果不报错,就可以认为 Gym 安装成功。 + +```python +$python +>>>import gym +``` ![](img/1.44.png) @@ -311,6 +343,10 @@ Q: 有模型学习和免模型学习有什么区别? ![](img/1.45.png) 在OpenAI Gym 里面有很经典的控制类游戏,比如说 Acrobot,就是把这个两节铁杖,然后甩了立起来。还有 CartPole,通过控制一个平板,让这个木棍立起来。还有 MountainCar 的一个例子,就通过前后移动这个车,让它到达这个旗子的位置。大家可以去[这个链接](https://gym.openai.com/envs/#classic_control)看一看这些环境。在刚开始测试强化学习的时候,可以选择这些简单环境,因为这些环境可能是在一两分钟之内你就可以见到一个效果。 + + +Gym 官方网站提供了一个简单的例子供我们了解接口(https://gym.openai.com/docs/)。通过这个例子,我们就能知道这个接口的使用方法。 + ![](img/1.46.png) 这里我们看一下 CartPole 的这个环境。对于这个环境,有两个动作,Cart 往左移还是往右移。这里得到了观测:它这个车当前的位置,Cart 当前的往左往右移的速度,这个杆的这个角度以及它的杆的最高点的这个速度。 @@ -328,12 +364,63 @@ observation, reward, done, info = env.step(action) 如果我们玩这个环境的话,就就直接可以 import gym,调入 CartPole 这个环境。然后这里就可以通过这个采样,然后来执行这个环境。 +```python +import gym # 导入Gym的Python接口环境包 +env = gym.make('CartPole-v0') # 构建实验环境 +env.reset() # 重置一个 episode +for _ in range(1000): + env.render() # 用于显示渲染的场景 + action = env.action_space.sample() + env.step(action) # 用于提交动作,括号内是具体的动作 +env.close() +``` +当你执行这段代码时,可能会很失望,因为机器人就像个醉汉,完全无视那根本该立起来的杆子,驾驶着小车朝某个方向一通跑,直到不见踪影。别着急,我们还没开始训练机器人呢。 + +Gym中的小游戏,大部分都可以用一个普通的实数或者向量来充当动作。打印 `env.action_space.sample()` 的返回值,能看到输出为1或者0。 + +`env.action_space.sample()`的含义是,在该游戏的所有动作空间里随机选择一个作为输出。在这个例子中,意思就是,动作只有两个,一个是 0,另一个是 1,一左一右。 + +`env.step()`这个方法的作用不止于此,它还有四个返回值,分别是`observation`、`reward`、`done`、`info`。 + +* `observation(object)`是状态信息,是在游戏中观测到的屏幕像素值或者盘面状态描述信息。 +* `reward(float)`是奖励值,即 Action 提交以后能够获得的奖励值。这个奖励值因游戏的不同而不同,但总体原则是,对完成游戏有帮助的动作会获得比较高的奖励值。 +* `done(boolean)`表示游戏是否已经完成。如果完成了,就需要重置游戏并开始一个新的Episode。 +* `info(dict)`是一些比较原始的用于诊断和调试的信息,或许对训练有帮助。不过,OpenAI团队在评价你提交的机器人时,是不允许使用这些信息的。 + +在每个训练中都要使用的返回值有observation、reward、done。但是,observation的结构会由于游戏的不同而发生变化。以CartPole-v0小游戏为例,我们修改下代码: + +```python +import gym # 导入Gym的Python接口环境包 +env = gym.make('CartPole-v0') # 构建实验环境 +env.reset() # 重置一个 episode +for _ in range(1000): + env.render() # 用于显示渲染的场景 + action = env.action_space.sample() + observation, reward, done, info = env.step(action) + print(observation) +env.close() +``` + +输出: + +``` +[ 0.01653398 0.19114579 0.02013859 -0.28050058] +[ 0.0203569 -0.00425755 0.01452858 0.01846535] +[ 0.02027175 -0.19958481 0.01489789 0.31569658] +...... +``` + +从输出可以看出这是一个四维的Observation。在其他游戏中会有维度很多的情况。 + +`env.step()`完成了一个完整的 $S \to A \to R \to S'$ 过程。我们只要不断观测这样的过程,并让机器在其中用相应的算法完成训练,就能得到一个高质量的强化学习模型。 + +## References + +* [百面深度学习](https://book.douban.com/subject/35043939/) +* [白话强化学习与PyTorch](https://book.douban.com/subject/34809676/) -## To-do -* Gym 安装过程 -* RL例子 diff --git a/docs/chapter1/img/1.35.png b/docs/chapter1/img/1.35.png index e6bd0f1a3779c48a774c53f6e72887bd6c1d1c66..3b3d8483a3d92864cf1bce4124a2d6519fbf5a38 100644 GIT binary patch literal 64598 zcmeFZg;!hAw>1iBC;^IFahKv1DDG0+S_tm$?(R;D6}Qqtad&qp?(SB+XbJL8dhdPT z@4hkKfABF*Motb%#@hR=z4lyl&6Nlx1u1k?VpIeK1a#>S;wlIT&wz7~GvqmNrOIrY z7I;E*QjroxC>td`0A9G8K&4IPGa|i+$kr)99{ubaNh)DAPo=YM!B0T$d9fW`o 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zaJ2d3YH2@LxXAZwnG?K{`5N7RWqr>t(UYX-hhCmjz)tso)nmM_E+m*b5mfu=h;w^j zDaKQ#8m=<;QV5JBUx-qSj&$xRI9G)a+jDrUQVpK_?{nX4m}n}pMUUGMiF zxB;{~9(qH;pFZN>QZ?;uxm$zoXcFq~PSuclP~@OrpOW9ER{mYWqQo+-~EjJlHjwVTfB+ve&8%*1u>jV& zq`9NMXqCHn-q)x82oHgZ|DM?*Tr3-4de0+qa{U3Zo){X+F^RqAzv?&;c@h77Qj`%R zFJ(1KVGe_H*JH~DRKqDWKh}uIYU%P(U@~tTRsEs;SvtYDmu=@~nA6tOcna>zyg#7& z*>LR4`y~*^(bQd0=Yf9>0a<-8k7}x(m$Idf}>hTH`bMlK8 zdWEBmqLZd8R2&hL);+HBw>I5%nEq=g;R<1_IGoqhG7s#tD_x^qw;H9Lk4z_60u#(+ z7Nf)%ZRJp%h(`ADoq7DEA+EIg*IMJ{P6pgepKpGRybEblFTX_^$tvw*>B3fzdco-G zs@aeM@*zAQp`fRBXg4mio6R{+vAa0Dnx|7vb)sO=loV0L_VgJ2)>Te`D5AsXhwVOS z(xrtY3jKjqZEJVOCx(UfVVW*5a=D?|B{II%u%BQvzolDV@Ao#+JyfXE)L=08-#OO% zVP 事实上,Q-learning 算法被提出的时间更早,Sarsa 算法是 Q-learning 算法的改进。 -但是 Q-learning 在更新 Q 表格的时候,它用到这个的 Q 值 $Q(S',a')$ 对应的那个 action ,它不一定是下一个 step 会执行的实际的 action,因为你下一个实际会执行的那个 action 可能会探索。Q-learning 默认的 action 不是通过 behavior policy 来选取的,它是默认 A' 为最优策略选的动作,所以 Q-learning 在学习的时候,不需要传入A',即 $a_{t+1}$ 的值。 ![](img/2.19.png) @@ -159,7 +164,52 @@ Sarsa 实际上都是用自己的策略产生了 S,A,R,S',A' 这一条轨迹。 然后Q-learning 的这个逐步的一个拆解的话,跟Sarsa 唯一一点不一样就是我并不需要提前知道我 $A_2$ ,我就能更新 $Q(S_1,A_1)$ 。在训练一个 episode 这个流程图当中,Q-leanring 在 learn 之前它也不需要去拿到 next action A',它只需要前面四个 $(S,A,R,S')$也就可以了。这一点就是跟 Sarsa 有一个很明显的区别。 +初始状态为$s$并进行动作 $a$,然后执行策略 $\pi$ 得到的期望总回报,称为`状态-动作值函数(State-Action Value Function)`: +$$ +Q^{\pi}(s, a)=\mathbb{E}_{s^{\prime} \sim p\left(s^{\prime} | s, a\right)}\left[r\left(s, a, s^{\prime}\right)+\gamma V^{\pi}\left(s^{\prime}\right)\right] \qquad(1) +$$ +状态-动作值函数也经常称为`Q 函数(Q-Function)`。 + +状态值函数$V^{\pi}(s)$ 是 Q-Function $Q^{\pi}(s,a)$ 关于动作 $a$ 的期望,即 +$$ +V^{\pi}(s)=\mathbb{E}_{a \sim \pi(a | s)}\left[Q^{\pi}(s, a)\right] \qquad(2) +$$ + +结合公式(1) 和公式(2),Q-Function可以写为 +$$ +Q^{\pi}(s, a)=\mathbb{E}_{s^{\prime} \sim p\left(s^{\prime} | s, a\right)}\left[r\left(s, a, s^{\prime}\right)+\gamma \mathbb{E}_{a^{\prime} \sim \pi\left(a^{\prime} | s^{\prime}\right)}\left[Q^{\pi}\left(s^{\prime}, a^{\prime}\right)\right]\right] +$$ +这是关于 Q-Function 的 Bellman Equation。 + +### Bellman Equation + +记策略 $\pi $ 的动作价值函数为 $Q^{\pi}(s_t,a_t)$,它表示在状态 $s_t$ 下,执行动作 $a_t$ 会带来的累积奖励 $G_t$ 的期望,具体公式为: +$$ +\begin{aligned} Q ^ { \pi } \left( s _ { t } , a _ { t } \right) & = \mathbb { E } \left[ G _ { t } \mid s _ { t } , a _ { t } \right] \\ & = \mathbb { E } \left[ r _ { t } + \gamma r _ { t + 1 } + \gamma ^ { 2 } r _ { t + 2 } + \cdots \mid s _ { t } , a _ { t } \right] \\ & = \mathbb { E } \left[ r _ { t } + \gamma \left( r _ { t + 1 } + \gamma r _ { t + 2 } + \cdots \right) \mid s _ { t } , a _ { t } \right] \\ & = \mathbb { E } \left[ r _ { t } + \gamma Q ^ { \pi } \left( s _ { t + 1 } , a _ { t + 1 } \right) \mid s _ { t } , a _ { t } \right] \end{aligned} +$$ +上式是马尔可夫决策过程中 Bellman 方程的基本形式。累积奖励 $G_t$ 的计算,不仅考虑当下 $t$ 时刻的动作 $a_t$ 的奖励 $r_t$,还会累积计算对之后決策带来的影响(公式中的 $\gamma$ 是后续奖励的衰减因子)。从上式可以看出,当前状态的动作价值 $Q^{\pi}(s_t,a_t)$ ,与当前动作的奖励 $r_t$ 以及下一状态的动作价值 $Q^{\pi}(s_{t+1},a_{t+1})$ 有关,因此,动作价值函数的计算可以通过动态规划算法来实现。 + +从另一方面考虑,在计算 $t$ 时刻的动作价值 $Q^{\pi}(s_t,a_t)$ 时,需要知道在 $t$、$t+1$、$t+2 \cdots \cdots$ 时刻的奖励,这样就不仅需要知道某一状态的所有可能出现的后续状态以及对应的奖励值,还要进行全宽度的回溯来更新状态的价值。这种方法无法在状态转移函数未知或者大规模问题中使用。因此,Q- learning 采用了浅层的时序差分采样学习,在计算累积奖励时,基于当前策略 $\pi$ 预测接下来发生的 $n$ 步动作($n$ 可以取 1 到 $+\infty$)并计算其奖励值。 + +具体来说,假设在状态 $s_t$ 下选择了动作 $a_t$,并得到了奖励 $r_t$ ,此时状态转移到 $s_{t+1}$,如果在此状态下根据同样的策略选择了动作 $a_{t+1}$ ,则 $Q^{\pi}(s_t,a_t)$ 可以表示为 +$$ +Q^{\pi}\left(s_{t}, a_{t}\right)=\mathbb{E}_{s_{t+1}, a_{t+1}}\left[r_{t}+\gamma Q^{\pi}\left(s_{t+1}, a_{t+1}\right) \mid s_{t}, a_{t}\right] +$$ + +Q-learning 算法在使用过程中,可以根据获得的累积奖励来选择策略,累积奖励的期望值越高,价值也就越大,智能体越倾向于选择这个动作。因此,最优策略 $\pi^*$ 对应的动作价值函数 $Q^*(s_t,a_t)$ 满足如下关系式: + +$$ +Q^{*}\left(s_{t}, a_{t}\right)=\max _{\pi} Q^{\pi}\left(s_{t}, a_{t}\right)=\mathbb{E}_{s_{t+1}}\left[r_{t}+\gamma \max _{a_{t+1}} Q\left(s_{t+1}, a_{t+1}\right) \mid s_{t}, a_{t}\right] +$$ + +Q-learning 算法在学习过程中会不断地更新 Q 值,但它并没有直接采用上式中的项进行更新,而是采用类似于梯度下降法的更新方式,即状态 $s_t$ 下的动作价值 $Q^*(s_t,a_t)$ 会朝着状态 $s_{t+1}$ 下的动作价值 $r_{t}+\gamma \max _{a_{t+1}} Q^{*}\left(s_{t+1}, a_{t+1}\right)$ 做一定比例的更新: +$$ +\begin{aligned} +Q^{*}\left(s_{t}, a_{t}\right) \leftarrow Q^{*}\left(s_{t}, a_{t}\right)+\alpha\left(r_{t}+\gamma \max _{a_{t+1}} Q^{*}\left(s_{t+1}, a_{t+1}\right)-Q^{*}\left(s_{t}, a_{t}\right)\right) +\end{aligned} +$$ +其中 $\alpha$ 是更新比例(学习速率)。这种渐进式的更新方式,可以减少策略估计造成的影响,并且最终会收敛至最优策略。 ![](img/2.20.png) @@ -167,14 +217,16 @@ Sarsa 实际上都是用自己的策略产生了 S,A,R,S',A' 这一条轨迹。 Sarsa 就是一个典型的 on-policy 策略,它只用一个 $\pi$ ,为了兼顾探索和利用,所以它训练的时候会显得有一点点胆小怕事。它在解决悬崖问题的时候,会尽可能地离悬崖边上远远的,确保说哪怕自己不小心探索了一点了,也还是在安全区域内不不至于跳进悬崖。Q-leanring 是一个比较典型的 off-policy 的策略,它有目标策略 target policy,一般用 $\pi$ 来表示。然后还有行为策略 behavior policy,用 $\mu$ 来表示。它分离了目标策略跟行为策略。Q-learning 就可以大胆的用 behavior policy 去探索得到的经验轨迹来去优化我的目标策略。这样子我更有可能去探索到最优的策略。 - - ![](img/2.21.png) 总结如上图所示。 +## References + +* [百面深度学习](https://book.douban.com/subject/35043939/) + diff --git a/docs/chapter5/chapter5.md b/docs/chapter5/chapter5.md index 9d7f58a..468fab3 100644 --- a/docs/chapter5/chapter5.md +++ b/docs/chapter5/chapter5.md @@ -25,7 +25,7 @@ Q-learning 是 `value-based` 的方法。在 value based 的方法里面,我 ![](img/5.3.png) -第二个方法是`Temporal-difference(时序差分)` 的方法, `即 TD based ` 的方法。在 MC based 的方法中,每次我们都要算 accumulated reward,也就是从某一个 state $s_a$ 一直玩到游戏结束的时候,得到的所有reward 的总和。所以你要 apply MC based 的 approach,你必须至少把这个游戏玩到结束。但有些游戏非常的长,你要玩到游戏结束才能够 update network,你可能根本收集不到太多的资料,花的时间太长了。所以我们会采用 TD based 的方法。TD based 的方法不需要把游戏玩到底,只要在游戏的某一个情况,某一个 state $s_t$ 的时候,采取 action $a_t$ 得到 reward $r_t$ ,跳到 state $s_{t+1}$,就可以 apply TD 的方法。 +第二个方法是`Temporal-difference(时序差分)` 的方法, `即 TD based ` 的方法。在 MC based 的方法中,每次我们都要算 accumulated reward,也就是从某一个 state $s_a$ 一直玩到游戏结束的时候,得到的所有 reward 的总和。所以你要 apply MC based 的 approach,你必须至少把这个游戏玩到结束。但有些游戏非常的长,你要玩到游戏结束才能够 update network,你可能根本收集不到太多的资料,花的时间太长了。所以我们会采用 TD based 的方法。TD based 的方法不需要把游戏玩到底,只要在游戏的某一个情况,某一个 state $s_t$ 的时候,采取 action $a_t$ 得到 reward $r_t$ ,跳到 state $s_{t+1}$,就可以 apply TD 的方法。 怎么 apply TD 的方法呢?这边是基于以下这个式子: $$ @@ -77,6 +77,18 @@ $$ 所以不同的方法考虑了不同的假设,运算结果不同。 +举个例子来解释时序差分强化学习和蒙特卡洛强化学习的区别, + +* 时序差分强化学习是指在不清楚马尔可夫状态转移概率的情况下,以采样的方式得到不完整的状态序列,估计某状态在该状态序列完整后可能得到的收益,并通过不断地采样持续更新价值。 +* 蒙特卡洛强化学习则需要经历完整的状态序列后,再来更新状态的真实价值。 + +例如,你想获得开车去公司的时间,每天上班开车的经历就是一次采样。假设今天在路口 A 遇到了堵车, + +* 时序差分强化学习会在路口 A 就开始更新预计到达路口 B、路口 C $\cdots \cdots$, 以及到达公司的时间; +* 而蒙特卡洛强化学习并不会立即更新时间,而是在到达公司后,再修改到达每个路口和公司的时间。 + +时序差分强化学习能够在知道结果之前就开始学习,相比蒙特卡洛强化学习,其更快速、灵活。 + ![](img/5.7.png) 还有另外一种critic,这种critic 叫做 `Q-function`。它又叫做`state-action value function`。