From 2c72bbf1d88823e6babf4234348054906bfbcfd7 Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Tue, 3 Nov 2020 16:42:38 +0800 Subject: [PATCH] improve explanation of model-free prediction and control --- docs/chapter3/chapter3.md | 379 +++++++++++++++++++-- docs/chapter3/img/3.15.png | Bin 277257 -> 532990 bytes docs/chapter3/img/3.16.png | Bin 532990 -> 277257 bytes docs/chapter3/img/MC_1.png | Bin 0 -> 310945 bytes docs/chapter3/img/MC_2.png | Bin 0 -> 192082 bytes docs/chapter3/img/MC_3.png | Bin 0 -> 221687 bytes docs/chapter3/img/MC_4.png | Bin 0 -> 242222 bytes docs/chapter3/img/MC_5.png | Bin 0 -> 239539 bytes docs/chapter3/img/MC_6.png | Bin 0 -> 195994 bytes docs/chapter3/img/TD_1.png | Bin 0 -> 156652 bytes docs/chapter3/img/TD_2.png | Bin 0 -> 245139 bytes docs/chapter3/img/TD_3.png | Bin 0 -> 305677 bytes docs/chapter3/img/TD_4.png | Bin 0 -> 249791 bytes docs/chapter3/img/TD_5.png | Bin 0 -> 207867 bytes docs/chapter3/img/TD_6.png | Bin 0 -> 200387 bytes 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docs/chapter3/img/model_free_1.png create mode 100644 docs/chapter3/img/model_free_2.png create mode 100644 docs/chapter3/img/model_free_control_1.png create mode 100644 docs/chapter3/img/model_free_control_2.png create mode 100644 docs/chapter3/img/model_free_control_3.png create mode 100644 docs/chapter3/img/model_free_control_4.png create mode 100644 docs/chapter3/img/model_free_control_5.png create mode 100644 docs/chapter3/img/model_free_control_6.png create mode 100644 docs/chapter3/img/model_free_control_7.png create mode 100644 docs/chapter3/img/model_free_control_8.png create mode 100644 docs/chapter3/img/model_free_control_9.png create mode 100644 docs/chapter3/img/n-step_sarsa.png diff --git a/docs/chapter3/chapter3.md b/docs/chapter3/chapter3.md index 893d36c..c015fee 100644 --- a/docs/chapter3/chapter3.md +++ b/docs/chapter3/chapter3.md @@ -1,6 +1,6 @@ # Tabular Methods -这节课我们通过最简单的`表格型的方法(tabular methods)`来讲解如何使用 value-based 方法去求解强化学习。 +本章我们通过最简单的`表格型的方法(tabular methods)`来讲解如何使用 value-based 方法去求解强化学习。 ## MDP @@ -14,9 +14,10 @@ 这样子的一个状态转移概率是具有`马尔可夫性质`的(系统下一时刻的状态仅由当前时刻的状态决定,不依赖于以往任何状态)。因为这个状态转移概率,它是下一时刻的状态是取决于当前的状态,它和之前的 $s_{t-1}$ 和 $s_{t-2}$ 都没有什么关系。然后再加上这个过程也取决于智能体跟环境交互的这个 $a_t$ ,所以有一个决策的一个过程在里面。我们就称这样的一个过程为`马尔可夫决策过程(Markov Decision Process, MDP)`。 - MDP 就是序列决策这样一个经典的表达方式。MDP 也是强化学习里面一个非常基本的学习框架。状态、动作、状态转移概率和奖励 $(S,A,P,R)$,这四个合集就构成了强化学习 MDP 的四元组,后面也可能会再加个衰减因子构成五元组。 +### Model-based + ![](img/3.2.png) @@ -25,25 +26,43 @@ MDP 就是序列决策这样一个经典的表达方式。MDP 也是强化学习 如上图所示,我们把这些可能的动作和可能的状态转移的关系画成一个树状图。它们之间的关系就是从 $s_t$ 到 $a_t$ ,再到 $s_{t+1}$ ,再到 $a_{t+1}$,再到 $s_{t+2}$ 这样子的一个过程。 -我们去跟环境交互,只能走完整的一条通路。这里面产生了一系列的一个决策的过程,就是我们跟环境交互产生了一个经验。我们会使用 `P 函数(probability function)`和 `R 函数(reward function)`来去描述环境。P 函数就是状态转移的概率,P 函数实际上反映的是环境的一个随机性。 +我们去跟环境交互,只能走完整的一条通路。这里面产生了一系列的一个决策的过程,就是我们跟环境交互产生了一个经验。**我们会使用 `P 函数(probability function)`和 `R 函数(reward function)`来去描述环境。**P 函数就是状态转移的概率,P 函数实际上反映的是环境的一个随机性。 + +**当我们知道 P 函数和 R 函数时,我们就说这个 MDP 是已知的,可以通过 policy iteration 和 value iteration 来找最佳的策略。** 比如,在熊发怒的情况下,我如果选择装死,假设熊看到人装死就一定会走的话,我们就称在这里面的状态转移概率就是 100%。但如果说在熊发怒的情况下,我选择跑路而导致可能跑成功以及跑失败,出现这两种情况。那我们就可以用概率去表达一下说转移到其中一种情况的概率大概 10%,另外一种情况的概率大概是 90% 会跑失败。 **如果知道这些状态转移概率和奖励函数的话,我们就说这个环境是已知的,因为我们是用这两个函数去描述环境的。**如果是已知的话,我们其实可以用动态规划去计算说,如果要逃脱熊,那么能够逃脱熊概率最大的最优策略是什么。很多强化学习的经典算法都是 model-free 的,就是环境是未知的。 -![](img/3.3.png) -因为现实世界中人类第一次遇到熊之前,我们根本不知道能不能跑得过熊,所以刚刚那个 10%、90% 的概率也就是虚构出来的概率。熊到底在什么时候会往什么方向去转变的话,我们经常是不知道的。我们是处在一个未知的环境里的,也就是这一系列的决策的 P 函数和 R 函数是未知的,这就是 model-based 跟 model-free 的一个最大的区别。强化学习就是可以用来解决用完全未知的和随机的环境。 +### Model-free -强化学习要像人类一样去学习,人类学习的话就是一条路一条路地去尝试一下,先走一条路,看看结果到底是什么。多试几次,只要能活命的。我们可以慢慢地了解哪个状态会更好, +![](img/3.3.png) +因为现实世界中人类第一次遇到熊之前,我们根本不知道能不能跑得过熊,所以刚刚那个 10%、90% 的概率也就是虚构出来的概率。熊到底在什么时候会往什么方向去转变的话,我们经常是不知道的。 + +**我们是处在一个未知的环境里的,也就是这一系列的决策的 P 函数和 R 函数是未知的,这就是 model-based 跟 model-free 的一个最大的区别。** + +强化学习就是可以用来解决用完全未知的和随机的环境。强化学习要像人类一样去学习,人类学习的话就是一条路一条路地去尝试一下,先走一条路,看看结果到底是什么。多试几次,只要能活命的。我们可以慢慢地了解哪个状态会更好, * 我们用价值函数 $V(s)$ 来代表这个状态是好的还是坏的。 -* 然后用 Q 函数来判断说在什么状态下做什么动作能够拿到最大奖励,用 Q 函数来表示这个状态-动作值。 +* 用 Q 函数来判断说在什么状态下做什么动作能够拿到最大奖励,用 Q 函数来表示这个状态-动作值。 + +### Model-based vs. Model-free + +![](img/model_free_1.png) + +* Policy iteration 和 value iteration 都需要得到环境的转移和奖励函数,所以在这个过程中,agent 没有跟环境进行交互。 +* 在很多实际的问题中,MDP 的模型有可能是未知的,也有可能模型太大了,不能进行迭代的计算。比如 Atari 游戏、围棋、控制直升飞机、股票交易等问题,这些问题的状态转移太复杂了。 + +![](img/model_free_2.png) + +* 在这种情况下,我们使用 model-free 强化学习的方法来解。 +* Model-free 没有获取环境的状态转移和奖励函数,我们让 agent 跟环境进行交互,采集到很多的轨迹数据,agent 从轨迹中获取信息来改进策略,从而获得更多的奖励。 ## Q-table ![](img/3.4.png) -接下来就会介绍 Q 函数。在多次尝试和熊打交道之后,人类就可以对熊的不同的状态去做出判断,我们可以用状态动作价值来表达说在某个状态下,为什么动作 1 会比动作 2 好,因为动作 1 的价值比动作 2 要高,这个价值就叫 `Q 函数`。 +接下来介绍下 Q 函数。在多次尝试和熊打交道之后,人类就可以对熊的不同的状态去做出判断,我们可以用状态动作价值来表达说在某个状态下,为什么动作 1 会比动作 2 好,因为动作 1 的价值比动作 2 要高,这个价值就叫 `Q 函数`。 **如果 `Q 表格`是一张已经训练好的表格的话,那这一张表格就像是一本生活手册。**我们就知道在熊发怒的时候,装死的价值会高一点。在熊离开的时候,我们可能偷偷逃跑的会比较容易获救。 @@ -98,11 +117,101 @@ $$ **`强化`就是我们可以用下一个状态的价值来更新当前状态的价值,其实就是强化学习里面 bootstrapping 的概念。**在强化学习里面,你可以每走一步更新一下 Q 表格,然后用下一个状态的 Q 值来更新这个状态的 Q 值,这种单步更新的方法叫做`时序差分`。 -## Temporal Difference +## Model-free Prediction + +在没法获取 MDP 的模型情况下,我们可以通过以下两种方法来估计某个给定策略的价值: + +* Monte Carlo policy evaluation +* Temporal Difference(TD) learning + +### Monte-Carlo Policy Evaluation + +![](img/MC_1.png) + +* `蒙特卡罗(Monte-Carlo,MC)`方法是基于采样的方法,我们让 agent 跟环境进行交互,就会得到很多轨迹。每个轨迹都有对应的 return: + +$$ +G_{t}=R_{t+1}+\gamma R_{t+2}+\gamma^{2} R_{t+3}+\ldots +$$ + +* 我们把每个轨迹的 return 进行平均,就可以知道某一个策略下面对应状态的价值。 + +* MC 是用 `empirical mean` return 的方法来估计。 + +* MC 方法不需要 MDP 的转移函数和奖励函数,并且不需要像动态规划那样用 bootstrapping 的方法。 + +* MC 的局限性:只能用在有终止的 MDP 。 + +![](img/MC_2.png) + +* 上图是 MC 算法的概括。 +* 为了得到评估 $v(s)$,我们进行了如下的步骤: + * 在每个 episode 中,如果在时间步 t 状态 s 被访问了,那么 + * 状态 s 的访问数 $N(s)$ 增加 1, + * 状态 s 的总的回报 $S(s)$ 增加 $G_t$。 + * 状态 s 的价值可以通过 return 的平均来估计,即 $v(s)=S(s)/N(s)$。 + +* 根据大数定律,只要我们得到足够多的轨迹,就可以趋近这个策略对应的价值函数。 + +假设现在有样本 $x_1,x_2,\cdots$,我们可以把 empirical mean 转换成 `incremental mean` 的形式,如下式所示: +$$ +\begin{aligned} +\mu_{t} &=\frac{1}{t} \sum_{j=1}^{t} x_{j} \\ +&=\frac{1}{t}\left(x_{t}+\sum_{j=1}^{t-1} x_{j}\right) \\ +&=\frac{1}{t}\left(x_{t}+(t-1) \mu_{t-1}\right) \\ +&=\frac{1}{t}\left(x_{t}+t \mu_{t-1}-\mu_{t-1}\right) \\ +&=\mu_{t-1}+\frac{1}{t}\left(x_{t}-\mu_{t-1}\right) +\end{aligned} +$$ +通过这种转换,我们就可以把上一时刻的平均值跟现在时刻的平均值建立联系,即: +$$ +\mu_t = \mu_{t-1}+\frac{1}{t}(x_t-\mu_{t-1}) +$$ +其中: + +* $x_t- \mu_{t-1}$ 是残差 +* $\frac{1}{t}$ 类似于 learning rate + +当我们得到 $x_t$,就可以用上一时刻的值来更新现在的值。 + +![](img/MC_3.png) + +我们可以把 Monte-Carlo 更新的方法写成 incremental MC 的方法: + +* 我们采集数据,得到一个新的轨迹。 +* 对于这个轨迹,我们采用增量的方法进行更新,如下式所示: + +$$ +\begin{array}{l} +N\left(S_{t}\right) \leftarrow N\left(S_{t}\right)+1 \\ +v\left(S_{t}\right) \leftarrow v\left(S_{t}\right)+\frac{1}{N\left(S_{t}\right)}\left(G_{t}-v\left(S_{t}\right)\right) +\end{array} +$$ + +* 我们可以直接把 $\frac{1}{N(S_t)}$ 变成 $\alpha$ (学习率),$\alpha$ 代表着更新的速率有多快,我们可以进行设置。 + +![](img/MC_4.png) + +**我们再来看一下 DP 和 MC 方法的差异。** + +* 动态规划也是常用的估计价值函数的方法。在动态规划里面,我们使用了 bootstrapping 的思想。bootstrapping 的意思就是我们基于之前估计的量来估计一个量。 + +* DP 就是用 Bellman expectation backup,就是通过上一时刻的值 $v_{i-1}(s')$ 来更新当前时刻 $v_i(s)$ 这个值,不停迭代,最后可以收敛。Bellman expectation backup 就有两层加和,内部加和和外部加和,算了两次 expectation,得到了一个更新。 + +![](img/MC_5.png) + +MC 是通过 empirical mean return (实际得到的收益)来更新它,对应树上面蓝色的轨迹,我们得到是一个实际的轨迹,实际的轨迹上的状态已经是决定的,采取的行为都是决定的。MC 得到的是一条轨迹,这条轨迹表现出来就是这个蓝色的从起始到最后终止状态的轨迹。现在只是更新这个轨迹上的所有状态,跟这个轨迹没有关系的状态都没有更新。 + +![](img/MC_6.png) + +* MC 可以在不知道环境的情况下 work,而 DP 是 model-based。 +* MC 只需要更新一条轨迹的状态,而 DP 则是需要更新所有的状态。状态数量很多的时候(比如一百万个,两百万个),DP 这样去迭代的话,速度是非常慢的。这也是 sample-based 的方法 MC 相对于 DP 的优势。 + +### Temporal Difference ![](img/3.10.png) -**为了让大家更好地理解时序差分这种更新方法,这边给出它的物理意义。**我们先理解一下巴普洛夫的条件反射实验,这个实验讲的是小狗会对盆里面的食物无条件产生刺激,分泌唾液。一开始小狗对于铃声这种中性刺激是没有反应的,可是我们把这个铃声和食物结合起来,每次先给它响一下铃,再给它喂食物,多次重复之后,当铃声响起的时候,小狗也会开始流口水。盆里的肉可以认为是强化学习里面那个延迟的 reward,声音的刺激可以认为是有 reward 的那个状态之前的一个状态。多次重复实验之后,最后的这个 reward 会强化小狗对于这个声音的条件反射,它会让小狗知道这个声音代表着有食物,这个声音对于小狗来说也就有了价值,它听到这个声音也会流口水。 +**为了让大家更好地理解`时序差分(Temporal Difference,TD)`这种更新方法,这边给出它的物理意义。**我们先理解一下巴普洛夫的条件反射实验,这个实验讲的是小狗会对盆里面的食物无条件产生刺激,分泌唾液。一开始小狗对于铃声这种中性刺激是没有反应的,可是我们把这个铃声和食物结合起来,每次先给它响一下铃,再给它喂食物,多次重复之后,当铃声响起的时候,小狗也会开始流口水。盆里的肉可以认为是强化学习里面那个延迟的 reward,声音的刺激可以认为是有 reward 的那个状态之前的一个状态。多次重复实验之后,最后的这个 reward 会强化小狗对于这个声音的条件反射,它会让小狗知道这个声音代表着有食物,这个声音对于小狗来说也就有了价值,它听到这个声音也会流口水。 ![](img/3.11.png) @@ -119,46 +228,250 @@ $$ ![](img/3.13.png) * 我们先初始化一下,然后开始时序差分的更新过程。 -* 在训练的过程中,你会看到这个小黄球在不断地试错,在探索当中会先迅速地发现有 reward 的地方。最开始的时候,只是这些有 reward 的格子 才有价值。当不断地重复走这些路线的时候,这些有价值的格子可以去慢慢地影响它附近的格子的价值。 -* 反复训练之后,这些有 reward 的格子周围的格子的状态就会慢慢地被强化。强化就是当它收敛到最后一个最优的状态了,就是把这些价值最终收敛到一个最优的情况之后,那个小黄球就会自动地知道,就是我一直往价值高的地方走,就能够走到能够拿到 reward 的地方。 - ![](img/3.14.png) +* 在训练的过程中,你会看到这个小黄球在不断地试错,在探索当中会先迅速地发现有 reward 的地方。最开始的时候,只是这些有 reward 的格子才有价值。当不断地重复走这些路线的时候,这些有价值的格子可以去慢慢地影响它附近的格子的价值。 +* 反复训练之后,这些有 reward 的格子周围的格子的状态就会慢慢地被强化。强化就是当它收敛到最后一个最优的状态了,这些价值最终收敛到一个最优的情况之后,那个小黄球就会自动地知道,就是我一直往价值高的地方走,就能够走到能够拿到 reward 的地方。 -如上图所示,这种强化方式可以用一行公式来表示,这种更新的方式叫做`时序差分(Temporal Difference)`。这个公式就是说可以拿下一步的 Q 值 $Q(S_{t+_1},A_{t+1})$ 来更新我这一步的 Q 值 $Q(S_t,A_t)$ 。 +![](img/TD_1.png) -为了理解这个公式,如上图所示,我们先把 $R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right.)$ 当作是一个目标值,就是 $Q(S_t,A_t)$ 想要去逼近的一个目标值。$R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right.)$ 被称为 `TD target`,TD target 是带衰减的未来收益的总和。 +* TD 是介于 MC 和 DP 之间的方法。 -我们想要计算的就是 $Q(S_t,A_t)$ 。因为最开始 Q 值都是随机初始化或者是初始化为零,它需要不断地去逼近它理想中真实的 Q 值(TD target),$\delta = R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right)-Q\left(S_{t}, A_{t}\right)$ 被称为 `TD error`。 +* TD 是 model free 的,不需要 MDP 的转移矩阵和奖励函数。 +* TD 可以从不完整的 episode 中学习,结合了 bootstrapping 的思想。 + +![](img/TD_2.png) + +* 上图是 TD 算法的框架。 +* 目的:对于某个给定的策略,在线(online)地算出它的价值函数,即一步一步地(step-by-step)算。 +* 最简单的算法是 `TD(0)`,每往前走一步,就做一步 bootstrapping,用得到的 estimated return 来更新上一时刻的值 +* Estimated return $R_{t+1}+\gamma v(S_{t+1})$ 被称为 `TD target`,TD target 是带衰减的未来收益的总和。TD target 由两部分组成: + * 走了某一步后得到的实际奖励:$R_{t+1}$, + * 我们利用了 bootstrapping 的方法,通过之前的估计来估计 $v(S_{t+1})$ ,然后加了一个折扣系数,即 $\gamma v(S_{t+1})$。 +* `TD error` $\delta=R_{t+1}+\gamma v(S_{t+1})-v(S_t)$。 +* 可以类比于 Incremental Monte-Carlo 的方法,写出如下的更新方法: -我们用 $G_t$ 来表示未来收益总和(return),并且对它做一下数学变化: $$ -\begin{aligned} -G_{t} &=R_{t+1}+\gamma R_{t+2}+\gamma^{2} R_{t+3}+\gamma^{3} R_{t+4}+\cdots \\ -&=R_{t+1}+\gamma\left(R_{t+2}+\gamma R_{t+3}+\gamma^{2} R_{t+4}+\cdots\right) \\ -&=R_{t+1}+\gamma G_{t+1} -\end{aligned} +v\left(S_{t}\right) \leftarrow v\left(S_{t}\right)+\alpha\left(R_{t+1}+\gamma v\left(S_{t+1}\right)-v\left(S_{t}\right)\right) $$ -就可以知道 $G_t = R_{t+1}+ \gamma G_{t+1}$。 + +> 上式体现了强化这个概念。 + +* 我们对比下 MC 和 TD: + * 在 MC 里面 $G_{i,t}$ 是实际得到的值(可以看成 target),因为它已经把一条轨迹跑完了,可以算每个状态实际的 return。 + * TD 没有等轨迹结束,往前走了一步,就可以更新价值函数。 + +![](img/TD_3.png) + +* TD 只执行了一步,状态的值就更新。 + +* MC 全部走完了之后,到了终止状态之后,再更新它的值。 + +![](img/TD_4.png) + +* TD 可以 online learning,每走一步就可以更新,效率高。 +* MC 必须等游戏结束才可以学习。 + + + +* TD 可以从不完整序列上进行学习。 +* TD 只能从完整的序列上进行学习。 + + + +* TD 可以在连续的环境下(没有终止)进行学习。 +* MC 只能在有终止的情况下学习。 + + + +* TD 利用了马尔可夫性质,在马尔可夫环境下有更高的学习效率。 +* MC 没有假设环境具有马尔可夫性质,利用采样的价值来估计某一个状态的价值,在不是马尔可夫的环境下更加有效 + + + +**举个例子来解释 TD 和 MC 的区别,** + +* TD 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。 +* MC 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, + +* TD 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 MC 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。 + +**TD 能够在知道结果之前就开始学习,相比 MC,其更快速、灵活。** + +![](img/TD_5.png) + +* 我们可以把 TD 进行进一步的推广。之前是只往前走一步,即 one-step TD,TD(0)。 + +* 我们可以调整步数,变成 `n-step TD`。比如 `TD(2)`,即往前走两步,然后利用两步得到的 return,使用 bootstrapping 来更新状态的价值。 + +* 这样就可以通过 step 来调整这个算法需要多少的实际奖励和 bootstrapping。 + +![](img/TD_6.png) + +* 通过调整步数,可以进行一个 MC 和 TD 之间的 trade-off,如果 $n=\infty$, 即整个游戏结束过后,再进行更新,TD 就变成了 MC。 +* n-step 的 TD target 如下式所示: + +$$ +G_{t}^{n}=R_{t+1}+\gamma R_{t+2}+\ldots+\gamma^{n-1} R_{t+n}+\gamma^{n} v\left(S_{t+n}\right) +$$ + +* 得到 TD target 之后,我们用 incremental learning 的方法来更新状态的价值: + +$$ +v\left(S_{t}\right) \leftarrow v\left(S_{t}\right)+\alpha\left(G_{t}^{n}-v\left(S_{t}\right)\right) +$$ + +### Bootstrapping and Sampling for DP,MC and TD + +![](img/comparison_1.png) + +* Bootstrapping:更新时使用了估计: + * MC 没用 bootstrapping,因为它是根据实际的 return 来更新。 + * DP 用了 bootstrapping。 + * TD 用了 bootstrapping。 + +* Sampling:更新时通过采样得到一个期望: + * MC 是纯 sampling 的方法。 + * DP 没有用 sampling,它是直接用 Bellman expectation equation 来更新状态价值的。 + * TD 用了 sampling。TD target 由两部分组成,一部分是 sampling,一部分是 bootstrapping。 + +![](img/comparison_2.png) + +DP 是直接算 expectation,把它所有相关的状态都进行加和。 + +![](img/comparison_3.png) + +MC 在当前状态下,采一个支路,在一个path 上进行更新,更新这个 path 上的所有状态。 + +![](img/comparison_4.png) + +TD 是从当前状态开始,往前走了一步,关注的是非常局部的步骤。 + +![](img/comparison_5.png) + +* 如果 TD 需要更广度的 update,就变成了 DP(因为 DP 是把所有状态都考虑进去来进行更新)。 +* 如果 TD 需要更深度的 update,就变成了 MC。 +* 右下角是穷举的方法(exhaustive search),穷举的方法既需要很深度的信息,又需要很广度的信息。 + +## Model-free Control + +Q: 当我们不知道 MDP 模型情况下,如何优化价值函数,得到最佳的策略? + +A: 我们可以把 policy iteration 进行一个广义的推广,使它能够兼容 MC 和 TD 的方法,即 `Generalized Policy Iteration(GPI) with MC and TD`。 + +![](img/model_free_control_1.png) + +Policy iteration 由两个步骤组成: + +1. 根据给定的当前的 policy $\pi$ 来估计价值函数; +2. 得到估计的价值函数后,通过 greedy 的方法来改进它的算法。 + +这两个步骤是一个互相迭代的过程。 + +![](img/model_free_control_2.png) + +得到一个价值函数过后,我们并不知道它的奖励函数和状态转移,所以就没法估计它的 Q 函数。所以这里有一个问题:当我们不知道奖励函数和状态转移时,如何进行策略的优化。 + +![](img/model_free_control_3.png) + +针对上述情况,我们引入了广义的 policy iteration 的方法。 + +我们对 policy evaluation 部分进行修改:用 MC 的方法代替 DP 的方法去估计 Q 函数。 + +当得到 Q 函数后,就可以通过 greedy 的方法去改进它。 + +![](img/model_free_control_4.png) + +上图是用 MC 估计 Q 函数的算法。 + +* 假设每一个 episode 都有一个 `exploring start`,exploring start 保证所有的状态和动作都在无限步的执行后能被采样到,这样才能很好地去估计。 +* 算法通过 MC 的方法产生了很多的轨迹,每个轨迹都可以算出它的价值。然后,我们可以通过 average 的方法去估计 Q 函数。Q 函数可以看成一个 Q-table,通过采样的方法把表格的每个单元的值都填上,然后我们使用 policy improvement 来选取更好的策略。 +* 算法核心:如何用 MC 方法来填 Q-table。 + +![](img/model_free_control_5.png) + +为了确保 MC 方法能够有足够的探索,我们使用了 $\varepsilon$-greedy exploration。 + +$\varepsilon\text{-greedy}$ 的意思是说,我们有 $1-\varepsilon$ 的概率会按照 Q-function 来决定 action,通常 $\varepsilon$ 就设一个很小的值, $1-\varepsilon$ 可能是 90%,也就是 90% 的概率会按照 Q-function 来决定 action,但是你有 10% 的机率是随机的。通常在实现上 $\varepsilon$ 会随着时间递减。在最开始的时候。因为还不知道那个 action 是比较好的,所以你会花比较大的力气在做 exploration。接下来随着 training 的次数越来越多。已经比较确定说哪一个 Q 是比较好的。你就会减少你的 exploration,你会把 $\varepsilon$ 的值变小,主要根据 Q-function 来决定你的 action,比较少做 random,这是 $\varepsilon\text{-greedy}$。 + +![](img/model_free_control_6.png) + +当我们使用 MC 和 $\varepsilon$-greedy 探索这个形式的时候,我们可以确保价值函数是单调的,改进的。 + +![](img/model_free_control_7.png)上图是 MC with $\varepsilon$-greedy exploration 算法的伪代码。 + +![](img/model_free_control_8.png) + +与 MC 相比,TD 有如下几个优势: + +* 低方差。 +* 能够在线学习。 +* 能够从不完整的序列学习。 + +所以我们可以把 TD 也放到 control loop 里面去估计 Q-table,再采取这个 $\varepsilon$-greedy improvement。这样就可以在 episode 没结束的时候来更新已经采集到的状态价值。 + +### Sarsa: On-policy TD Control + +![](img/model_free_control_9.png) + +TD 是给定了一个策略,然后我们去估计它的价值函数。接着我们要考虑怎么用 TD 这个框架来估计 Q-function。 + +![](img/3.14.png) + +Sarsa 跟 TD Prediction 也是类似的,它是直接估计 Q-table。得到这个 Q-table 后,就可以更新这个策略。 + +如上图所示,我们使用时序差分的方式来进行更新,如下式所示: +$$ +Q\left(S_{t}, A_{t}\right) \leftarrow Q\left(S_{t}, A_{t}\right)+\alpha\left[R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right)-Q\left(S_{t}, A_{t}\right)\right] +$$ +这个公式就是说可以拿下一步的 Q 值 $Q(S_{t+_1},A_{t+1})$ 来更新我这一步的 Q 值 $Q(S_t,A_t)$ 。 + +为了理解这个公式,如上图所示,我们先把 $R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right.)$ 当作是一个目标值,就是 $Q(S_t,A_t)$ 想要去逼近的一个目标值。$R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right.)$ 就是 TD target。 + +我们想要计算的就是 $Q(S_t,A_t)$ 。因为最开始 Q 值都是随机初始化或者是初始化为零,它需要不断地去逼近它理想中真实的 Q 值(TD target),$R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right)-Q\left(S_{t}, A_{t}\right)$ 就是 TD error。 也就是说,我们拿 $Q(S_t,A_t)$ 来逼近 $G_t$,那 $Q(S_{t+1},A_{t+1})$ 其实就是近似 $G_{t+1}$。我就可以用 $Q(S_{t+1},A_{t+1})$ 近似 $G_{t+1}$,然后把 $R_{t+1}+Q(S_{t+1},A_{t+1})$ 当成目标值。 $Q(S_t,A_t)$ 就是要逼近这个目标值,我们用软更新的方式来逼近。软更新的方式就是每次我只更新一点点,$\alpha$ 类似于学习率。最终的话,Q 值都是可以慢慢地逼近到真实的 target 值。这样我们的更新公式只需要用到当前时刻的 $S_{t},A_t$,还有拿到的 $R_{t+1}, S_{t+1},A_{t+1}$ 。 -**该算法由于每次更新值函数需要知道当前的状态(state)、当前的动作(action)、奖励(reward)、下一步的状态(state)、下一步的动作(action),即 $(S_{t}, A_{t}, R_{t+1}, S_{t+1}, A_{t+1})$ 这几个值 ,由此得名 `Sarsa` 算法**。它走了一步之后,拿到了 $(S_{t}, A_{t}, R_{t+1}, S_{t+1}, A_{t+1})$ 之后,就可以做一次更新。 +**该算法由于每次更新值函数需要知道当前的状态(state)、当前的动作(action)、奖励(reward)、下一步的状态(state)、下一步的动作(action),即 $(S_{t}, A_{t}, R_{t+1}, S_{t+1}, A_{t+1})$ 这几个值 ,由此得名 `Sarsa` 算法**。它走了一步之后,拿到了 $(S_{t}, A_{t}, R_{t+1}, S_{t+1}, A_{t+1})$ 之后,就可以做一次更新。 ![](img/3.15.png) +我们直接看这个框框里面的更新公式, 和之前的公式是一样的。$S'$ 就是 $S_{t+1}$ 。我们就是拿下一步的 Q 值 $Q(S',A')$ 来更新这一步的 Q 值 $Q(S,A)$,不断地强化每一个 Q。 + +![](img/n-step_sarsa.png)Sarsa 属于单步更新法,也就是说每执行一个动作,就会更新一次价值和策略。如果不进行单步更新,而是采取 $n$ 步更新或者回合更新,即在执行 $n$ 步之后再来更新价值和策略,这样就得到了 `n 步 Sarsa(n-step Sarsa)`。 + +比如 2-step Sarsa,就是执行两步后再来更新 Q 的值。 + +具体来说,对于 Sarsa,在 $t$ 时刻其价值的计算公式为 +$$ +q_{t}=R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right) +$$ +而对于 $n$ 步 Sarsa,它的 $n$ 步 Q 收获为 +$$ +q_{t}^{(n)}=R_{t+1}+\gamma R_{t+2}+\ldots+\gamma^{n-1} R_{t+n}+\gamma^{n} Q\left(S_{t+n}, A_{t+n}\right) +$$ + +如果给 $q_t^{(n)}$ 加上衰减因子 $\lambda$ 并进行求和,即可得到 Sarsa($\lambda$) 的 Q 收获: +$$ +q_{t}^{\lambda}=(1-\lambda) \sum_{n=1}^{\infty} \lambda^{n-1} q_{t}^{(n)} +$$ +因此,$n$ 步 Sarsa($\lambda$)的更新策略可以表示为 +$$ +Q\left(S_{t}, A_{t}\right) \leftarrow Q\left(S_{t}, A_{t}\right)+\alpha\left(q_{t}^{\lambda}-Q\left(S_{t}, A_{t}\right)\right) +$$ +总的来说,Sarsa 和 Sarsa($\lambda$) 的差别主要体现在价值的更新上。 + +![](img/3.16.png) + 我们看看用代码去怎么去实现。了解单步更新的一个基本公式之后,代码实现就很简单了。右边是环境,左边是 agent 。我们每次跟环境交互一次之后呢,就可以 learn 一下,向环境输出 action,然后从环境当中拿到 state 和 reward。Agent 主要实现两个方法: * 一个就是根据 Q 表格去选择动作,输出 action。 * 另外一个就是拿到 $(S_{t}, A_{t}, R_{t+1}, S_{t+1}, A_{t+1})$ 这几个值去更新我们的 Q 表格。 -## Sarsa: On-policy TD Control - -![](img/3.16.png) - -我们直接看这个框框里面的更新公式, 和之前的公式是一模一样的。$S'$ 就是 $S_{t+1}$ 。我们就是拿下一步的 Q 值 $Q(S',A')$ 来更新这一步的 Q 值 $Q(S,A)$,不断地强化每一个 Q。 - -## Q-learning: Off-policy TD Control +### Q-learning: Off-policy TD Control ![](img/3.17.png) @@ -191,8 +504,6 @@ $$ $$ Behavior policy $\mu$ 可以是一个随机的 policy,但我们采取 $\varepsilon\text{-greedy}$,让 behavior policy 不至于是完全随机的,它是基于 Q-table 逐渐改进的。 -> $\varepsilon\text{-greedy}$ 的意思是说,我们有 $1-\varepsilon$ 的概率会按照 Q-function 来决定 action,通常 $\varepsilon$ 就设一个很小的值, $1-\varepsilon$ 可能是 90%,也就是 90% 的概率会按照 Q-function 来决定 action,但是你有 10% 的机率是随机的。通常在实现上 $\varepsilon$ 会随着时间递减。在最开始的时候。因为还不知道那个 action 是比较好的,所以你会花比较大的力气在做 exploration。接下来随着 training 的次数越来越多。已经比较确定说哪一个 Q 是比较好的。你就会减少你的 exploration,你会把 $\varepsilon$ 的值变小,主要根据 Q-function 来决定你的 action,比较少做 random,这是 $\varepsilon\text{-greedy}$。 - 我们可以构造 Q-learning target,Q-learning 的 next action 都是通过 arg max 操作来选出来的,于是我们可以代入 arg max 操作,可以得到下式: $$ \begin{aligned} @@ -245,10 +556,10 @@ Sarsa 是用自己的策略产生了 S,A,R,S',A' 这一条轨迹。然后拿着 ## References +* [Intro to Reinforcement Learning (强化学习纲要)](https://github.com/zhoubolei/introRL) * [Reinforcement Learning: An Introduction (second edition)](https://book.douban.com/subject/30323890/) * [百面深度学习](https://book.douban.com/subject/35043939/) * [神经网络与深度学习](https://nndl.github.io/) -* [Intro to Reinforcement Learning (强化学习纲要)](https://github.com/zhoubolei/introRL) * [机器学习](https://book.douban.com/subject/26708119//) diff --git a/docs/chapter3/img/3.15.png b/docs/chapter3/img/3.15.png index 03d2c768557e41666cc7614da8d60752ca052fa1..6ca2f3f56618eca3dc5f7d7390d5cc241971106a 100644 GIT binary patch literal 532990 zcmeFZcT`i`*EWg;L6o9Wqy;%1l#WW15~_mIYv@g+28f8XKtMo1MT$yMdXb*cLl0Fz zO6W--bm^fJLJQ?`zVaUY#=YPB&;9eq7(08j#!A-ObFVq)^UOKd{-C3!N=w5=LqS17 z`~2AxJqn5|3ltQWA5&jBYa#t`t~fhg@X%9LqA2R+SU&s5$IkeaS!$^~AEOaG(&A_Yab6UF8K)ko{>{Lf#^+40YJ{^$8p`i1}1`>ZYf z;{Wb_Wg-32|GvKb_@A$l-_OLKovyxmX5v9X!OZi|apAe%-r=&Se>Bq z=1@&sKd9oMnPd+J>@f0t)Kf>eA(O^XGBO_ayQ=|1K!=S*zIr4=UdM##i za;yNq7zr2rTcJPe*~Oz=gFLv{=g*D&$syDFH$FcS;t@LO38uJ6edq1} z+Ea=Pml+iPzdwS0P&AwCg?Rp7J6*V3@VCMPBOc?A|IN?c;Zt~!Fl^2F-;BwBm7&M* zYW#~E|CfoK{lqgGh8`YqW7Yrf+5a<6y4uCd9)`i$#{bPNd^zj(oEH1vjG6c4OdfI8 ze9ZsNE&NApe+cM5V*5Xo>_4mR4|V*HlKvm2?mxEe4}}T(k4yN&o+|vulm4kN{{bL> z*i(xC;I=R#ib!LXw=P} 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