From 339b097cca65d4a59c388e8dc9bb625b7f69da13 Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Sun, 18 Oct 2020 22:18:15 +0800 Subject: [PATCH] fix some typos and add explanation for prediction and control --- docs/chapter2/chapter2.md | 41 ++++++++++++++++++++--- docs/chapter2/img/control_example.png | Bin 0 -> 219817 bytes docs/chapter2/img/prediction_example.png | Bin 0 -> 189113 bytes 3 files changed, 36 insertions(+), 5 deletions(-) create mode 100644 docs/chapter2/img/control_example.png create mode 100644 docs/chapter2/img/prediction_example.png diff --git a/docs/chapter2/chapter2.md b/docs/chapter2/chapter2.md index f6d1f2d..d2438b4 100644 --- a/docs/chapter2/chapter2.md +++ b/docs/chapter2/chapter2.md @@ -140,7 +140,7 @@ $$ 这里迭代的方法就有几种, -* 比如说我们可以通过动态规划的方法, +* 我们可以通过动态规划的方法, * 也可以通过蒙特卡罗的办法,就通过采样的办法去计算它, * 也可以通过 Temporal-Difference Learning 的办法。这个 `Temporal-Difference Learning` 叫 `TD Leanring`,它是动态规划和蒙特卡罗的一个结合。 @@ -152,6 +152,8 @@ $$ ![](img/2.17.png)**我们也可以用这个动态规划的办法**,一直去迭代它的 Bellman Equation,让它最后收敛,我们就可以得到它的一个状态。所以在这里算法二就是一个迭代的算法,通过 bootstrapping(拔靴自助) 的办法,然后去不停地迭代这个 Bellman Equation。当这个最后更新的状态跟你上一个状态变化并不大的时候,更新就可以停止,我们就可以输出最新的 $V'(s)$ 作为它当前的状态。所以这里就是把 Bellman Equation 变成一个 Bellman Update,这样就可以得到它的一个价值。 +>动态规划的方法基于后继状态值的估计来更新状态值的估计(算法二中的第 3 行用 $V'$ 来更新 $V$ )。也就是说,它们根据其他估算值来更新估算值。我们称这种基本思想为 bootstrapping。 + ## Markov Decision Process(MDP) ![](img/2.18.png) @@ -160,7 +162,7 @@ $$ ![](img/2.19.png) -Policy 定义了在某一个状态应该采取什么样的行为。当我们知道当前状态过后,可以带入这个 policy function,那我们会得到一个概率,概率就代表了在所有可能的行为里面怎样去采取行动。就可能有 0.7 的概率往左走,有 0.3 的概率往右走,这样是一个概率的表示。另外这个策略也可能是确定的,它有可能是直接输出一个值,或者就直接告诉你当前应该采取什么样的行为,而不是一个行为的概率。然后这里我们有一个假设,就是这个概率函数应该是静态的(stationary),不同时间点,你采取的行为其实都是对这个 policy function 进行采样。 +**Policy 定义了在某一个状态应该采取什么样的行为。**当我们知道当前状态过后,可以带入这个 policy function,那我们会得到一个概率,概率就代表了在所有可能的行为里面怎样去采取行动。就可能有 0.7 的概率往左走,有 0.3 的概率往右走,这样是一个概率的表示。另外这个策略也可能是确定的,它有可能是直接输出一个值,或者就直接告诉你当前应该采取什么样的行为,而不是一个行为的概率。然后这里我们有一个假设,就是这个概率函数应该是静态的(stationary),不同时间点,你采取的行为其实都是对这个 policy function 进行采样。 ![](img/2.20.png) @@ -227,15 +229,37 @@ MDP,你其实可以把它想象成一个摆渡的人在这个船上面,她 $$ v_{t}^{\pi}(s)=\sum_{a} P(\pi(s)=a)\left(r(s, a)+\gamma \sum_{s^{\prime} \in S} P\left(s^{\prime} \mid s, a\right) v_{t-1}^{\pi}\left(s^{\prime}\right)\right) $$ -然后就会得到它的状态。另外一个练习的例子,就是说我们现在采取的 policy 在每个状态,我们有 0.5 的概率往左走,有 0.5 的概率往右走,那么放到这个状态里面去如何计算。其实也是把这个 Bellman Expectation Equation 拿出来,然后进行迭代就可以算出来了,就当我们开始的时候,我们可以初始化。初始化这个不同的 $v(s')$ 都会有一个值,那么放到这个里面去迭代,最后它的 $v$ ,然后就会算出来。 +然后就会得到它的状态。 + +另外一个练习的例子,就是说我们现在采取的 policy 在每个状态,我们有 0.5 的概率往左走,有 0.5 的概率往右走,那么放到这个状态里面去如何计算。其实也是把这个 Bellman Expectation Equation 拿出来,然后进行迭代就可以算出来了,就当我们开始的时候,我们可以初始化。初始化这个不同的 $v(s')$ 都会有一个值,那么放到这个里面去迭代,最后它的 $v$ ,然后就会算出来。 ![](img/2.31.png) MDP 的 `prediction` 和 `control` 是 MDP 里面的核心问题。 -* Prediction 是说给定一个 MDP 以及一个 policy $\pi$ ,去计算它的 value function,就等于每个状态它的价值函数是多少。 -* Control 是说我们去寻找一个最佳的策略,它的 input 就是 MDP,输出是通过去寻找它的最佳策略,然后同时输出它的最佳价值函数(optimal value function)以及它的这个最佳策略(optimal policy)。 +* **Prediction 是说给定一个 MDP 以及一个 policy $\pi$ ,去计算它的 value function,就对于每个状态,它的价值函数是多少。** +* **Control 是说我们去寻找一个最佳的策略:** + * **它的 input 就是 MDP,** + * **输出是通过去寻找它的最佳策略,然后同时输出它的最佳价值函数(optimal value function)以及它的最佳策略(optimal policy)。** * 在 MDP 里面,prediction 和 control 都可以通过这个动态规划去解决。 +* 要强调的是,这两者的区别就在于, + * 预测问题是**给定一个 policy**,我们要确定它的 value function 是多少。 + * 而控制问题是在**没有 policy 的前提下**,我们要确定最优的 value function 以及对应的决策方案。 +* **实际上,这两者是递进的关系,在强化学习中,我们通过解决预测问题,进而解决控制问题。** + +![](img/prediction_example.png) + +**举一个例子来说明 prediction 与 control 的区别。** + +首先是**预测问题**:在上图的方格中,我们规定从 A $\to$ A' 可以得到 +10 的奖励,从 B $\to$ B' 可以得到 +5 的奖励,其它步骤的奖励为 -1。现在,我们给定一个 policy:在任何状态中,它的行为模式都是随机的,也就是上下左右的概率各 25%。那么,预测问题要做的就是,在这种决策模式下,我们的 value function 是什么,也就是如上图 b 所示。 + +![](img/control_example.png) + + + +接着是**控制问题**:在控制问题中,问题背景与预测问题相同,唯一的区别就是:不再限制 policy。也就是说行为模式是未知的,我们要自己确定。所以我们通过解决控制问题,求得每一个状态的最优的 value function(如上图 b 所示),也得到了最优的 policy(如上图 c 所示)。 + +简单总结下,控制问题要做的问题就是,给定同样的条件,在所有可能的策略下最优的价值函数是什么?最优策略是什么? ![](img/2.32.png) @@ -419,3 +443,10 @@ $$ * 如果采取的算法是 policy iteration,那这里用的是 Bellman Expectation Equation 。把它分成两步,先上它的这个价值函数,再去优化它的策略,然后不停迭代。这里用到的只是 Bellman Expectation Equation。 * 如果采取的算法是 value iteration,那这里用到的 Bellman Equation 就是 Bellman Optimality Equation,通过 arg max 这个过程,不停地去 arg max 它,最后它就会达到最优的状态。 +## References + +* [强化学习基础 David Silver 笔记](https://zhuanlan.zhihu.com/c_135909947) +* [Reinforcement Learning: An Introduction (second edition)](https://book.douban.com/subject/30323890/) +* [David Silver 强化学习公开课中文讲解及实践](https://zhuanlan.zhihu.com/reinforce) +* [UCL Course on RL(David Silver)](https://www.davidsilver.uk/teaching/) + diff --git a/docs/chapter2/img/control_example.png b/docs/chapter2/img/control_example.png new file mode 100644 index 0000000000000000000000000000000000000000..edf9037112b848d8adbe60f5af7313c351989485 GIT binary patch literal 219817 zcmeFYg$z(L_PPl0`wm zNPhSLdFHsni5mq4ht^J7T3ta}npWM-+1k$03I#OMQ9y2kSfO2XZb?A3sK=;L*mEi=eZ3^`@E@4QB?+QeD00+lMb-2Jr@? zW8sx&UQ^9ihQY&&Hk*qTp1~;cGL90G*w4^u)6*Caygkjr!^3!8J5j~rWm3@yE!;IP 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