From 4ab449e2ec836c6c8c6f81f1bee8cf40e7927f31 Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Tue, 27 Oct 2020 14:01:44 +0800 Subject: [PATCH] fix ch2 typos --- docs/chapter2/chapter2.md | 40 +++++++++++++++--- docs/chapter2/img/q_function_backup.png | Bin 0 -> 201309 bytes .../img/state_value_function_backup.png | Bin 0 -> 107074 bytes 3 files changed, 34 insertions(+), 6 deletions(-) create mode 100644 docs/chapter2/img/q_function_backup.png create mode 100644 docs/chapter2/img/state_value_function_backup.png diff --git a/docs/chapter2/chapter2.md b/docs/chapter2/chapter2.md index 874bc46..aabb53e 100644 --- a/docs/chapter2/chapter2.md +++ b/docs/chapter2/chapter2.md @@ -85,7 +85,7 @@ $$ 这里我们进一步定义一些概念。 -* `Horizon` 说明了同一个 episode 或者是整个一个轨迹的长度,它是由有限个步数决定的。 +* `Horizon` 是指一个 episode 的长度(每个 episode 最大的时间步数),它是由有限个步数决定的。 * `Return(回报)` 说的是把奖励进行折扣后所获得的收益。Return 可以定义为奖励的逐步叠加,如下式所示: @@ -366,7 +366,13 @@ $$ ![](img/2.25.png) -这里有一个概念叫 `Backup`。Backup 类似于 bootstrapping 之间这个迭代关系,就对于某一个状态,它的当前价值是跟它未来价值线性相关的。 +这里有一个概念叫 `backup`。Backup 类似于 bootstrapping 之间这个迭代关系,就对于某一个状态,它的当前价值是跟它的未来价值线性相关的。 + +我们把上面这样的图称为 `backup diagram(备份图)`,因为它们图示的关系构成了更新或备份操作的基础,而这些操作是强化学习方法的核心。这些操作将价值信息从一个状态(或状态-动作对)的后继状态(或状态-动作对)转移回它。 + +每一个空心圆圈代表一个状态,每一个实心圆圈代表一个状态-动作对。 + + $$ v^{\pi}(s)=\sum_{a \in A} \pi(a \mid s)\left(R(s, a)+\gamma \sum_{s^{\prime} \in S} P\left(s^{\prime} \mid s, a\right) v^{\pi}\left(s^{\prime}\right)\right) \tag{12} $$ @@ -375,11 +381,22 @@ $$ * 第一层加和就是这个叶子节点,往上走一层的话,我们就可以把未来的价值($s'$ 的价值) backup 到黑色的节点。 * 第二层加和是对 action 进行加和。得到黑色节点的价值过后,再往上 backup 一层,就会推到根节点的价值,即当前状态的价值。 -**所以 `Backup Diagram` 定义了未来下一时刻的状态价值跟上一时刻的状态价值之间的关联。** +![](img/state_value_function_backup.png) -> 我们把上面这样的图称为 backup diagram(备份图),因为它们图示的关系构成了更新或备份操作的基础,而这些操作是强化学习方法的核心。这些操作将价值信息从一个状态(或状态-动作对)的后继状态(或状态-动作对)转移回它。 -> -> 每一个空心圆圈代表一个状态,每一个实心圆圈代表一个状态-动作对。 +上图是状态-价值函数的计算分解图,上图 B 计算公式为 +$$ +v^{\pi}(s)=\sum_{a \in A} \pi(a \mid s) q^{\pi}(s, a) \tag{i} +$$ +上图 B 给出了状态-价值函数与 Q 函数之间的关系。上图 C 计算 Q 函数为 +$$ +q^{\pi}(s,a)=R(s, a)+\gamma \sum_{s^{\prime} \in S} P\left(s^{\prime} \mid s, a\right) v^{\pi}\left(s^{\prime}\right) \tag{ii} +$$ + +将式 (ii) 代入式 (i) 可得: +$$ +v^{\pi}(s)=\sum_{a \in A} \pi(a \mid s)\left(R(s, a)+\gamma \sum_{s^{\prime} \in S} P\left(s^{\prime} \mid s, a\right) v^{\pi}\left(s^{\prime}\right)\right) +$$ +**所以 backup diagram 定义了未来下一时刻的状态-价值函数跟上一时刻的状态-价值函数之间的关联。** ![](img/2.26.png) @@ -392,6 +409,16 @@ $$ * 第一层加和是先把这个叶子节点从黑色节点推到这个白色的节点,进了它的这个状态。 * 当我们到达某一个状态过后,再对这个白色节点进行一个加和,这样就把它重新推回到当前时刻的一个 Q 函数。 +![](img/q_function_backup.png) + +在上图 C 中, +$$ +v^{\pi}\left(s^{\prime}\right)=\sum_{a^{\prime} \in A} \pi\left(a^{\prime} \mid s^{\prime}\right) q^{\pi}\left(s^{\prime}, a^{\prime}\right) \tag{iii} +$$ +将式 (iii) 代入式 (ii) 可得到 Q 函数: +$$ +q^{\pi}(s, a)=R(s, a)+\gamma \sum_{s^{\prime} \in S} P\left(s^{\prime} \mid s, a\right) \sum_{a^{\prime} \in A} \pi\left(a^{\prime} \mid s^{\prime}\right) q^{\pi}\left(s^{\prime}, a^{\prime}\right) +$$ **所以这个等式就决定了未来 Q 函数跟当前 Q 函数之间的这个关联。** ### Policy Evaluation @@ -750,4 +777,5 @@ $$ * [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/) * [Derivation of Bellman’s Equation](https://jmichaux.github.io/_notebook/2018-10-14-bellman/) +* [深入浅出强化学习:原理入门](https://book.douban.com/subject/27624485//) diff --git a/docs/chapter2/img/q_function_backup.png b/docs/chapter2/img/q_function_backup.png new file mode 100644 index 0000000000000000000000000000000000000000..b8f488d1b0fdbdc9ea9f699c3c138b1287147df9 GIT binary patch literal 201309 zcmZs@byyo+*F6j&xCMvcuEo6+cXuo9Qrw;3?q1woiWEw5FYeGni@UoOK6+o@?|q-U z&-+KN$t1aE&YW3$@3q$1lSpMnDO4md5)>2^s*JR_DijoKHWU>>4^*sUR22Ap7$}teHx3C@5hl8F3MHPw1l#_!J8%x6}8uehsr%S7s%N3BXU?qN2md zz_eV@&H$-uk zK=S$n1V=)5Qh~{HUaEzBJImbo<<;8HUk9Dh)qYHFWd7i5PwybZTIbs`D^y+FVPiyx zA))ZE@bRlIB{xFA2fQBPe>OyT(CeGHX=2@dHX@)P%xGdU-ni4Xw-=;TaJJ#B=}=L9 zY%S5t`LS!#O(QmH<7I&P1Q8A3Mpbn5!Ku&w-b z?sHj`O@nS&D}h 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