From c65ffbb2d0671607f1cee7511985a24e0e325341 Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Fri, 19 Feb 2021 18:27:21 +0800 Subject: [PATCH] fix ch2 typos --- docs/chapter2/chapter2.md | 42 +++++++++++++++++++++++++++---------- docs/chapter2/img/2.13.png | Bin 282216 -> 187187 bytes docs/chapter2/img/2.25.png | Bin 154121 -> 104548 bytes 3 files changed, 31 insertions(+), 11 deletions(-) diff --git a/docs/chapter2/chapter2.md b/docs/chapter2/chapter2.md index fdf42d3..3eded60 100644 --- a/docs/chapter2/chapter2.md +++ b/docs/chapter2/chapter2.md @@ -207,13 +207,39 @@ $$ >Bellman Equation 就是当前状态与未来状态的迭代关系,表示当前状态的值函数可以通过下个状态的值函数来计算。Bellman Equation 因其提出者、动态规划创始人 Richard Bellman 而得名 ,也叫作“动态规划方程”。 +**Bellman Equation 定义了状态之间的迭代关系,如下式所示。** +$$ +V(s)=R(s)+\gamma \sum_{s^{\prime} \in S} P\left(s^{\prime} \mid s\right) V\left(s^{\prime}\right) +$$ ![](img/2.13.png) -**Bellman Equation 定义了状态之间的迭代关系。**假设有一个马尔可夫转移矩阵是右边这个样子。Bellman Equation 描述的就是当前状态到未来状态的一个转移。假设我们当前是在 $s_1$, 那么它只可能去到三个未来的状态:有 0.1 的概率留在它当前这个位置,有 0.2 的概率去到 $s_2$ 状态,有 0.7 的概率去到 $s_4$ 的状态,所以我们要把这个转移乘以它未来的状态的价值,再加上它的 immediate reward 就会得到它当前状态的价值。**所以 Bellman Equation 定义的就是当前状态跟未来状态的一个迭代的关系。** +假设有一个马尔可夫转移矩阵是右边这个样子,Bellman Equation 描述的就是当前状态到未来状态的一个转移。假设我们当前是在 $s_1$, 那么它只可能去到三个未来的状态:有 0.1 的概率留在它当前这个位置,有 0.2 的概率去到 $s_2$ 状态,有 0.7 的概率去到 $s_4$ 的状态,所以我们要把这个转移乘以它未来的状态的价值,再加上它的 immediate reward 就会得到它当前状态的价值。**所以 Bellman Equation 定义的就是当前状态跟未来状态的一个迭代的关系。** -![](img/2.14.png) - -我们可以把 Bellman Equation 写成一种矩阵的形式。首先有这个转移矩阵。我们当前这个状态是一个向量 $[V(s_1),V(s_2),\cdots,V(s_N)]^T$。我们可以写成迭代的形式。我们每一行来看的话,$V$ 这个向量乘以了转移矩阵里面的某一行,再加上它当前可以得到的 reward,就会得到它当前的价值。 +我们可以把 Bellman Equation 写成一种矩阵的形式,如下式所示。 +$$ +\left[\begin{array}{c} +V\left(s_{1}\right) \\ +V\left(s_{2}\right) \\ +\vdots \\ +V\left(s_{N}\right) +\end{array}\right]=\left[\begin{array}{c} +R\left(s_{1}\right) \\ +R\left(s_{2}\right) \\ +\vdots \\ +R\left(s_{N}\right) +\end{array}\right]+\gamma\left[\begin{array}{cccc} +P\left(s_{1} \mid s_{1}\right) & P\left(s_{2} \mid s_{1}\right) & \ldots & P\left(s_{N} \mid s_{1}\right) \\ +P\left(s_{1} \mid s_{2}\right) & P\left(s_{2} \mid s_{2}\right) & \ldots & P\left(s_{N} \mid s_{2}\right) \\ +\vdots & \vdots & \ddots & \vdots \\ +P\left(s_{1} \mid s_{N}\right) & P\left(s_{2} \mid s_{N}\right) & \ldots & P\left(s_{N} \mid s_{N}\right) +\end{array}\right]\left[\begin{array}{c} +V\left(s_{1}\right) \\ +V\left(s_{2}\right) \\ +\vdots \\ +V\left(s_{N}\right) +\end{array}\right] +$$ +首先有这个转移矩阵。我们当前这个状态是一个向量 $[V(s_1),V(s_2),\cdots,V(s_N)]^T$。我们可以写成迭代的形式。我们每一行来看的话,$V$ 这个向量乘以了转移矩阵里面的某一行,再加上它当前可以得到的 reward,就会得到它当前的价值。 当我们把 Bellman Equation 写成矩阵形式后,可以直接求解: $$ @@ -298,8 +324,6 @@ $$ ### Value function for MDP -![](img/2.22.png) - 顺着 MDP 的定义,我们可以把 `状态-价值函数(state-value function)`,就是在 MDP 里面的价值函数也进行一个定义,它的定义是跟 MRP 是类似的,如式 (3) 所示: $$ v^{\pi}(s)=\mathbb{E}_{\pi}\left[G_{t} \mid s_{t}=s\right] \tag{3} @@ -333,8 +357,6 @@ $$ ### Bellman Expectation Equation -![](img/2.23.png) - **我们可以把状态-价值函数和 Q 函数拆解成两个部分:即时奖励(immediate reward) 和后续状态的折扣价值(discounted value of successor state)。** 通过对状态-价值函数进行一个分解,我们就可以得到一个类似于之前 MRP 的 Bellman Equation,这里叫 `Bellman Expectation Equation`,如式 (6) 所示: @@ -347,9 +369,7 @@ q^{\pi}(s, a)=E_{\pi}\left[R_{t+1}+\gamma q^{\pi}\left(s_{t+1}, A_{t+1}\right) \ $$ **Bellman expectation equation 定义了你当前状态跟未来状态之间的一个关联。** -![](img/2.24.png) - -**那我们进一步进行一个简单的分解。** +我们进一步进行一个简单的分解。 我们先给出等式 (8): $$ diff --git a/docs/chapter2/img/2.13.png b/docs/chapter2/img/2.13.png index 0e2dceabce14322360a74660c180bc74ca1ad49f..5d6d52ee6e677497994f20e384b569a70499668e 100644 GIT binary patch literal 187187 zcmeFZbwE_@*1!u$ip03J3#8rywC6(%s#6 zbI$p`@AcgC-v91@2R5*0KfRw?&sx7l_;Xb`oQD(-k&uvZ6y%{ANJwbtNJuD-n80r& zd+AYZEl!Go6{dh6)mr7c&yl+h8Q5OW@JlbtEKLZX~2lV^&MFr^zaF2?vNCbDgXCg1Nrg4 z;-DZQg;^t^{wvOXP42(qfS>y`|NQ&lJ@WtM_#WkdVxytIfABx|D315ZGOg(DOU8O5 zuj`D2M8b6cgRG#zxDT}UwY8Rxi;jx2h?#>Or-`|PDU8#@?#+EFBvB6$;MNZ2VnXL( zXKU{);vok9M+_0*{{Ar+nC>4@Tx`U^Ix5fUq#c}Kbo`u8Ik~~&59#RWM4iknL^PnX 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