From 1adddca46d9009adec12a61a4ba80d586c9603a4 Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Tue, 21 Sep 2021 11:06:09 +0800 Subject: [PATCH] fix ch8 --- docs/chapter8/chapter8.md | 57 +++++++++++++++++++++++--------------- docs/chapter8/img/8.1.png | Bin 692394 -> 0 bytes docs/chapter8/img/8.2.png | Bin 812631 -> 493994 bytes docs/chapter8/img/8.3.png | Bin 991925 -> 402692 bytes 4 files changed, 35 insertions(+), 22 deletions(-) delete mode 100644 docs/chapter8/img/8.1.png diff --git a/docs/chapter8/chapter8.md b/docs/chapter8/chapter8.md index 234e6d7..c7154e0 100644 --- a/docs/chapter8/chapter8.md +++ b/docs/chapter8/chapter8.md @@ -1,41 +1,54 @@ -# Q-learning for Continuous Actions +# 针对连续动作的 DQN -## Solution 1 & Solution 2 +## 方案 1 & 方案 2 +跟基于策略梯度的方法比起来,DQN 是比较稳的。策略梯度是没有太多游戏是玩得起来的,策略梯度比较不稳,在没有 近端策略优化 之前,我们很难用策略梯度做什么事情。DQN 相对而言是比较稳的。最早 DeepMind 的论文拿深度强化学习来玩雅达利的游戏,用的就是 DQN。DQN 比较容易训练的一个理由是:在 DQN 里面,你只要能够估计出Q函数,就保证你一定可以找到一个比较好的策略。也就是你只要能够估计出Q函数,就保证你可以改进策略。而估计Q函数这件事情,是比较容易的,因为它就是一个回归问题。在回归问题里面, 你可以轻易地知道模型学习得是不是越来越好,只要看那个回归的损失有没有下降,你就知道说模型学习得好不好,所以估计Q函数相较于学习一个策略是比较容易的。你只要估计Q函数,就可以保证说现在一定会得到比较好的策略。所以一般而言 DQN 比较容易操作。 -![](img/8.1.png) +DQN 其实存在一些问题,最大的问题是它不太容易处理连续动作。很多时候动作是连续的,比如我们玩雅达利的游戏,智能体只需要决定比如说上下左右,这种动作是离散的。那很多时候动作是连续的。举例来说假设智能体要做的事情是开自驾车,它要决定说它方向盘要左转几度, 右转几度,这是连续的。假设智能体是一个机器人,它身上有 50 个 关节,它的每一个动作就对应到它身上的这 50 个关节的角度。而那些角度也是连续的。所以很多时候动作并不是一个离散的东西,它是一个向量。在这个向量里面,它的每一个维度都有一个对应的值,都是实数,它是连续的。假设动作是连续的,做 DQN 就会有困难。因为在做 DQN 里面一个很重要的一步是你要能够解这个优化问题。估计出 Q函数$Q(s,a)$ 以后,必须要找到一个 $a$,它可以让 $Q(s,a)$ 最大,如下式所示。 -继续讲一下 Q-learning,其实跟 policy gradient based 方法比起来,Q-learning 是比较稳的。policy gradient 是没有太多游戏是玩得起来的,policy gradient 比较不稳,尤其在没有 PPO 之前,你很难用 policy gradient 做什么事情。Q-learning 相对而言是比较稳的。最早 DeepMind 的 paper 拿 deep reinforcement learning 来玩 Atari 的游戏,用的就是 Q-learning。Q-learning 比较容易 train 的一个理由是:在 Q-learning 里面,你只要能够 estimate 出 Q-function,就保证你一定可以找到一个比较好的 policy。也就是你只要能够 estimate 出 Q-function,就保证你可以 improve 你的 policy。而 estimate Q-function 这件事情,是比较容易的,因为它就是一个 regression problem。在这个 regression problem 里面, 你可以轻易地知道 model learn 得是不是越来越好,只要看那个 regression 的 loss 有没有下降,你就知道说你的 model learn 得好不好,所以 estimate Q-function 相较于 learn 一个 policy 是比较容易的。你只要 estimate Q-function,就可以保证说现在一定会得到比较好的 policy。所以一般而言 Q-learning 比较容易操作。 +$$ + a=\arg \max _{a} Q(s, a) +$$ -Q: Q-learning 有什么问题呢? +假设$a$是离散的,即$a$的可能性都是有限的。举例来说,雅达利的小游戏里面,$a$ 就是上下左右跟开火,它是有限的,我们可以把每一个可能的动作都带到 Q 里面算它的 Q 值。但假如$a$是连续的,你无法穷举所有可能的连续动作,试试看哪一个连续动作可以让 Q 的值最大。 -A: **最大的问题是它不太容易处理 continuous action**。很多时候 action 是 continuous 的。什么时候你的 action 会是 continuous 的呢?我们玩 Atari 的游戏,你的 agent 只需要决定比如说上下左右,这种 action 是 discrete 的。那很多时候你的 action 是 continuous 的。举例来说假设你的 agent 要做的事情是开自驾车,它要决定说它方向盘要左转几度, 右转几度,这是 continuous 的。假设 agent 是一个机器人,它身上有 50 个 关节,它的每一个 action 就对应到它身上的这 50 个关节的角度。而那些角度也是 continuous 的。所以很多时候 action 并不是一个 discrete 的东西,它是一个 vector。在这个 vector 里面,它的每一个 dimension 都有一个对应的 value,都是 real number,它是 continuous 的。假设 action 是 continuous 的,做 Q-learning 就会有困难。因为在做 Q-learning 里面一个很重要的一步是你要能够解这个 optimization problem。你 estimate 出 Q-function $Q(s,a)$ 以后,必须要找到一个 a,它可以让 $Q(s,a)$ 最大。假设 a 是 discrete 的,那 a 的可能性都是有限的。举例来说,Atari 的小游戏里面,a 就是上下左右跟开火,它是有限的,你可以把每一个可能的 action 都带到 Q 里面算它的 Q value。但假如 a 是 continuous 的,你无法穷举所有可能的 continuous action,试试看哪一个 continuous action 可以让 Q 的 value 最大。 +怎么解这个问题呢?就有各种不同的方案。 -所以怎么办呢?在概念上,我们就是要能够解这个问题。怎么解这个问题呢?就有各种不同的 solution。 +第一个方案是假设你不知道怎么解这个问题,因为$a$是没有办法穷举的,怎么办?我们可以采样出 $N$ 个可能的 $a$:$\left\{a_{1}, a_{2}, \cdots, a_{N}\right\}$ ,一个一个带到 Q函数里面,看谁最大。这个方法其实也不会太不高效, 因为你在运算的时候会使用 GPU,一次会把 $N$ 个连续动作都丢到 Q函数里面,一次得到 $N$ 个 Q 值,然后看谁最大。当然这不是一个非常精确的做法,因为你没有办法做太多的采样, 所以你估计出来的 Q 值,最后决定的动作可能不是非常的精确,这是第一个方案。 -**第一个 solution 是假设你不知道怎么解这个问题,因为 a 是没有办法穷举的。怎么办?用 sample 的。Sample 出 N 个 可能的 a,一个一个带到 Q-function 里面,看谁最快。**这个方法其实也不会太不 efficient, 因为你真的在运算的时候,你会用 GPU,一次会把 N 个 continuous action 都丢到 Q-function 里面,一次得到 N 个 Q value,然后看谁最大。当然这不是一个非常精确的做法,因为你没有办法做太多的 sample, 所以你 estimate 出来的 Q value,你最后决定的 action 可能不是非常的精确, 这是第一个 solution。 +第二个方案是什么呢?既然要解的是一个优化问题(optimization problem),其实是要最大化目标函数(objective function),要最大化一个东西, 就可以用梯度上升。我们就把$a$当作是参数,然后要找一组$a$去最大化Q函数,就用梯度上升去更新 $a$ 的值,最后看看能不能找到一个$a$去最大化Q函数,也就是目标函数。当然这样子你会遇到全局最大值(global maximum)的问题, 就不见得能够真的找到最优的结果,而且这个运算量显然很大, 因为你要迭代地更新 $a$。我们训练一个网络就很花时间了。如果你用梯度上升的方法来处理连续的问题, 等于是你每次要决定采取哪一个动作的时候,都还要做一次训练网络的过程,显然运算量是很大的。这是第二个方案。 -**第二个 solution 是什么呢?既然要解的是一个 optimization problem,其实是要 maximize objective function,要 maximize 一个东西, 就可以用 gradient ascent。**你就把 a 当作是 parameter,然后你要找一组 a 去 maximize 你的 Q-function,你就用 gradient ascent 去 update a 的 value,最后看看你能不能找到一个 a 去 maximize 你的 Q-function,也就是你的 objective function。当然这样子你会遇到 global maximum 的问题, 就不见得能够真的找到 optimal 的结果,而且这个运算量显然很大, 因为你要迭代地 update a。我们 train 一个 network 就很花时间了。如果你用 gradient ascent 的方法来处理 continuous 的 problem, 等于是你每次要决定 take 哪一个 action 的时候,你都还要做一次 train network 的 process,显然运算量是很大的。这是第二个 solution。 -## Solution 3: Design a network +## 方案 3:设计网络 + +第三个方案是特别设计一个网络的架构,特别设计Q函数,使得解 arg max 的问题变得非常容易。也就是这边的Q函数不是一个一般的Q函数,特别设计一下它的样子,让你要找让这个Q函数最大的 $a$ 的时候非常容易。 + +下图是一个例子,这边有Q函数,这个Q函数的做法是这样。 + 通常输入状态 $s$ 就是一个图像,可以用一个向量或一个矩阵来表示。 + 输入 $s$,Q函数会输出 3 个东西。它会输出 $\mu(s)$,这是一个向量。它会输出 $\Sigma(s)$ ,这是一个矩阵。它会输出 $V(s)$,这是一个标量。 + 输出这 3 个东西以后,我们知道Q函数其实是吃一个$s$跟 $a$,然后决定一个值。Q函数意思是说在某一个状态,采取某一个动作的时候,你期望的奖励有多大。到目前为止这个Q函数只吃 $s$,它还没有吃$a$进来,$a$ 在哪里呢?当这个Q函数吐出 $\mu$、 $\Sigma$ 跟 $V$ 的时候,我们才把$a$引入,用$a$跟 $\mu(s)、\Sigma(s)、V$ 互相作用一下,你才算出最终的 Q 值。 ![](img/8.2.png) -**第三个 solution 是特别 design 一个 network 的架构,特别 design 你的 Q-function,使得解 arg max 的 problem 变得非常容易**。也就是这边的 Q-function 不是一个 general 的 Q-function,特别设计一下它的样子,让你要找让这个 Q-function 最大的 a 的时候非常容易。 + $a$怎么和这 3 个东西互相作用呢?实际上 $Q(s,a)$,Q函数的运作方式是先输入 $s$,让你得到 $\mu,\Sigma$ 跟 $V$。然后再输入 $a$,然后接下来把$a$跟 $\mu$ 相减。注意一下$a$现在是连续的动作,所以它也是一个向量。假设你现在是要操作机器人的话,这个向量的每一个维度,可能就对应到机器人的某一个关节,它的数值就是关节的角度,所以$a$是一个向量。把向量 $a$ 减掉向量 $\mu$,取转置,所以它是一个横的向量。$\Sigma$ 是一个矩阵。然后$a$减掉 $\mu(s)$ ,$a$ 和 $\mu(s)$ 都是向量,减掉以后还是一个竖的向量。所以 $-(a-\mu(s))^{T} \Sigma(s)(a-\mu(s))+V(s)$ 是一个标量,这个数值就是 Q 值 $Q(s,a)$。 -上图是一个例子,这边有我们的 Q-function,这个 Q-function 的做法是这样。 + 假设 $Q(s,a)$ 定义成这个样子,我们要怎么找到一个$a$去最大化这个 Q 值呢?这个方案非常简单。因为 $(a-\mu(s))^{T} \Sigma(s)(a-\mu(s))$ 一定是正的,它前面乘上一个负号,所以第一项就假设我们不看这个负号的话,第一项的值越小,最终的 Q 值就越大。因为我们是把 $V(s)$ 减掉第一项,所以第一项的值越小,最后的 Q 值就越大。怎么让第一项的值最小呢?你直接把$a$代入 $\mu$ 的值,让它变成 0,就会让第一项的值最小。 -* Input state s,通常它就是一个 image,可以用一个向量或一个 matrix 来表示。 -* Input 这个 s,Q-function 会 output 3 个东西。它会 output $\mu(s)$,这是一个 vector。它会 output $\Sigma(s)$ ,这是一个 matrix。它会 output $V(s)$,这是一个 scalar。 -* output 这 3 个东西以后,我们知道 Q-function 其实是吃一个 s 跟 a,然后决定一个 value。Q-function 意思是说在某一个 state,take 某一个 action 的时候,你 expected 的 reward 有多大。到目前为止这个 Q-function 只吃 s,它还没有吃 a 进来,a 在哪里呢?当这个 Q-function 吐出 $\mu$、 $\Sigma$ 跟 $V$ 的时候,我们才把 a 引入,用 a 跟 $\mu(s)、\Sigma(s)、V$ 互相作用一下,你才算出最终的 Q value。 -* a 怎么和这 3 个东西互相作用呢?实际上 $Q(s,a)$,你的 Q-function 的运作方式是先 input s,让你得到 $\mu,\Sigma$ 跟 V。然后再 input a,然后接下来把 a 跟 $\mu$ 相减。注意一下 a 现在是 continuous 的 action,所以它也是一个 vector。假设你现在是要操作机器人的话,这个 vector 的每一个 dimension,可能就对应到机器人的某一个关节,它的数值就是关节的角度,所以 a 是一个 vector。把 a 的这个 vector 减掉 $\mu$ 的这个 vector,取 transpose,所以它是一个横的 vector。$\Sigma$ 是一个 matrix。然后 a 减掉 $\mu(s)$ ,a 和 $\mu(s)$ 都是 vector,减掉以后还是一个竖的 vector。所以 $-(a-\mu(s))^{T} \Sigma(s)(a-\mu(s))+V(s)$ 是一个 scalar,这个数值就是 Q value $Q(s,a)$,。 -* 假设 $Q(s,a)$ 定义成这个样子,我们要怎么找到一个 a 去 maximize 这个 Q value 呢?这个 solution 非常简单,什么样的 a, 可以让这一个 Q-function 最终的值最大呢?。因为 $(a-\mu(s))^{T} \Sigma(s)(a-\mu(s))$ 一定是正的,它前面乘上一个负号,所以第一项就假设我们不看这个负号的话,第一项的值越小,最终的 Q value 就越大。因为我们是把 V(s) 减掉第一项,所以第一项的值越小,最后的 Q value 就越大。怎么让第一项的值最小呢?你直接把 a 代入 $\mu$ 的值,让它变成 0,就会让第一项的值最小。 -* $\Sigma$ 一定是正定的。因为这个东西就像是 Gaussian distribution,所以 $\mu$ 就是 Gaussian 的 mean,$\Sigma$ 就是 Gaussian 的 variance。但 variance 是一个 positive definite 的 matrix,怎么样让这个 $\Sigma$ 一定是 positive definite 的 matrix 呢?其实在 $Q^{\pi}$ 里面,它不是直接 output $\Sigma$,如果直接 output 一个 $\Sigma$, 它不一定是 positive definite 的 matrix。它其实是 output 一个 matrix,然后再把那个 matrix 跟另外一个 matrix 做 transpose 相乘, 然后可以确保 $\Sigma $ 是 positive definite 的。这边要强调的点就是说,实际上它不是直接 output 一个 matrix。你再去那个 paper 里面 check 一下它的 trick,它可以保证说 $\Sigma$ 是 positive definite 的。 -* 你把 a 代入 $\mu(s)$ 以后,你可以让 Q 的值最大。所以假设要你 arg max 这个东西,虽然一般而言,若 Q 是一个 general function, 你很难算,但是我们这边 design 了 Q 这个 function,a 只要设 $\mu(s)$,我们就得到最大值。你在解这个 arg max 的 problem 的时候就变得非常容易。所以 Q-learning 也可以用在 continuous 的 case,只是有一些局限,就是 function 不能够随便乱设,它必须有一些限制。 + $\Sigma$ 一定是正定的。因为这个东西就像是高斯分布(Gaussian distribution),所以 $\mu$ 就是高斯分布的均值,$\Sigma$ 就是高斯分布的方差。但方差是一个正定(positive definite)的矩阵,怎么样让这个 $\Sigma$ 一定是正定的矩阵呢?其实在 $Q^{\pi}$ 里面,它不是直接输出 $\Sigma$,如果直接输出 一个 $\Sigma$, 它不一定是正定的矩阵。它其实是输出 一个矩阵,然后再把那个矩阵跟另外一个矩阵做转置相乘, 然后可以确保 $\Sigma$ 是正定的。这边要强调的点就是说,实际上它不是直接输出一个矩阵。你再去那个论文里面查看一下它的技巧,它可以保证说 $\Sigma$ 是正定的。 -## Solution 4: Don't use Q-learning + 你把$a$代入 $\mu(s)$ 以后,你可以让 Q 的值最大。所以假设要你 arg max Q 函数,如下式所示。 +$$ +\mu(s)=\arg \max _{a} Q(s, a) +$$ + +虽然一般而言,若 Q 是一个一般的函数, 你很难算,但是我们这边设计了 Q 这个函数,$a$ 只要设 $\mu(s)$,我们就得到最大值。你在解这个 arg max 的问题的时候就变得非常容易。所以 DQN 也可以用在连续的情况,只是有一些局限,就是函数不能够随便乱设,它必须有一些限制。 + +## 方案 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