From a71a2e2d04dd54bf4b3bc3d27e5acf20b3fedd7a Mon Sep 17 00:00:00 2001 From: qiwang067 Date: Mon, 19 Sep 2022 20:30:56 +0800 Subject: [PATCH] update ch8 --- docs/chapter8/chapter8.md | 70 ++++++++++++++++++-------------------- docs/chapter8/img/8.2.png | Bin 493994 -> 0 bytes docs/chapter8/img/8.3.png | Bin 402692 -> 0 bytes docs/img/ch8/8.2.png | Bin 0 -> 123041 bytes docs/img/ch8/8.3.png | Bin 0 -> 413406 bytes 5 files changed, 34 insertions(+), 36 deletions(-) delete mode 100644 docs/chapter8/img/8.2.png delete mode 100644 docs/chapter8/img/8.3.png create mode 100644 docs/img/ch8/8.2.png create mode 100644 docs/img/ch8/8.3.png diff --git a/docs/chapter8/chapter8.md b/docs/chapter8/chapter8.md index c7154e0..85f63a8 100644 --- a/docs/chapter8/chapter8.md +++ b/docs/chapter8/chapter8.md @@ -1,54 +1,52 @@ -# 针对连续动作的 DQN +# 第8章 针对连续动作的深度Q网络 -## 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。 +与基于策略梯度的方法相比,深度Q网络比较稳定,策略梯度比较不稳定,玩大部分游戏不能使用策略梯度。 -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。 +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。最早 DeepMind 的论文拿深度强化学习来玩雅达利的游戏,用的就是深度Q网络。深度Q网络比较容易训练的一个原因是:在深度Q网络里面,我们只要能够估计出Q函数,就保证一定可以找到一个比较好的策略。也就是我们只要能够估计出Q函数,就保证可以改进策略。而估计Q函数是比较容易的,因为它就是一个回归问题。在回归问题里面,我们可以通过观察回归的损失有没有下降,就可以知道模型学习得好不好,所以估计Q函数相较于学习一个策略是比较容易的。我们只要估计Q函数,就可以保证现在一定会得到比较好的策略,所以一般而言深度Q网络比较容易操作。 + +但深度Q网络其实存在一些问题,最大的问题是它很难处理连续动作。很多时候动作是连续的,比如我们玩雅达利的游戏时,智能体只需要决定如上、下、左、右这4个动作,这种动作是离散的。很多时候动作是连续的,例如,假设智能体要开车,它要决定方向盘要左转几度、右转几度,这种动作就是连续的。假设智能体是一个机器人,身上有 50 个 关节,它的每一个动作就对应身上 50 个关节的角度,而这些角度也是连续的。所以很多时候动作并不是离散的,它是一个向量,这个向量的每一个维度都有一个对应的值,这些值都是实数,它是连续的。如果动作是连续的,我们使用深度Q网络就会有困难。因为在使用深度Q网络时很重要的一步是我们要能够解决优化问题,也就是估计出 Q函数$Q(s,a)$ 以后,我们必须要找到一个 $a$,它可以让 $Q(s,a)$ 最大,即 $$ - a=\arg \max _{a} Q(s, a) + \label{eq:max_q_ch8} + a=\underset{a}{\arg \max} Q(s, a) $$ -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。 +假设$a$是离散的,即$a$的可能性是有限的。例如,在雅达利的小游戏里面,$a$ 就是上、下、左、右与开火,它是有限的,我们可以把每一个可能的动作都代入 Q 里面算它的 Q 值。但假如$a$是连续的,我们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 值最大。 -怎么解这个问题呢?就有各种不同的方案。 +怎么解决这个问题呢?我们有多种不同的方案,下面一一介绍。 +## 8.1 方案 1:对动作进行采样 +第1个方案是什么呢?我们可以采样出 $N$ 个可能的 $a$:$\left\{a_{1}, a_{2}, \cdots, a_{N}\right\}$ ,把它们一个一个地代入 Q函数,看谁的Q值最大。这个方案不会太低效,因为我们在运算的时候会使用 GPU,一次把 $N$ 个连续动作都代入 Q函数,一次得到 $N$ 个 Q 值,看谁最大。当然这不是一个非常精确的方案,因为我们没有办法进行太多的采样, 所以估计出来的 Q 值、最后决定的动作可能不是非常精确。 +## 8.2 方案 2:梯度上升 +第2个方案是什么呢?既然要解决的是一个优化问题(optimization problem),我们就要最大化目标函数(objective function)。要最大化目标函数, 我们就可以用梯度上升。我们把$a$当作参数,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$a$最大化Q函数(目标函数)。但我们会遇到全局最大值(global maximum)的问题,不一定能够找到最优的结果,而且运算量显然很大, 因为要迭代地更新 $a$,训练一个网络就很花时间了。如果我们使用梯度上升的方案来处理连续的问题,每次决定采取哪一个动作的时候,还要训练一次网络,显然运算量是很大的。 -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。 +## 8.3 方案 3:设计网络架构 +第3个方案是特别设计网络的架构,特别设计Q函数来使得解决 arg max 操作的问题变得非常容易。 -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。 - - -## 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) - - $a$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 $Q(s,a)$。 - - 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。 - - $\Sigma$ 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。 - - 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。 +如图 8.1 所示,通常输入状态 $\boldsymbol{s}$ 是图像,我们可以用向量或矩阵来表示它。输入 $\boldsymbol{s}$,Q函数会输出向量$\pmb{\mu}(\boldsymbol{s})$、矩阵$\pmb{\varSigma}(\boldsymbol{s})$ 和标量 $V(\boldsymbol{s})$。Q函数根据输入$\boldsymbol{s}$与 $\boldsymbol{a}$ 来决定输出值。到目前为止,Q函数只有输入 $\boldsymbol{s}$,它还没有输入$\boldsymbol{a}$,$\boldsymbol{a}$ 在哪里呢?接下来我们可以输入 $\boldsymbol{a}$,用$\boldsymbol{a}$与 $\pmb{\mu}(\boldsymbol{s})$、$\pmb{\varSigma}(\boldsymbol{s})$和$V(\boldsymbol{s})$ 互相作用。Q函数$Q(\boldsymbol{s},\boldsymbol{a})$可定义为 $$ -\mu(s)=\arg \max _{a} Q(s, a) + \label{eq:} + Q(\boldsymbol{s},\boldsymbol{a})=-(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))^{\mathrm{T}} \pmb{\varSigma}(\boldsymbol{s})(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))+V(\boldsymbol{s}) $$ -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。 +注意,$\boldsymbol{a}$现在是连续的动作,所以它是一个向量。假设我们要操作机器人,向量$\boldsymbol{a}$的每一个维度可能就对应机器人的每一个关节,它的数值就是关节的角度。假设 $\boldsymbol{a}$ 和 $\pmb{\mu}(\boldsymbol{s})$ 是列向量,那么 $(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))^{\mathrm{T}}$ 是一个行向量。$\pmb{\varSigma}(\boldsymbol{s})$ 是一个正定矩阵(positive-definite matrix),因为 $\pmb{\varSigma}(\boldsymbol{s}) = \boldsymbol{L}\boldsymbol{L}^{\mathrm{T}}$,其中 $\boldsymbol{L}$ 为下三角矩阵(lower-triangular matrix)。 $\boldsymbol{a}$-$\pmb{\mu}(\boldsymbol{s})$也是一个列向量。所以Q值即 $-(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))^{\mathrm{T}} \pmb{\varSigma}(\boldsymbol{s})(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))+V(\boldsymbol{s})$ 是标量。 -## 方案 4:不使用DQN -第 4 招就是不要用 DQN。用 DQN 处理连续动作还是比较麻烦。 -基于策略的方法 PPO 和基于价值的方法 DQN,这两者其实是可以结合在一起的,如下图所示,也就是演员-评论员的方法。 - -![](img/8.3.png) +我们要怎么找到一个$\boldsymbol{a}$来最大化 Q 值呢?因为 $(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))^{\mathrm{T}} \pmb{\varSigma}(\boldsymbol{s})(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))$ 一定是正的,它前面有一个负号,假设我们不看负号,所以第一项 $(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))^{\mathrm{T}} \pmb{\varSigma}(\boldsymbol{\boldsymbol{s}})(\boldsymbol{a}-\pmb{\mu}(\boldsymbol{s}))$ 的值越小,最终的 Q 值就越大。因为我们是把 $V(\boldsymbol{s})$ 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?我们直接令 $\pmb{\mu}(\boldsymbol{s})$ 等于$\boldsymbol{a}$,让第一项变成 0,就可以让第一项的值最小。 因此,令 $\pmb{\mu}(\boldsymbol{s})$ 等于$\boldsymbol{a}$,我们就可以得到最大值,解决 arg max 操作的问题就变得非常容易。所以深度Q网络也可以用在连续的情况中,只是有一些局限:函数不能随便设置。 + +>如果 $n$阶对称矩阵$\boldsymbol{A}$ 对于任意非零的$n$维向量$\boldsymbol{x}$都有 $\boldsymbol{x}^\mathrm{T}\boldsymbol{A}\boldsymbol{x}>0$,则称矩阵$\boldsymbol{A}$为正定矩阵。 +
+ +
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图 8.1 方案 3:设计网络架构
+ +## 8.4 方案 4:不使用深度Q网络 + +第4个方案就是不使用深度Q网络,用深度Q网络处理连续动作是比较麻烦的。如图 8.2 所示,我们将基于策略的方法————PPO 和基于价值的方法————深度Q网络结合在一起,就可以得到演员-评论员的方法。 +
+ +
+
图 8.2 方案 4:不使用深度Q网络
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