From e21d857edd1be42c1dd2f939ea3bfc8558d4fcb4 Mon Sep 17 00:00:00 2001 From: johnjim0816 Date: Tue, 1 Aug 2023 20:35:03 +0800 Subject: [PATCH] update --- docs/ch10/main.md | 58 +++++++++++------- docs/ch11/main.md | 46 +++++++------- docs/ch2/main.md | 2 +- docs/ch8/main.md | 6 +- docsify.md | 6 +- ...\347\255\226\350\277\207\347\250\213.docx" | Bin 183236 -> 182557 bytes ...\345\236\213\346\216\247\345\210\266.docx" | Bin 0 -> 610326 bytes 7 files changed, 67 insertions(+), 51 deletions(-) create mode 100644 "words/\347\254\2545\347\253\240 \345\205\215\346\250\241\345\236\213\346\216\247\345\210\266.docx" diff --git a/docs/ch10/main.md b/docs/ch10/main.md index 2503e2a..72e25e0 100644 --- a/docs/ch10/main.md +++ b/docs/ch10/main.md @@ -1,116 +1,130 @@ # Actor-Critic 算法 -在策略梯度的章节中,实际上我们已经开了一部分 `Actor-Critic` 算法的头了,这一章我们将继续深入探讨 `Actor-Critic` 算法。 +$\qquad$ 在策略梯度的章节中,实际上我们已经开了一部分 $\text{Actor-Critic}$ 算法的头了,这一章我们将继续深入探讨 $\text{Actor-Critic}$ 算法。 ## 策略梯度算法的缺点 -这里策略梯度算法特指蒙特卡洛策略梯度算法,相比于 `DQN` 之类的基于价值的算法,策略梯度算法有以下优点: +$\qquad$ 这里策略梯度算法特指蒙特卡洛策略梯度算法,相比于 $\text{DQN}$ 之类的基于价值的算法,策略梯度算法有以下优点: * **适配连续动作空间**。在将策略函数设计的时候我们已经展开过,这里不再赘述。 * **适配随机策略**。由于策略梯度算法是基于策略函数的,因此可以适配随机策略,而基于价值的算法则需要一个确定的策略。此外其计算出来的策略梯度是无偏的,而基于价值的算法则是有偏的。 -但同样的,策略梯度算法也有其缺点: +$\qquad$ 但同样的,策略梯度算法也有其缺点: * **采样效率低**。由于使用的是蒙特卡洛估计,与基于价值算法的时序差分估计相比其采样速度必然是要慢很多的,这个问题在前面相关章节中也提到过。 * **高方差**。虽然跟基于价值的算法一样都会导致高方差,但是策略梯度算法通常是在估计梯度时蒙特卡洛采样引起的高方差,这样的方差甚至比基于价值的算法还要高。 * **收敛性差**。容易陷入局部最优,策略梯度方法并不保证全局最优解,因为它们可能会陷入局部最优点。策略空间可能非常复杂,存在多个局部最优点,因此算法可能会在局部最优点附近停滞。 * **难以处理高维离散动作空间**:对于离散动作空间,采样的效率可能会受到限制,因为对每个动作的采样都需要计算一次策略。当动作空间非常大时,这可能会导致计算成本的急剧增加。 -而结合了策略梯度和值函数的 `Actor-Critic` 算法则能同时兼顾两者的优点,并且甚至能缓解两种方法都很难解决的高方差问题。可能读者会奇怪为什么各自都有高方差的问题,结合了之后反而缓解了这个问题呢?我们再仔细分析一下两者高方差的根本来源,策略梯度算法是因为直接对策略参数化,相当于既要利用策略去与环境交互采样,又要利用采样去估计策略梯度,而基于价值的算法也是需要与环境交互采样来估计值函数的,因此也会有高方差的问题。而结合之后呢,`Actor` 部分还是负责估计策略梯度和采样,但 `Critic` 即原来的值函数部分就不需要采样而只负责估计值函数了,并且由于它估计的值函数指的是策略函数的值,相当于带来了一个更稳定的估计,来指导 `Actor` 的更新,反而能够缓解策略梯度估计带来的方差。当然尽管 `Actor-Critic` 算法能够缓解方差问题,但并不能彻底解决问题,在接下来的章节中我们也会展开介绍一些改进的方法。 +$\qquad$ 结合了策略梯度和值函数的 $\text{Actor-Critic}$ 算法则能同时兼顾两者的优点,并且甚至能缓解两种方法都很难解决的高方差问题。可能读者会奇怪为什么各自都有高方差的问题,结合了之后反而缓解了这个问题呢?我们再仔细分析一下两者高方差的根本来源,策略梯度算法是因为直接对策略参数化,相当于既要利用策略去与环境交互采样,又要利用采样去估计策略梯度,而基于价值的算法也是需要与环境交互采样来估计值函数的,因此也会有高方差的问题。 +$\qquad$ 而结合之后呢,$\text{Actor}$ 部分还是负责估计策略梯度和采样,但 $\text{Critic}$ 即原来的值函数部分就不需要采样而只负责估计值函数了,并且由于它估计的值函数指的是策略函数的值,相当于带来了一个更稳定的估计,来指导 $\text{Actor}$ 的更新,反而能够缓解策略梯度估计带来的方差。当然尽管 $\text{Actor-Critic}$ 算法能够缓解方差问题,但并不能彻底解决问题,在接下来的章节中我们也会展开介绍一些改进的方法。 ## Q Actor-Critic 算法 -在策略梯度章节中,我们其实已经对 `Actor-Critic` 算法的目标函数进行过推导了,这里就不详细展开,只是简单回顾一下目标函数,如下: +$\qquad$ 在策略梯度章节中,我们其实已经对 $\text{Actor-Critic}$ 算法的目标函数进行过推导了,这里就不详细展开,只是简单回顾一下目标函数,如式 $\text(10.1)$ 所示。 $$ +tag{10.1} \begin{aligned} \nabla_\theta J(\theta) \propto \mathbb{E}_{\pi_{\theta}}\left[Q^\pi(s, a) \nabla_\theta \log \pi_\theta(a \mid s)\right] \end{aligned} $$ -在 `REINFORCE` 算法中,我们使用蒙特卡洛估计来表示当前状态-动作对 $(s_t,a_t)$ 的价值。而这里其实可以类比于 $Q$ 函数,用 $Q^\pi(s_t, a_t)$ 来估计当前的价值,注意这里的输入是状态和动作,而不单单是状态,输出的是单个值,也可以用 $Q_{\phi}(s_t, a_t)$ 表示,其中 $\phi$ 表示 Critic 网络的参数。这样我们就可以将目标函数写成: +$\qquad$ 在 `REINFORCE` 算法中,我们使用蒙特卡洛估计来表示当前状态-动作对 $(s_t,a_t)$ 的价值。而这里其实可以类比于 $Q$ 函数,用 $Q^\pi(s_t, a_t)$ 来估计当前的价值,注意这里的输入是状态和动作,而不单单是状态,输出的是单个值,也可以用 $Q_{\phi}(s_t, a_t)$ 表示,其中 $\phi$ 表示 $\text{Critic}$ 网络的参数。这样我们就可以将目标函数写成如式 $\text(10.2)$ 所示的形式。 $$ +tag{10.2} \begin{aligned} \nabla_\theta J(\theta) \propto \mathbb{E}_{\pi_{\theta}}\left[Q_{\phi}(s_t, a_t) \nabla_\theta \log \pi_\theta(a_t \mid s_t)\right] \end{aligned} $$ -这样的算法通常称之为 `Q Actor-Critic` 算法,这也是最简单的 `Actor-Critic` 算法,现在我们一般都不用这个算法了,但是这个算法的思想是很重要的,因为后面的算法都是在这个算法的基础上进行改进的。 +$\qquad$ 这样的算法通常称之为 $\text{Q Actor-Critic}$ 算法,这也是最简单的 $\text{Actor-Critic}$ 算法,现在我们一般都不用这个算法了,但是这个算法的思想是很重要的,因为后面的算法都是在这个算法的基础上进行改进的。 + +$\qquad$ 如图 $\text{10.1}$ 所示,我们通常将 $\text{Actor}$ 和 $\text{Critic}$ 分别用两个模块来表示,即图中的 策略函数( $\text{Policy}$ )和价值函数( $\text{Value Function}$ )。$\text{Actor}$ 与环境交互采样,然后将采样的轨迹输入 $\text{Critic}$ 网络,$\text{Critic}$ 网络估计出当前状态-动作对的价值,然后再将这个价值作为 $\text{Actor}$ 网络的梯度更新的依据,这也是所有 $\text{Actor-Critic}$ 算法的基本通用架构。
-
图 10.1 Actor-Critic 算法架构
+
图 $\text{10.1}$ $\text{Actor-Critic}$ 算法架构
-如图所示,我们通常将 `Actor` 和 `Critic` 分别用两个模块来表示,即图中的 策略函数(`Policy`)和价值函数(`Value Function`)。`Actor` 与环境交互采样,然后将采样的轨迹输入 `Critic` 网络,`Critic` 网络估计出当前状态-动作对的价值,然后再将这个价值作为 `Actor` 网络的梯度更新的依据,这也是所有 `Actor-Critic` 算法的基本通用架构。 ## A2C 与 A3C 算法 -我们知道 `Actor-Critic` 架构是能够缓解策略梯度算法的高方差问题的,但是并不能彻底解决问题。为了进一步缓解高方差问题,我们引入一个优势函数(Advantage Function)$A^\pi(s_t, a_t)$,用来表示当前状态-动作对相对于平均水平的优势,即: +$\qquad$ 我们知道 $\text{Actor-Critic}$ 架构是能够缓解策略梯度算法的高方差问题的,但是并不能彻底解决问题。为了进一步缓解高方差问题,我们引入一个优势函数( $\text{advantage function}$ )$A^\pi(s_t, a_t)$,用来表示当前状态-动作对相对于平均水平的优势,即式 $\text(10.3)$ 。 $$ +\tag{10.3} \begin{aligned} A^\pi(s_t, a_t) = Q^\pi(s_t, a_t) - V^\pi(s_t) \end{aligned} $$ -这里优势函数相当于减去了一个基线,这个基线可以自由设计,但是通常我们会选择状态价值函数 $V^\pi(s_t)$ 作为基线,减去这个基线会让梯度估计更稳定。有读者可能会奇怪,减去基线真的能减少方差吗?比如 $\{1,2,3\}$ 这个数列,都减去均值 $2$ 之后的数列 $\{-1,0,1\}$ 的方差不还是一样约等于 $0.67$ 吗? 这里其实犯了一个错误,就是我们讲的基线是指在同一个状态下的基线,而不是在整个数列上的均值,这里的基线是指 $V^\pi(s_t)$,而不是 $V^\pi(s)$ 的均值。 +$\qquad$ 这里优势函数相当于减去了一个基线,这个基线可以自由设计,但是通常我们会选择状态价值函数 $V^\pi(s_t)$ 作为基线,减去这个基线会让梯度估计更稳定。有读者可能会奇怪,减去基线真的能减少方差吗?比如 $\{1,2,3\}$ 这个数列,都减去均值 $2$ 之后的数列 $\{-1,0,1\}$ 的方差不还是一样约等于 $0.67$ 吗? 这里其实犯了一个错误,就是我们讲的基线是指在同一个状态下的基线,而不是在整个数列上的均值,这里的基线是指 $V^\pi(s_t)$,而不是 $V^\pi(s)$ 的均值。 + +$\qquad$ 另一方面来讲,优势函数可以理解为在给定状态 $s_t$ 下,选择动作 $a_t$ 相对于平均水平的优势。如果优势为正,则说明选择这个动作比平均水平要好,反之如果为负则说明选择这个动作比平均水平要差。换句话说,原先对于每一个状态-动作对只能以自己为参照物估计,现在可以平均水平为参照物估计了,这样就能减少方差。 -另一方面来讲,优势函数可以理解为在给定状态 $s_t$ 下,选择动作 $a_t$ 相对于平均水平的优势。如果优势为正,则说明选择这个动作比平均水平要好,反之如果为负则说明选择这个动作比平均水平要差。换句话说,原先对于每一个状态-动作对只能以自己为参照物估计,现在可以平均水平为参照物估计了,这样就能减少方差。这就好比我们练习马拉松,原先的做法是我们只关注于每天跑了多少,并不知道之前几天跑了多少,这很容易导致我们盲目追求每天跑动的距离,而忽略了自己的身体状况,导致受伤,也就得到了一个较差的跑步策略。而引入优势函数之后我们就可以知道之前几天的平均跑步距离,这样就能更好的控制自己的身体状况,避免受伤,并且更好地达到马拉松的目标。 +$\qquad$ 这就好比我们练习马拉松,原先的做法是我们只关注于每天跑了多少,并不知道之前几天跑了多少,这很容易导致我们盲目追求每天跑动的距离,而忽略了自己的身体状况,导致受伤,也就得到了一个较差的跑步策略。而引入优势函数之后我们就可以知道之前几天的平均跑步距离,这样就能更好的控制自己的身体状况,避免受伤,并且更好地达到马拉松的目标。 -有了优势函数之后,我们就可以将目标函数写成: +$\qquad$ 有了优势函数之后,我们就可以将目标函数写成式 $\text(10.4)$ 的形式。 $$ +\tag{10.4} \begin{aligned} \nabla_\theta J(\theta) \propto \mathbb{E}_{\pi_{\theta}}\left[A^\pi(s_t, a_t) \nabla_\theta \log \pi_\theta(a_t \mid s_t)\right] \end{aligned} $$ -这就是 `Advantage Actor-Critic` 算法,通常简称为 `A2C` 算法。然而 `A2C` 算法并不是由一篇单独的论文提出来的,而是在 `Asynchronous Advantage Actor-Critic` 算法(简称 `A3C`)的论文中提到的。`A3C` 算法,顾名思义就是异步的 `A2C` 算法,在算法原理上跟 `A2C` 算法是一模一样的,只是引入了多进程的概念提高了训练效率。 +$\qquad$ 这就是 $\text{Advantage Actor-Critic}$ 算法,通常简称为 $\text{A2C}$ 算法。然而 $\text{A2C}$ 算法并不是由一篇单独的论文提出来的,而是在异步形式的 $\text{A2C}$ 算法( $\text{asynchronous advantage actor-critic,A3C}$ )的论文中提出来的。它在算法原理上跟 $\text{A2C}$ 算法是一模一样的,只是引入了多进程的概念提高了训练效率。 + +$\qquad$ 如图 $\text{10.2}$ 所示,原先的 $\text{A2C}$ 算法相当于只有一个全局网络并持续与环境交互更新。而 $\text{A3C}$ 算法中增加了多个进程,每一个进程都拥有一个独立的网络和环境以供交互,并且每个进程每隔一段时间都会将自己的参数同步到全局网络中,这样就能提高训练效率。这种训练模式也是比较常见的多进程训练模式,也能用于其他算法中,也包括前面讲到的基于价值的算法。
-
图 10.2 A3C 算法架构
- -如图所示,原先的 `A2C` 算法相当于只有一个全局网络并持续与环境交互更新。而 `A3C` 算法中增加了多个进程,每一个进程都拥有一个独立的网络和环境以供交互,并且每个进程每隔一段时间都会将自己的参数同步到全局网络中,这样就能提高训练效率。这种训练模式也是比较常见的多进程训练模式,也能用于其他算法中,也包括前面讲到的基于价值的算法。 +
图 $\text{10.2}$ $\text{A3C}$ 算法架构
## 广义优势估计 -上一小节中,我们通过引入优势函数来缓解梯度估计带来的高方差问题,但由于优势函数通本质上来说还是使用蒙特卡洛估计,因此尽管减去了基线,有时候还是会产生高方差,从而导致训练过程不稳定。这时候有读者可能会想到一句话,即“知识一般是通过螺旋式的规律来学习的,也是会螺旋式升级的”,这句话的意思是我们在学某些知识时可能不会马上用到,但是会暂时埋下一个种子,等到后面深入使用的时候会回忆起来并且加深相关知识的理解。当然这句话不是某个名人说的,而是笔者自己总结出来,也是想传达给读者的学习思路。回到正题,既然蒙特卡洛估计一定会带来高方差问题,那么读者可能回到想到前面章节中在讲到蒙特卡洛和时序差分方法的差异时,会发现这两个方法是互补的,时序差分能有效解决高方差问题但是是有偏估计,而蒙特卡洛是无偏估计但是会带来高方差问题,因此通常会结合这两个方法形成一种新的估计方式,即 TD($\lambda$) 估计。类似地,在这里我们也可以引入 $\lambda$,结合多步(n-step)的折扣回报来改进优势函数,形成一种新的估计方式,我们称之为广义优势估计(Generalized Advantage Estimation,简称 GAE)。公式如下: +$\qquad$ 上一小节中,我们通过引入优势函数来缓解梯度估计带来的高方差问题,但由于优势函数通本质上来说还是使用蒙特卡洛估计,因此尽管减去了基线,有时候还是会产生高方差,从而导致训练过程不稳定。这时候有读者可能会想到一句话,即“知识一般是通过螺旋式的规律来学习的,也是会螺旋式升级的”,这句话的意思是我们在学某些知识时可能不会马上用到,但是会暂时埋下一个种子,等到后面深入使用的时候会回忆起来并且加深相关知识的理解。当然这句话不是某个名人说的,而是笔者自己总结出来,也是想传达给读者的学习思路。 + +$\qquad$ 回到正题,既然蒙特卡洛估计一定会带来高方差问题,那么读者可能回到想到前面章节中在讲到蒙特卡洛和时序差分方法的差异时,会发现这两个方法是互补的,时序差分能有效解决高方差问题但是是有偏估计,而蒙特卡洛是无偏估计但是会带来高方差问题,因此通常会结合这两个方法形成一种新的估计方式,即 $\text{TD}(\lambda)$ 估计。类似地,在这里我们也可以引入 $\lambda$ ,结合多步( $\text{n-step}$ )的折扣回报来改进优势函数,形成一种新的估计方式,我们称之为广义优势估计( $\text{generalized advantage estimation,GAE}$ ),其形式如式 $\text(10.5)$ 所示。 $$ +\tag{10.5} \begin{aligned} A^{\mathrm{GAE}(\gamma, \lambda)}(s_t, a_t) &= \sum_{l=0}^{\infty}(\gamma \lambda)^l \delta_{t+l} \\ &= \sum_{l=0}^{\infty}(\gamma \lambda)^l \left(r_{t+l} + \gamma V^\pi(s_{t+l+1}) - V^\pi(s_{t+l})\right) \end{aligned} $$ -其中 $\delta_{t+l}$ 表示时间步 $t+l$ 时的 TD 误差,即: +其中 $\delta_{t+l}$ 表示时间步 $t+l$ 时的 $\text{TD}$ 误差,如式 $\text(10.6)$ 所示。 $$ +\tag{10.6} \begin{aligned} \delta_{t+l} = r_{t+l} + \gamma V^\pi(s_{t+l+1}) - V^\pi(s_{t+l}) \end{aligned} $$ -当 $\lambda = 0$ 时,GAE 退化为单步 TD 误差,即: +当 $\lambda = 0$ 时,GAE 退化为单步 $\text{TD}$ 误差,如式 $\text(10.7)$ 所示。 $$ +\tag{10.7} \begin{aligned} A^{\mathrm{GAE}(\gamma, 0)}(s_t, a_t) = \delta_t = r_t + \gamma V^\pi(s_{t+1}) - V^\pi(s_t) \end{aligned} $$ -当 $\lambda = 1$ 时,GAE 退化为蒙特卡洛估计,即: +当 $\lambda = 1$ 时,$\text{GAE}$ 退化为蒙特卡洛估计,如式 $\text(10.8)$ 所示。 $$ +\tag{10.8} \begin{aligned} A^{\mathrm{GAE}(\gamma, 1)}(s_t, a_t) = \sum_{l=0}^{\infty}(\gamma \lambda)^l \delta_{t+l} = \sum_{l=0}^{\infty}(\gamma)^l \delta_{t+l} \end{aligned} $$ -如何选择合适的 $\lambda$ 还请读者回看前面时序差分的相关章节内容,这里就不再赘述。到这里,我们就将 `Actor-Critic` 算法的基本原理讲完了,注意广义优势估计并不是 `Actor-Critic` 算法的必要组成部分,只是一种改进的方法。相反地,它更像是一种通用的模块,在实践中可以用在任何需要估计优势函数的地方,比如后面章节要讲的 `PPO` 算法中就用到了这种估计方法。 +如何选择合适的 $\lambda$ 还请读者回看前面时序差分的相关章节内容,这里就不再赘述。到这里,我们就将 $\text{Actor-Critic}$ 算法的基本原理讲完了,注意广义优势估计并不是 $\text{Actor-Critic}$ 算法的必要组成部分,只是一种改进的方法。相反地,它更像是一种通用的模块,在实践中可以用在任何需要估计优势函数的地方,比如后面章节要讲的 $\text{PPO}$ 算法中就用到了这种估计方法。 ## 实战:A2C 算法 diff --git a/docs/ch11/main.md b/docs/ch11/main.md index 10772cf..c286179 100644 --- a/docs/ch11/main.md +++ b/docs/ch11/main.md @@ -1,34 +1,34 @@ # DDPG 与 TD3 算法 -本章开始到接下来的几章,我们将介绍一些非常典型的基于策略梯度的算法,包括 `DDPG`、`PPO`、`SAC` 等。这些方法的实现方式各不相同,也各有各的特点,因此每个算法都单独形成一个章节来展开。同时它们是目前实践中最最常用的一些算法,甚至有 “遇事不决PPO” 之类的说法,因此读者一定要认真学习。那么本章将 `DDPG` 算法作为开篇,严格来说,原论文作者提出 `DDPG` 算法的初衷其实是 `DQN` 算法的一个扩展,或者说一种连续动作版本的 `DQN` 算法。但是由于当时 `Actor-Critic` 架构还没有被提出(`A3C` 算法是在 2016 年发表的,比 `DDPG` 算法晚了一年),只是后来我们回看 `DDPG` 算法的时候发现其在形式上更像 `Actor-Critic` 的架构,因此我们就将其归为 `Actor-Critic` 算法的一种。 +本章开始到接下来的几章,我们将介绍一些非常典型的基于策略梯度的算法,包括 $\text{DDPG}$、$\text{PPO}$、$\text{SAC}$ 等。这些方法的实现方式各不相同,也各有各的特点,因此每个算法都单独形成一个章节来展开。同时它们是目前实践中最最常用的一些算法,甚至有 “遇事不决PPO” 之类的说法,因此读者一定要认真学习。那么本章将 $\text{DDPG}$ 算法作为开篇,严格来说,原论文作者提出 $\text{DDPG}$ 算法的初衷其实是 $\text{DQN}$ 算法的一个扩展,或者说一种连续动作版本的 $\text{DQN}$ 算法。但是由于当时 $\text{Actor-Critic}$ 架构还没有被提出( $\text{A3C}$ 算法是在 $\text{2016}$ 年发表的,比 $\text{DDPG}$ 算法晚了一年),只是后来我们回看 $\text{DDPG}$ 算法的时候发现其在形式上更像 $\text{Actor-Critic}$ 的架构,因此我们就将其归为 $\text{Actor-Critic}$ 算法的一种。 ## DPG 方法 -`DDPG` 算法,英文全称为 `Deep Deterministic Policy Gradient` ,中文全称为深度确定性策略梯度算法,是一种确定性的策略梯度算法。为了让读者更好地理解 `DDPG` 算法,我们先把“深度”这两个字去掉,即先介绍一下 `DPG` 算法,后面我们也会发现 `DPG` 算法其实就是 `DDPG` 算法的核心,也就是精髓所在。 +深度确定性策略梯度算法( $\text{deep deterministic policy gradient,DDPG}$),是一种确定性的策略梯度算法。为了让读者更好地理解 $\text{DDPG}$ 算法,我们先把 “深度“ 这两个字去掉,即先介绍一下 $\text{DPG}$ 算法,后面我们也会发现 $\text{DPG}$ 算法其实就是 $\text{DDPG}$ 算法的核心,也就是精髓所在。 -虽然 `DDPG` 原论文作者是基于 `DQN` 算法来展开的,目前很多相关资料也是如此。但是笔者认为,有了 `Actor-Critic` 算法的铺垫之后,从策略梯度的角度来理解 DPG 算法是更容易的。首先我们知道 `DQN` 算法的一个主要缺点就是不能用于连续动作空间,这是因为在 `DQN` 算法中动作是通过贪心策略或者说 `argmax` 的方式来从 `Q` 函数间接得到的。要想适配连续动作空间,我们干脆就将选择动作的过程变成一个直接从状态映射到具体动作的函数 $\mu_\theta (s)$,其中 $\theta$ 表示模型的参数,这样一来就把求解 `Q` 函数、贪心选择动作这两个过程合并成了一个函数,也就是我们常说的 `Actor`。有读者可能会问,为什么这里用 $\mu_\theta (s)$ 来表示策略而不是 `Actor-Critic` 章节中提到的 $\pi_{\theta}(a|s)$ 呢?注意,这里 $\mu_\theta (s)$ 输出的是一个值,而 $\pi_{\theta}(a|s)$ 通常输出的是一个概率分布,这是两者的本质区别,与输出概率分布的随机性策略(`Stochastic Policy`)不同,这里输出一个值的策略我们就称作确定性策略(`Deterministic Policy`)。有了这个策略函数之后,类似地,我们也可以推导出 `DPG` 算法的目标函数,也就是策略梯度公式,如下: +虽然 $\text{DDPG}$ 原论文作者是基于 $\text{DQN}$ 算法来展开的,目前很多相关资料也是如此。但是笔者认为,有了 $\text{Actor-Critic}$ 算法的铺垫之后,从策略梯度的角度来理解 DPG 算法是更容易的。首先我们知道 $\text{DQN}$ 算法的一个主要缺点就是不能用于连续动作空间,这是因为在 $\text{DQN}$ 算法中动作是通过贪心策略或者说 `argmax` 的方式来从 $Q$ 函数间接得到的。要想适配连续动作空间,我们干脆就将选择动作的过程变成一个直接从状态映射到具体动作的函数 $\mu_\theta (s)$,其中 $\theta$ 表示模型的参数,这样一来就把求解 $Q$ 函数、贪心选择动作这两个过程合并成了一个函数,也就是我们常说的 `Actor`。有读者可能会问,为什么这里用 $\mu_\theta (s)$ 来表示策略而不是 $\text{Actor-Critic}$ 章节中提到的 $\pi_{\theta}(a|s)$ 呢?注意,这里 $\mu_\theta (s)$ 输出的是一个值,而 $\pi_{\theta}(a|s)$ 通常输出的是一个概率分布,这是两者的本质区别,与输出概率分布的随机性策略(`Stochastic Policy`)不同,这里输出一个值的策略我们就称作确定性策略( $\text{deterministic policy}$ )。有了这个策略函数之后,类似地,我们也可以推导出 $\text{DPG}$ 算法的目标函数,也就是策略梯度公式,如下: $$ \nabla_\theta J(\theta) \approx \mathbb{E}_{s_t \sim \rho^\beta}\left[\left.\nabla_a Q\left(s_t, a\right)\right|_{a=\mu_\theta\left(s_t\right)} \nabla_\theta \mu_\theta\left(s_t\right)\right] $$ -其中 $\rho^\beta$ 是策略的初始分布,用于探索状态空间,在实际应用中相当于网络模型的初始参数,读者可以不用关注。另外注意这里的 $Q(s_t, a)$ 表示的不是 `Q`函数,跟 `Actor-Critic` 算法一样,即作为一个 `Critic` 网络,将状态和动作作为输入,并且输出一个值。`DDPG` 策略梯度公式的推导过程与 `Actor-Critic` 算法的推导过程虽然有所区别,但整体上是非常相似的,感兴趣的读者可以查询相关资料或者自行推导一下,这里就不再赘述了。 +其中 $\rho^\beta$ 是策略的初始分布,用于探索状态空间,在实际应用中相当于网络模型的初始参数,读者可以不用关注。另外注意这里的 $Q(s_t, a)$ 表示的不是 $Q$函数,跟 $\text{Actor-Critic}$ 算法一样,即作为一个 $\text{Critic}$ 网络,将状态和动作作为输入,并且输出一个值。$\text{DDPG}$ 策略梯度公式的推导过程与 $\text{Actor-Critic}$ 算法的推导过程虽然有所区别,但整体上是非常相似的,感兴趣的读者可以查询相关资料或者自行推导一下,这里就不再赘述了。 ## DDPG 算法 -有了 `DPG` 算法的铺垫之后,我们再来看看 `DDPG` 算法。`DDPG` 算法的 “深度” 其实很简单,它就体现在于 `DPG` 算法中使用 `Actor` 和 `Critic` 两个网络来分别表示策略函数(确定性策略)和值函数,它们可以是线性模型或浅层神经网络。而DDPG算法采用深度神经网络来近似 `Actor` 和`Critic`,使得算法可以处理更复杂的任务和高维的状态空间。用现在更流行的话讲,`DDPG` 算法同 `DPG` 算法相比,就是使用了更大的模型。除此之外,`DDPG` 算法还有一些其他的改进,例如使用了经验回放、目标网络等 `DQN` 算法中的一些技巧。因此,`DDPG` 算法更像是将 `DPD` 思路引入到了 `DQN` 算法中,而不是简单的将 `DPG` 算法扩展到了深度神经网络上。另外,`DDPG` 算法还有一项重要的贡献,就是引入了噪声网络来增加策略的探索性,即在训练过程中采用了一种确定性策略加噪声的方式。即在Actor网络的输出动作上加入一定的噪声,使得策略在训练过程中具有一定的探索性,有助于探索更广泛的动作空间。 +有了 $\text{DPG}$ 算法的铺垫之后,我们再来看看 $\text{DDPG}$ 算法。$\text{DDPG}$ 算法的 “深度” 其实很简单,它就体现在于 $\text{DPG}$ 算法中使用 `Actor` 和 $\text{Critic}$ 两个网络来分别表示策略函数(确定性策略)和值函数,它们可以是线性模型或浅层神经网络。而DDPG算法采用深度神经网络来近似 `Actor` 和$\text{Critic}$,使得算法可以处理更复杂的任务和高维的状态空间。用现在更流行的话讲,$\text{DDPG}$ 算法同 $\text{DPG}$ 算法相比,就是使用了更大的模型。除此之外,$\text{DDPG}$ 算法还有一些其他的改进,例如使用了经验回放、目标网络等 $\text{DQN}$ 算法中的一些技巧。因此,$\text{DDPG}$ 算法更像是将 $\text{DPG}$ 思路引入到了 $\text{DQN}$ 算法中,而不是简单的将 $\text{DPG}$ 算法扩展到了深度神经网络上。另外,$\text{DDPG}$ 算法还有一项重要的贡献,就是引入了噪声网络来增加策略的探索性,即在训练过程中采用了一种确定性策略加噪声的方式。即在Actor网络的输出动作上加入一定的噪声,使得策略在训练过程中具有一定的探索性,有助于探索更广泛的动作空间。 -这里读者顺便插一句题外话,回顾到目前为止的所有章节内容,不知道读者有没有发现,其实我们发现在强化学习基础算法的研究改进当中,无外乎几个亘古不变的主题:一是**如何提高对值函数的估计**,保证其准确性,即尽量无偏且低方差,例如最开始的用深度神经网络替代简单的 `Q` 表、结合蒙特卡洛和时序差分的 $TD(\lambda)$ 、引入目标网络以及广义优势估计等等;二是**如何提高探索以及平衡探索-利用的问题**,尤其在探索性比较差的确定性策略中,例如`DQN`和`DDPG`算法都会利用各种技巧来提高探索,例如经验回放、$\varepsilon$-greedy 策略、噪声网络等等。这两个问题是强化学习算法的基础核心问题,希望能够给读者在学习和研究的过程中带来一定的启发。 +这里读者顺便插一句题外话,回顾到目前为止的所有章节内容,不知道读者有没有发现,其实我们发现在强化学习基础算法的研究改进当中,无外乎几个亘古不变的主题:一是**如何提高对值函数的估计**,保证其准确性,即尽量无偏且低方差,例如最开始的用深度神经网络替代简单的 $Q$ 表、结合蒙特卡洛和时序差分的 $TD(\lambda)$ 、引入目标网络以及广义优势估计等等;二是**如何提高探索以及平衡探索-利用的问题**,尤其在探索性比较差的确定性策略中,例如$\text{DQN}$和$\text{DDPG}$算法都会利用各种技巧来提高探索,例如经验回放、$\varepsilon$-greedy 策略、噪声网络等等。这两个问题是强化学习算法的基础核心问题,希望能够给读者在学习和研究的过程中带来一定的启发。 -回到正题,由于目标网络、经验回放前面章节都讲过了,这里就略过,我们讲讲 `DDPG` 引入的噪声。其实引入噪声的方法在前面 `Noisy DQN` 算法中就讲到了,只是 `Noisy DQN` 算法是在网络中引入噪声,而 `DDPG` 算法是在输出动作上引入噪声。本质上来讲,引入噪声的作用就是为了在不破坏系统的前提下,提高系统运行的抗干扰性。这跟我们生活中打疫苗是类似的,通常我们会将灭活的病毒也就是疫苗注入到体内,引发免疫系统的警觉,从而提高免疫系统的抗干扰性,即提高我们身体的免疫力。这里疫苗就相当于轻微的噪声,如果免疫系统一直没见过这种噪声,那么一旦遇到真正的病毒之后是很有可能崩溃的,反之如果经常接触这种轻微的噪声,那么免疫系统就会逐渐适应,从而提高抗干扰性。又好比我们平时做的消防演练,虽然平时的演练都不是真正意义上的灾害,但经过熟练的演练之后一旦遇到真正的灾害不说从容应对,至少也不会过于慌乱了。再次回归正题,`DDPG` 算法是在输出动作上引入噪声的,由于 $\mu_\theta (s)$ 输出的是单个值,其实最简单的方式就是在输出的值上加上一个随机数,这个随机数可以是正态分布的(即高斯噪声),也可以是均匀分布的,只要能够保证这个随机数的值不要过大就行。 +回到正题,由于目标网络、经验回放前面章节都讲过了,这里就略过,我们讲讲 $\text{DDPG}$ 引入的噪声。其实引入噪声的方法在前面 `Noisy DQN` 算法中就讲到了,只是 `Noisy DQN` 算法是在网络中引入噪声,而 $\text{DDPG}$ 算法是在输出动作上引入噪声。本质上来讲,引入噪声的作用就是为了在不破坏系统的前提下,提高系统运行的抗干扰性。这跟我们生活中打疫苗是类似的,通常我们会将灭活的病毒也就是疫苗注入到体内,引发免疫系统的警觉,从而提高免疫系统的抗干扰性,即提高我们身体的免疫力。这里疫苗就相当于轻微的噪声,如果免疫系统一直没见过这种噪声,那么一旦遇到真正的病毒之后是很有可能崩溃的,反之如果经常接触这种轻微的噪声,那么免疫系统就会逐渐适应,从而提高抗干扰性。又好比我们平时做的消防演练,虽然平时的演练都不是真正意义上的灾害,但经过熟练的演练之后一旦遇到真正的灾害不说从容应对,至少也不会过于慌乱了。再次回归正题,$\text{DDPG}$ 算法是在输出动作上引入噪声的,由于 $\mu_\theta (s)$ 输出的是单个值,其实最简单的方式就是在输出的值上加上一个随机数,这个随机数可以是正态分布的(即高斯噪声),也可以是均匀分布的,只要能够保证这个随机数的值不要过大就行。 -当然简单的噪声引入除了简单这一个优点之外,可能剩下的全都是缺点了,因此在 `DDPG` 算法中使用的其实是一种叫做 `Ornstein-Uhlenbeck` 的噪声,简称 `OU` 噪声。`OU` 噪声是一种具有回归特性的随机过程,其与高斯噪声相比的优点在于: +当然简单的噪声引入除了简单这一个优点之外,可能剩下的全都是缺点了,因此在 $\text{DDPG}$ 算法中使用的其实是一种叫做 `Ornstein-Uhlenbeck` 的噪声,简称 `OU` 噪声。`OU` 噪声是一种具有回归特性的随机过程,其与高斯噪声相比的优点在于: * **探索性**:`OU` 噪声具有持续的、自相关的特性。相比于独立的高斯噪声,`OU` 噪声更加平滑,并且在训练过程中更加稳定。这种平滑特性使得OU噪声有助于探索更广泛的动作空间,并且更容易找到更好的策略。 -* **控制幅度**:`OU` 噪声可以通过调整其参数来控制噪声的幅度。在 `DDPG` 算法中,可以通过调整 `OU` 噪声的方差来控制噪声的大小,从而平衡探索性和利用性。较大的方差会增加探索性,而较小的方差会增加利用性。 -* **稳定性**:`OU` 噪声的回归特性使得噪声在训练过程中具有一定的稳定性。相比于纯粹的随机噪声,在 `DDPG` 算法中使用`OU` 噪声可以更好地保持动作的连续性,避免剧烈的抖动,从而使得训练过程更加平滑和稳定。 +* **控制幅度**:`OU` 噪声可以通过调整其参数来控制噪声的幅度。在 $\text{DDPG}$ 算法中,可以通过调整 `OU` 噪声的方差来控制噪声的大小,从而平衡探索性和利用性。较大的方差会增加探索性,而较小的方差会增加利用性。 +* **稳定性**:`OU` 噪声的回归特性使得噪声在训练过程中具有一定的稳定性。相比于纯粹的随机噪声,在 $\text{DDPG}$ 算法中使用`OU` 噪声可以更好地保持动作的连续性,避免剧烈的抖动,从而使得训练过程更加平滑和稳定。 * **可控性**:由于`OU` 噪声具有回归特性,它在训练过程中逐渐回归到均值,因此可以控制策略的探索性逐渐减小。这种可控性使得在训练的早期增加探索性,然后逐渐减小探索性,有助于更有效地进行训练。 -总的来说,`OU` 噪声作为 `DDPG` 算法中的一种探索策略,具有平滑、可控、稳定等优点,使得算法能够更好地在连续动作空间中进行训练,探索更广泛的动作空间,并找到更优的策略。它是 `DDPG` 算法成功应用于连续动作空间问题的重要因素之一。虽然它有这么多的优点,实际上在简单的环境中,它跟使用简单的高斯噪声甚至不用噪声的效果是差不多的,只有在复杂的环境中才会体现出来区别。因此,如果读者在实际应用中面临的问题比较简单,可以不用OU噪声,而是使用高斯噪声或者不用噪声即可,这样可以减少算法的复杂度,加快算法的收敛速度,正所谓 “杀鸡焉用牛刀”。 +总的来说,`OU` 噪声作为 $\text{DDPG}$ 算法中的一种探索策略,具有平滑、可控、稳定等优点,使得算法能够更好地在连续动作空间中进行训练,探索更广泛的动作空间,并找到更优的策略。它是 $\text{DDPG}$ 算法成功应用于连续动作空间问题的重要因素之一。虽然它有这么多的优点,实际上在简单的环境中,它跟使用简单的高斯噪声甚至不用噪声的效果是差不多的,只有在复杂的环境中才会体现出来区别。因此,如果读者在实际应用中面临的问题比较简单,可以不用OU噪声,而是使用高斯噪声或者不用噪声即可,这样可以减少算法的复杂度,加快算法的收敛速度,正所谓 “杀鸡焉用牛刀”。 `OU` 噪声主要由两个部分组成:随机高斯噪声和回归项,其数学定义如下: @@ -36,11 +36,11 @@ $$ d x_t=\theta\left(\mu-x_t\right) d t+\sigma d W_t $$ -其中 $x_t$ 是OU过程在时间 $t$ 的值,即当前的噪声值,这个 $t$ 也是强化学习中的时步 (time step)。$\mu$ 是回归到的均值,表示噪声在长时间尺度上的平均值。$\theta$ 是 `OU` 过程的回归速率,表示噪声向均值回归的速率。$\sigma$ 是 `OU` 过程的扰动项,表示随机高斯噪声的标准差。$dW_t$ 是布朗运动(Brownian motion)或者维纳过程(Wiener process),是一个随机项,表示随机高斯噪声的微小变化。在实际应用中,我们只需要调整 $\mu$ 和 $\sigma$ 就可以了,$\theta$ 通常是固定的,而 $dW_t$ 是随机项,我们也不需要关注。尽管如此,需要调整的参数还是有点多,这也是为什么 `DDPG` 算法的调参比较麻烦的原因之一。 +其中 $x_t$ 是OU过程在时间 $t$ 的值,即当前的噪声值,这个 $t$ 也是强化学习中的时步 (time step)。$\mu$ 是回归到的均值,表示噪声在长时间尺度上的平均值。$\theta$ 是 `OU` 过程的回归速率,表示噪声向均值回归的速率。$\sigma$ 是 `OU` 过程的扰动项,表示随机高斯噪声的标准差。$dW_t$ 是布朗运动(Brownian motion)或者维纳过程(Wiener process),是一个随机项,表示随机高斯噪声的微小变化。在实际应用中,我们只需要调整 $\mu$ 和 $\sigma$ 就可以了,$\theta$ 通常是固定的,而 $dW_t$ 是随机项,我们也不需要关注。尽管如此,需要调整的参数还是有点多,这也是为什么 $\text{DDPG}$ 算法的调参比较麻烦的原因之一。 ## DDPG 算法的优缺点 -总的来说,`DDPG` 算法的优点主要有: +总的来说,$\text{DDPG}$ 算法的优点主要有: * **适用于连续动作空间**:DDPG算法采用了确定性策略来选择动作,这使得它能够直接处理连续动作空间的问题。相比于传统的随机策略,确定性策略更容易优化和学习,因为它不需要进行动作采样,h缓解了在连续动作空间中的高方差问题。 * **高效的梯度优化**:DDPG算法使用策略梯度方法进行优化,其梯度更新相对高效,并且能够处理高维度的状态空间和动作空间。同时,通过Actor-Critic结构,算法可以利用值函数来辅助策略的优化,提高算法的收敛速度和稳定性。 @@ -50,23 +50,23 @@ $$ * **只适用于连续动作空间**:这既是优点,也是缺点。 -* **高度依赖超参数**:`DDPG` 算法中有许多超参数需要进行调整,除了一些 `DQN`的算法参数例如学习率、批量大小、目标网络的更新频率等,还需要调整一些 `OU` 噪声的参数 +* **高度依赖超参数**:$\text{DDPG}$ 算法中有许多超参数需要进行调整,除了一些 $\text{DQN}$的算法参数例如学习率、批量大小、目标网络的更新频率等,还需要调整一些 `OU` 噪声的参数 调整这些超参数并找到最优的取值通常是一个挑战性的任务,可能需要大量的实验和经验。 -* **高度敏感的初始条件**:`DDPG` 算法对初始条件非常敏感。初始策略和值函数的参数设置可能会影响算法的收敛性和性能,需要仔细选择和调整。 +* **高度敏感的初始条件**:$\text{DDPG}$ 算法对初始条件非常敏感。初始策略和值函数的参数设置可能会影响算法的收敛性和性能,需要仔细选择和调整。 * 容易陷入局部最优:由于采用了确定性策略,可能会导致算法陷入局部最优,难以找到全局最优策略。为了增加探索性,需要采取一些措施,如加入噪声策略或使用其他的探索方法。 -可以看到,`DDPG` 算法的优点可能掩盖不了它众多的缺点,尤其对于初学者来说,调参是一个非常头疼的问题,因此在实际应用中,同样情况下可能会选择更加简单的 `PPO` 算法会来得更加容易一些。当然,对于一些熟练的调参大侠来说,`DDPG` 算法以及相关改进版本的算法也是值得尝试的,毕竟它们在实际应用中的效果还是非常不错的。 +可以看到,$\text{DDPG}$ 算法的优点可能掩盖不了它众多的缺点,尤其对于初学者来说,调参是一个非常头疼的问题,因此在实际应用中,同样情况下可能会选择更加简单的 $\text{PPO}$ 算法会来得更加容易一些。当然,对于一些熟练的调参大侠来说,$\text{DDPG}$ 算法以及相关改进版本的算法也是值得尝试的,毕竟它们在实际应用中的效果还是非常不错的。 ## TD3 算法 -我们知道 `DDPG` 算法的缺点太多明显,因此后来有人对其进行了改进,这就是我们接下来要介绍的 `TD3` 算法。`TD3` 算法,英文全称为 `Twin Delayed DDPG`,中文全称为双延迟确定性策略梯度算法。相对于 `DDPG` 算法,`TD3` 算法的改进主要做了三点重要的改进,一是 双 `Q` 网络,体现在名字中的 `Twin`,二是 延迟更新,体现在名字中的 `Delayed`,三是 躁声正则(`Noise Regularisation`),下面我们分别来看一下。 +我们知道 $\text{DDPG}$ 算法的缺点太多明显,因此后来有人对其进行了改进,这就是我们接下来要介绍的 `TD3` 算法。`TD3` 算法,英文全称为 `Twin Delayed DDPG`,中文全称为双延迟确定性策略梯度算法。相对于 $\text{DDPG}$ 算法,`TD3` 算法的改进主要做了三点重要的改进,一是 双 $Q$ 网络,体现在名字中的 `Twin`,二是 延迟更新,体现在名字中的 `Delayed`,三是 躁声正则(`Noise Regularisation`),下面我们分别来看一下。 ### 双 Q 网络 -双 `Q` 网络的思想其实很简单,就是在 `DDPG` 算法中的 `Critic` 网络上再加一层,这样就形成了两个 `Critic` 网络,分别记为 $Q_{\omega_1}$ 和 $Q_{\omega_2}$,其中 $\omega_1$ 和 $\omega_2$ 分别表示两个网络的参数。这样一来,我们就可以得到两个 `Q` 值,分别记为 $Q_{\omega_1}(s_t, a_t)$ 和 $Q_{\omega_2}(s_t, a_t)$, -然后我们在计算 `TD` 误差的时候,就可以取两个 `Q` 值中较小的那个,即: +双 $Q$ 网络的思想其实很简单,就是在 $\text{DDPG}$ 算法中的 $\text{Critic}$ 网络上再加一层,这样就形成了两个 $\text{Critic}$ 网络,分别记为 $Q_{\omega_1}$ 和 $Q_{\omega_2}$,其中 $\omega_1$ 和 $\omega_2$ 分别表示两个网络的参数。这样一来,我们就可以得到两个 $Q$ 值,分别记为 $Q_{\omega_1}(s_t, a_t)$ 和 $Q_{\omega_2}(s_t, a_t)$, +然后我们在计算 `TD` 误差的时候,就可以取两个 $Q$ 值中较小的那个,即: $$ \begin{aligned} @@ -75,7 +75,7 @@ y_t &= r_{t+1} + \gamma \min_{i=1,2} Q_{\omega_i}(s_{t+1}, \mu_{\theta}(s_{t+1}) \end{aligned} $$ -其中 $L$ 表示损失函数,同理,我们也可以得到另一个 `Critic` 网络的损失函数 $L(\omega_2)$,即: +其中 $L$ 表示损失函数,同理,我们也可以得到另一个 $\text{Critic}$ 网络的损失函数 $L(\omega_2)$,即: $$ \begin{aligned} @@ -84,15 +84,15 @@ y_t &= r_{t+1} + \gamma \min_{i=1,2} Q_{\omega_i}(s_{t+1}, \mu_{\theta}(s_{t+1}) \end{aligned} $$ -细心的读者会发现,这跟 `Double DQN` 的原理是一样的,只不过 `Double DQN` 是在 `Q` 网络上做文章,而 `TD3` 算法是在 `Critic` 网络上做文章。这样做的好处是可以减少 `Q` 值的过估计,从而提高算法的稳定性和收敛性。 +细心的读者会发现,这跟 `Double DQN` 的原理是一样的,只不过 `Double DQN` 是在 $Q$ 网络上做文章,而 `TD3` 算法是在 $\text{Critic}$ 网络上做文章。这样做的好处是可以减少 $Q$ 值的过估计,从而提高算法的稳定性和收敛性。 ### 延迟更新 -延迟更新更像是一种实验技巧,即在训练中 `Actor` 的更新频率要低于 `Critic` 的更新频率。在学习过程中,`Critic` 是不断更新的,可以想象一下,假设在某个时刻 `Actor` 好不容易达到一个最高点,这个时候 `Critic` 又更新了,那么 `Actor` 的最高点就被打破了,这样一来 `Actor` 就会不断地追逐 `Critic`,这样就会造成误差的过分累积,进而导致 `Actor` 的训练不稳定,甚至可能会发散。因此,为了避免这种情况,我们就可以在训练中让 `Actor` 的更新频率低于 `Critic` 的更新频率,这样一来 `Actor` 的更新就会比较稳定,不会受到 `Critic` 的影响,从而提高算法的稳定性和收敛性。举个不太恰当的例子,`Critic` 就好比领导,`Actor`则好比员工,领导不断给员工下达目标,员工不断地去完成目标,如果领导的决策经常失误,那么员工就很容易像无头苍蝇一样不知道该完成哪些目标。因此,一个好的解决方式就是让领导学得比员工更快,领导学得更快了之后下达目标的失误率就会更低,这样员工就能够更好地完成目标了,从而提高整个团队的效率。在实践中,`Actor` 的更新频率一般要比 `Critic` 的更新频率低一个数量级,例如 `Critic` 每更新 10 次,`Actor` 只更新 1 次。 +延迟更新更像是一种实验技巧,即在训练中 `Actor` 的更新频率要低于 $\text{Critic}$ 的更新频率。在学习过程中,$\text{Critic}$ 是不断更新的,可以想象一下,假设在某个时刻 `Actor` 好不容易达到一个最高点,这个时候 $\text{Critic}$ 又更新了,那么 `Actor` 的最高点就被打破了,这样一来 `Actor` 就会不断地追逐 $\text{Critic}$,这样就会造成误差的过分累积,进而导致 `Actor` 的训练不稳定,甚至可能会发散。因此,为了避免这种情况,我们就可以在训练中让 `Actor` 的更新频率低于 $\text{Critic}$ 的更新频率,这样一来 `Actor` 的更新就会比较稳定,不会受到 $\text{Critic}$ 的影响,从而提高算法的稳定性和收敛性。举个不太恰当的例子,$\text{Critic}$ 就好比领导,`Actor`则好比员工,领导不断给员工下达目标,员工不断地去完成目标,如果领导的决策经常失误,那么员工就很容易像无头苍蝇一样不知道该完成哪些目标。因此,一个好的解决方式就是让领导学得比员工更快,领导学得更快了之后下达目标的失误率就会更低,这样员工就能够更好地完成目标了,从而提高整个团队的效率。在实践中,`Actor` 的更新频率一般要比 $\text{Critic}$ 的更新频率低一个数量级,例如 $\text{Critic}$ 每更新 10 次,`Actor` 只更新 1 次。 ### 躁声正则 -更准确地说,在原论文中不叫噪声正则,而是目标策略平滑正则化(`Target Policy Smoothing Regularization`),意思都是一样的,只是笔者在表述的时候做了简化。在延迟更新中,我们的主要思想是通过提高 `Critic` 的更新频率来减少值函数的估计误差,也就是“降低领导决策的失误率”。但是这样其实是治标不治本的做法,因为它只是让 `Critic` 带来的误差不要过分地影响到了 `Actor`,而没有考虑改进 `Critic` 本身的稳定性。因此,我们也可以给 `Critic` 引入一个噪声提高其抗干扰性,这样一来就可以在一定程度上提高 `Critic` 的稳定性,从而进一步提高算法的稳定性和收敛性。注意,这里的噪声是在 `Critic` 网络上引入的,而不是在输出动作上引入的,因此它跟 `DDPG` 算法中的噪声是不一样的。具体来说,我们可以在计算 `TD` 误差的时候,给目标值 $y$ 加上一个噪声,并且为了让噪声不至于过大,还增加了一个裁剪(clip),即: +更准确地说,在原论文中不叫噪声正则,而是目标策略平滑正则化(`Target Policy Smoothing Regularization`),意思都是一样的,只是笔者在表述的时候做了简化。在延迟更新中,我们的主要思想是通过提高 $\text{Critic}$ 的更新频率来减少值函数的估计误差,也就是“降低领导决策的失误率”。但是这样其实是治标不治本的做法,因为它只是让 $\text{Critic}$ 带来的误差不要过分地影响到了 `Actor`,而没有考虑改进 $\text{Critic}$ 本身的稳定性。因此,我们也可以给 $\text{Critic}$ 引入一个噪声提高其抗干扰性,这样一来就可以在一定程度上提高 $\text{Critic}$ 的稳定性,从而进一步提高算法的稳定性和收敛性。注意,这里的噪声是在 $\text{Critic}$ 网络上引入的,而不是在输出动作上引入的,因此它跟 $\text{DDPG}$ 算法中的噪声是不一样的。具体来说,我们可以在计算 `TD` 误差的时候,给目标值 $y$ 加上一个噪声,并且为了让噪声不至于过大,还增加了一个裁剪(clip),即: $$ y=r+\gamma Q_{\theta^{\prime}}\left(s^{\prime}, \pi_{\phi^{\prime}}\left(s^{\prime}\right)+\epsilon\right) \epsilon \sim \operatorname{clip}(N(0, \sigma),-c, c) diff --git a/docs/ch2/main.md b/docs/ch2/main.md index 9370a77..17130fd 100644 --- a/docs/ch2/main.md +++ b/docs/ch2/main.md @@ -4,7 +4,7 @@ ## 马尔可夫决策过程 -**马尔可夫决策过程(Markov decision process,MDP)是强化学习的基本问题模型之一**,它能够以数学的形式来描述智能体在与环境交互的过程中学到一个目标的过程。这里智能体充当的是作出决策或动作,并且在交互过程中学习的角色,环境指的是智能体与之交互的一切外在事物,不包括智能体本身。举个例子,比如我们要学习弹钢琴,在这个过程中充当决策者和学习者的我们人本身就是智能体,而我们的交互主体即钢琴就是环境。当我们执行动作也就是弹的时候会观测到一些信息,例如琴键的位置等,这就是状态。此外当我们弹下去的时候会收到钢琴发出的声音,也就是反馈,我们通过钢琴发出的声音来判断自己弹得好不好然后反思并纠正下一次弹的动作。当然如果这时候有一个钢琴教师在旁边指导我们,那样其实钢琴和教师就同时组成了环境,我们也可以交互过程中接收教师的反馈来提高自己的钢琴水平。 +**马尔可夫决策过程**( $\text{Markov decision process,MDP}$)是强化学习的基本问题模型之一,它能够以数学的形式来描述智能体在与环境交互的过程中学到一个目标的过程。这里智能体充当的是作出决策或动作,并且在交互过程中学习的角色,环境指的是智能体与之交互的一切外在事物,不包括智能体本身。举个例子,比如我们要学习弹钢琴,在这个过程中充当决策者和学习者的我们人本身就是智能体,而我们的交互主体即钢琴就是环境。当我们执行动作也就是弹的时候会观测到一些信息,例如琴键的位置等,这就是状态。此外当我们弹下去的时候会收到钢琴发出的声音,也就是反馈,我们通过钢琴发出的声音来判断自己弹得好不好然后反思并纠正下一次弹的动作。当然如果这时候有一个钢琴教师在旁边指导我们,那样其实钢琴和教师就同时组成了环境,我们也可以交互过程中接收教师的反馈来提高自己的钢琴水平。
diff --git a/docs/ch8/main.md b/docs/ch8/main.md index f8817ef..cb0167d 100644 --- a/docs/ch8/main.md +++ b/docs/ch8/main.md @@ -1,13 +1,13 @@ # DQN 算法进阶 -本章开始介绍一些基于 DQN 改进的一些算法,由于本章的算法都是网络模型、经验回放等层面的改进,整体的训练模式跟 DQN 算法是一模一样的,因此出于简洁考虑,本章算法不再贴出对应的伪代码。另一方面来讲,建议读者在阅读本章之前,牢牢掌握 DQN 算法的原理。 +本章开始介绍一些基于 $\text{DQN}$ 改进的一些算法,由于本章的算法都是网络模型、经验回放等层面的改进,整体的训练模式跟 $\text{DQN}$ 算法是一模一样的,因此出于简洁考虑,本章算法不再贴出对应的伪代码。另一方面来讲,建议读者在阅读本章之前,牢牢掌握 $\text{DQN}$ 算法的原理。 ## Double DQN 算法 -Double DQN 算法是谷歌 DeepMind 于 2015 年 12 月提出的一篇解决 $Q$ 值过估计(overestimate)的论文。 +$\text{Double DQN}$ 算法是谷歌 DeepMind 于 2015 年 12 月提出的一篇解决 $Q$ 值过估计(overestimate)的论文。 > ① 论文链接:http://papers.neurips.cc/paper/3964-double-q-learning.pdf -回忆一下 DQN 算法的更新公式,如下: +回忆一下 $\text{DQN}$ 算法的更新公式,如下: $$ Q(s_t,a_t) \leftarrow Q(s_t,a_t)+\alpha[r_t+\gamma\max _{a}Q^{\prime}(s_{t+1},a)-Q(s_t,a_t)] $$ diff --git a/docsify.md b/docsify.md index ea68bca..59fa233 100644 --- a/docsify.md +++ b/docsify.md @@ -36,9 +36,11 @@ latex 公式显示问题 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