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CS代写|强化学习代写Reinforcement learning代考|CSE546 Properties of the Random Trajectory

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CS代写|强化学习代写Reinforcement learning代考|Properties of the Random Trajectory

The Bellman equation is made possible by two fundamental properties of the Markov decision process, time homogeneity and the Markov property. More precisely, these are properties of the random trajectory $\left(X_t, R_t, A_t\right){t \geq 0}$, whose probability distribution is described by the generative equations of Section 2.3. To understand the distribution of the random trajectory $\left(X_t, A_t, R_t\right){t \geq 0}$, it is helpful to depict its possible outcomes as an infinite tree (Figure 2.4). The root of the tree is the initial state $X_0$, and each level of consists of a state-actionreward triple, drawn according to the generative equations. Each branch of the tree then corresponds to a realisation of the trajectory. Most of the quantities that we are interested in can be extracted by “slicing” this tree in particular ways. For example, the random return corresponds to the sum of the rewards along each branch, discounted according to their level.

In order to relate the probability distributions of different parts of the random trajectory, let us introduce the notation $\mathcal{D}(Z)$ to denote the probability distribution of a random variable $Z$. When $Z$ is real-valued we have, for $S \subseteq \mathbb{R}$,
$$\mathcal{D}(Z)(S)=\mathbb{P}(Z \in S) .$$
One advantage of the $\mathcal{D}(\cdot)$ notation is that it often avoids unnecessarily introducing (and formally characterising) such a subset $S$, and is more easily extended to other kinds of random variables. Importantly, we write $\mathcal{D}\pi$ to refer to the distribution of random variables derived from the joint distribution $\mathbb{P}\pi \cdot{ }^{14}$ For example, $\mathcal{D}\pi\left(R_0+R_1\right)$ and $\mathcal{D}\pi\left(X_2\right)$ are the probability distribution of the sum $R_0+R_1$ and of the third state in the random trajectory, respectively.

CS代写|强化学习代写Reinforcement learning代考|The Random-Variable Bellman Equation

The Bellman equation characterises the expected value of the random return from any state $x$ compactly, allowing us to reduce the generative equations (an infinite sequence) to the sample transition model. In fact, we can leverage time-homogeneity and the Markov property to characterise all aspects of the random return in this manner. Consider again the definition of this return as a discounted sum of random rewards:
$$G=\sum_{t=0}^{\infty} \gamma^t R_t$$
As with value functions, we would like to relate the return from the initial state to the random returns that occur downstream in the trajectory. To this end, let us define the return function
$$G^\pi(x)=\sum_{t=0}^{\infty} \gamma^t R_t, \quad X_0=x,$$
which describes the return obtained when acting according to $\pi$ starting from a given state $x$. Note that in this definition, the notation $X_0=x$ again modifies the initial state distribution $\xi_0$. Equation $2.13$ is thus understood as “the discounted sum of random rewards described by the generative equations with $\xi_0=\delta_x$ “. Formally, $G^\pi$ is a collection of random variables indexed by an initial state $x$, each generated by a random trajectory $\left(X_t, A_t, R_t\right){t \geq 0}$ under the distribution $\mathbb{P}\pi(\cdot \mathrm{I}$ $X_0=x$ ). Because Equation $2.13$ is concerned with random variables, we will sometimes find it convenient to be more precise and call it the return-variable function. 15

CS代写|强化学习代写强化学习代考|随机轨迹的属性

. c

Bellman方程是由马尔可夫决策过程的两个基本性质，时间同质性和马尔可夫性质而成为可能的。更准确地说，这些是随机轨迹$\left(X_t, R_t, A_t\right){t \geq 0}$的性质，其概率分布由2.3节的生成方程描述。为了理解随机轨迹$\left(X_t, A_t, R_t\right){t \geq 0}$的分布，将其可能的结果描述为无限树是有帮助的(图2.4)。树的根是初始状态$X_0$，每个关卡都包含一个状态-行为-奖励三元组，根据生成方程绘制。树的每个分支都对应着轨迹的实现。我们感兴趣的大部分量都可以通过以特定的方式“切片”这棵树来提取。例如，随机回报对应于每个分支的奖励之和，根据其级别进行折现

$$\mathcal{D}(Z)(S)=\mathbb{P}(Z \in S) .$$
$\mathcal{D}(\cdot)$表示法的一个优点是，它常常避免不必要地引入(并正式地描述)这样一个子集$S$，并且更容易扩展到其他类型的随机变量。重要的是，我们写$\mathcal{D}\pi$是指由联合分布$\mathbb{P}\pi \cdot{ }^{14}$导出的随机变量的分布。例如，$\mathcal{D}\pi\left(R_0+R_1\right)$和$\mathcal{D}\pi\left(X_2\right)$分别是$R_0+R_1$和随机轨迹中第三状态的和的概率分布。

CS代写|强化学习代写强化学习代考|随机变量Bellman方程

. c

Bellman方程简洁地描述了来自任何状态$x$的随机返回的期望值，允许我们将生成方程(一个无限序列)简化为样本转移模型。事实上，我们可以利用时间同质性和马尔可夫性质，以这种方式描述随机回报的各个方面。再次考虑将此回报定义为随机奖励的折现和:
$$G=\sum_{t=0}^{\infty} \gamma^t R_t$$

$$G^\pi(x)=\sum_{t=0}^{\infty} \gamma^t R_t, \quad X_0=x,$$
，它描述了从给定状态$x$开始按照$\pi$操作时获得的返回值。注意，在这个定义中，表示法$X_0=x$再次修改了初始状态分布$\xi_0$。方程$2.13$因此被理解为“由$\xi_0=\delta_x$生成方程所描述的随机奖励的折现和”。形式上，$G^\pi$是由初始状态$x$索引的随机变量集合，每个变量由分布$\mathbb{P}\pi(\cdot \mathrm{I}$$X_0=x$下的随机轨迹$\left(X_t, A_t, R_t\right){t \geq 0}$生成。因为方程$2.13$涉及的是随机变量，我们有时会发现更精确的称呼它为返回变量函数更方便。15

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。