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# CS代写|强化学习代写Reinforcement learning代考|LSML22 The Monte Carlo Method

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## CS代写|强化学习代写Reinforcement learning代考|The Monte Carlo Method

Birds such as the pileated woodpecker follow a feeding routine that regularly takes them back to the same foraging grounds. The success of this routine can be measured in terms of the total amount of food obtained during a fixed period of time, say a single day. As part of a field study, it may be desirable to predict the success of a particular bird’s routine on the basis of a limited set of observations; for example, to assess its survival chances at the beginning of winter based on feeding observations from the summer months. In reinforcement learning terms, we view this as the problem of learning to predict the expected return (total food per day) of a given policy $\pi$ (the feeding routine). Here, variations in weather, human activity, and other foraging animals are but a few of the factors that affect the amount of food obtained on any particular day.

In our example, the problem of learning to predict is abstractly a problem of statistical estimation. To this end, let us model the woodpecker’s feeding routine as a Markov decision process. ${ }^{17}$ We associate each day with a sample trajectory or episode, corresponding to measurements made at regular intervals about the bird’s location $x$, behaviour $a$, and per-period food intake $r$. Suppose that we have observed a set of $K$ sample trajectories,
$$\left{\left(x_{k, t}, a_{k, t}, r_{k, t}\right){t=0}^{T_k-1}\right}{k=1}^K,$$
where we use $k$ to index the trajectory and $t$ to index time, and where $T_k$ denotes the number of measurements taken each day. In this example, it is most sensible to assume a fixed number of measurements $T_k=T$, but in the general setting $T_k$ may be random and possibly dependent on the trajectory, often corresponding to the time when a terminal state is first reached. For now, let us also assume that there is a unique starting state $x_0$, such that $x_{k, 0}=x_0$ for all $k$. We are interested in the problem of estimating the expected return
$$\mathbb{E}\pi\left[\sum{t=0}^{T-1} \gamma^t R_t\right]=V^\pi\left(x_0\right),$$
corresponding to the expected per-day food intake of our bird. ${ }^{18}$

## CS代写|强化学习代写Reinforcement learning代考|Incremental Learning

Both in practice and in theory, it is useful to consider a learning model under which sample trajectories are processed sequentially, rather than all at once. Algorithms that operate in this fashion are called incremental algorithms, as they maintain a running value function estimate $V \in \mathbb{R}^{\mathcal{X}}$ which they improve with each sample. ${ }^{19}$ Under this model, we now consider an infinite sequence of sample trajectories
$$\left(\left(x_{k, t}, a_{k, t}, r_{k, t}\right){t=0}^{T_k-1}\right){k \geq 0},$$
presented one at a time to the learning algorithm. In addition, we consider the more general setting in which the initial states $\left(x_{k, 0}\right)_{k \geq 0}$ may be different; we call these states the source states, as with the sample transition model (Section 2.6). As in the previous section, a minimum requirement for learning $V^\pi$ is that every state $x \in \mathcal{X}$ should be the source state of some trajectories.

## CS代写|强化学习代写Reinforcement learning代考|The Monte Carlo Method

. The蒙特卡罗方法

$$\left{\left(x_{k, t}, a_{k, t}, r_{k, t}\right){t=0}^{T_k-1}\right}{k=1}^K,$$
，其中我们使用$k$来索引轨迹，使用$t$来索引时间，其中$T_k$表示每天进行的测量数量。在本例中，假设固定数量的测量值$T_k=T$是最明智的，但在一般设置中，$T_k$可能是随机的，可能依赖于轨迹，通常对应于第一次到达终端状态的时间。现在，我们还假设有一个唯一的起始状态$x_0$，这样所有$k$都有$x_{k, 0}=x_0$。我们感兴趣的问题是估计预期收益
$$\mathbb{E}\pi\left[\sum{t=0}^{T-1} \gamma^t R_t\right]=V^\pi\left(x_0\right),$$

## CS代写|强化学习代写Reinforcement learning代考|Incremental learning

.增量学习

$$\left(\left(x_{k, t}, a_{k, t}, r_{k, t}\right){t=0}^{T_k-1}\right){k \geq 0},$$

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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。