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# CS代写|强化学习代写Reinforcement learning代考|CS394R Categorical Temporal-Difference Learning

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## CS代写|强化学习代写Reinforcement learning代考|Categorical Temporal-Difference Learning

Categorical dynamic programming (CDP) computes a sequence $\left(\eta_k\right){k \geq 0}$ of return-distribution functions, defined by iteratively applying the projected distributional Bellman operator $\Pi{\mathrm{C}} \mathcal{T}^\pi$ to an initial return-distribution function $\eta_0:$
$$\eta_{k+1}=\Pi_{\mathrm{C}} \mathcal{T}^\pi \eta_k .$$
As we established in Section 5.9, the sequence generated by CDP converges to the fixed point $\hat{\eta}{\mathrm{C}}^\pi$. Let us express this fixed point in terms of a collection of probabilities $\left(\left(p_i^\pi(x)\right){i=1}^m: x \in \mathcal{X}\right)$ associated with $m$ particles located at $\theta_1, \ldots, \theta_m$
$$\hat{\eta}{\mathrm{C}}^\pi(x)=\sum{i=1}^m p_i^\pi(x) \delta_{\theta_i} .$$
To derive an incremental algorithm from the categorical-projection Bellman operator, let us begin by expressing the projected distributional operator $\Pi_{\mathrm{C}} \mathcal{T}^\pi$ in terms of an expectation over the sample transition $\left(X=x, A, R, X^{\prime}\right)$ :
$$\left(\Pi_{\mathrm{C}} \mathcal{T}^\pi \eta\right)(x)=\Pi_{\mathrm{C}} \mathbb{E}\pi\left[\left(\mathrm{b}{R, \gamma}\right)_{#} \eta^\pi\left(X^\gamma\right) \mid X=x\right]$$
Following the line of reasoning from Section 6.2, in order to construct an unbiased sample target by substituting $R$ and $X^{\prime}$ with their realisations, we need to rewrite Equation $6.8$ with the expectation outside of the projection $\Pi_C$. The following establishes the validity of exchanging the order of these two operations.

## CS代写|强化学习代写Reinforcement learning代考|Quantile Temporal-Difference Learning

Quantile regression is a method for determining the quantiles of a probability distribution incrementally and from samples. ${ }^{47}$ In this section, we develop an algorithm that aims to find the fixed point $\hat{\eta}_{\mathrm{Q}}^\pi$ of the quantile-projected Bellman operator $\Pi_Q \mathcal{T}^\pi$ via quantile regression.

To begin, suppose that given $\tau \in(0,1)$ we are interested in estimating the $\tau^{\text {th }}$ quantile of a distribution $\nu$, corresponding to $F_\nu^{-1}(\tau)$. Quantile regression maintains an estimate $\theta$ of this quantile. Given a sample $z$ drawn from $\nu$, it adjusts $\theta$ according to
$$\theta \leftarrow \theta+\alpha\left(\tau-\mathbb{1}{{z<\theta}}\right) .$$ One can show that quantile regression follows the negative gradient of the quantile $\operatorname{loss}^{48}$ \begin{aligned} \mathcal{L}\tau(\theta) &=\left(\tau-\mathbb{1}{z<\theta}}\right)(z-\theta) \ &=\left|\mathbb{1}{{z<\theta}}-\tau\right| \times|z-\theta| . \end{aligned} In Equation 6.12, the term $\left|\mathbb{1}{{z<\theta}}-\tau\right|$ is an asymmetric step size which is either $\tau$ or $1-\tau$, according to whether the sample $z$ is greater or smaller than $\theta$, respectively. When $\tau<0.5$, samples greater than $\theta$ have a lesser effect on it than samples smaller than $\theta$; the effect is reversed when $\tau>0.5$. The update rule in Equation $6.11$ will continue to adjust the estimate until the equilibrium point $\theta^$ is reached (Exercise $6.4$ asks you to visualise the behaviour of quantile regression with different distributions). This equilibrium point is the location at which smaller and larger samples have an equal effect in expectation. At that point, letting $Z \sim \nu$, we have \begin{aligned} 0 &=\mathbb{E}\left[\tau-\mathbb{1}{\left{Z<\theta^\right}}\right] \ &=\tau-\mathbb{E}\left[\mathbb{1}{\left{Z<\theta^\right}}\right] \ &=\tau-\mathbb{P}\left(Z<\theta^\right) \ \Longrightarrow \mathbb{P}\left(Z<\theta^\right) &=\tau \ \Longrightarrow \theta^ &=F\nu^{-1}(\tau) . \end{aligned}

## CS代写|强化学习代写REINFORCEMENT LEARNING代 考|CATEGORICAL TEMPORAL-DIFFERENCE LEARNING

$$\eta_{k+1}=\Pi_{\mathrm{C}} \mathcal{T}^\pi \eta_k .$$

$$\hat{\eta} \mathrm{C}^\pi(x)=\sum i=1^m p_i^\pi(x) \delta_{\theta_i} .$$

## CS代写|强化学习代写REINFORCEMENT LEARNING代考|QUANTILE TEMPORAL-DIFFERENCE LEARNING

$$\theta \leftarrow \theta+\alpha(\tau-1 z<\theta) .$$ 可以证明分位数回归邅循分位数的负梯度loss ${ }^{48}$ 在公式 $6.12$ 中，项 $|1 z<\theta-\tau|$ 是一个不对称的步长，它是 $\tau$ 或者 $1-\tau$ ，根据样品是否 $z$ 大于或小于 $\theta$ ，分别。什么时候 $\tau<0.5$, 样本大于 $\theta$ 对它的影响小于小于的 样本 $\theta$; 效果相反时 $\tau>0.5$. 方程中的更新规则 $6.11$ 将继续调整估计直到平衡点 $\backslash \theta^{\wedge}$ 到达了
Exercise\$6.4\$asksyoutovisualisethebehaviourofquantileregressionwithdifferentdistributions. 这个平衡点是较小和较大样本在预期中具有相同效果 的位置。那时，让 $Z \sim \nu$ ，我们有

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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。