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## CS代写|强化学习代写Reinforcement learning代考|Relationship With Distributional Dynamic Programming

In Chapter 5 we introduced distributional dynamic programming (DDP) as a class of methods that operates over return-distribution functions. In fact, every statistical functional dynamic programming is also a DDP algorithm (but not the other way around; see Exercise 8.8). This relationship is established by considering the implied representation
$$\mathscr{F}=\left{\iota(s): s \in I_\psi\right} \subseteq \mathscr{P}(\mathbb{R})$$
and the projection $\Pi_{\mathscr{F}}=\iota \circ \psi$ (see Figure 8.3).

From this correspondence, we may establish the relationship between Bellman closedness and the notion of a diffusion-free projection developed in Chapter 5.

Proposition 8.17. Let $\psi$ be a Bellman-closed sketch. Then for any choice of exact imputation strategy $\iota: I_\psi \rightarrow \mathscr{P}\psi(\mathbb{R})$, the projection operator $\Pi{\mathscr{F}}=$ $\iota \psi$ is diffusion-free.
$\triangle$
Proof. We may directly check the diffusion-free property (omitting parentheses for conciseness):
$$\Pi_{\mathscr{F}} \mathcal{T}^\pi \Pi_{\mathscr{F}}=\iota \psi \mathcal{T}^\pi \iota \psi \stackrel{(a)}{=} \iota \mathcal{T}\psi^\pi \psi \iota \psi \stackrel{(b)}{=} \iota \mathcal{T}\psi^\pi \psi \stackrel{(a)}{=} \iota \psi \mathcal{T}^\pi=\Pi_{\mathscr{F}} \mathcal{T}^\pi .$$
where steps marked (a) follow from the identity $\psi \mathcal{T}^\pi=\mathcal{T}_\psi^\pi \psi$, and (b) follows from the identity $\psi \iota \psi=\psi$ for any exact imputation strategy $\iota$ for $\psi$.

## CS代写|强化学习代写Reinforcement learning代考|Expectile Dynamic Programming

Expectiles form a family of statistical functionals parametrised by a level $\tau \in(0,1)$. They extend the notion of the mean of a distribution ( $\tau=0.5)$ similar to how quantiles extend the notion of a median. Expectiles have classically found application in econometrics and finance as a form of risk measure (see the bibliographical remarks for further details). Based on the principles of statistical functional dynamic programming, expectile dynamic programming ${ }^{65}$ uses an approximate imputation strategy in order to iteratively estimate the expectiles of the return function.

Definition 8.18. For a given $\tau \in(0,1)$, the $\tau$-expectile of a distribution $\nu \in$ $\mathscr{P}2(\mathbb{R})$ is $$\psi\tau^{\mathrm{E}}(\nu)=\underset{z \in \mathbb{R}}{\arg \min } \mathrm{ER}\tau(z ; \nu),$$ where $$\mathbb{E R}\tau(z ; \nu)=\underset{Z \sim \nu}{\mathbb{E}}\left[\left|\mathbb{Y}_{{Z<z}}-\tau\right| \times(Z-z)^2\right]$$
is the expectile loss.
The loss appearing in Definition $8.18$ is strongly convex [Boyd and Vandenberghe, 2004] and bounded below by 0 . As a consequence, Equation $8.12$ has a unique minimiser for a given $\tau$; this verifies that the corresponding expectile is uniquely defined.

## CS代写|强化学习代写|强化学习代考|与分布式动态编程的关系

lleft的缺失或未被识别的分隔符

$triangle$

$$\Pi_{mathscr{F}}. \Pi_{T}^pi\Pi_{mathscr{F}}=iota \psi \mathcal{T}^\pi \iota \psi \stackrel{(a)}{=}。\iota\mathcal{T}^pi \psi^pi\psi \iota \psi \stackrel{(b)}{=}。\iota\mathcal{T} \psi^pi \psi \stackrel{(a)}{=}。\iota \psi \mathcal{T}^\pi=\Pi_{mathscr{F}}。\mathcal{T}^pi$$

## CS代写|强化学代可强化学习代考|Expectile Dynamic编程

$$\psi \tau^{mathrm{E}}(nu)=underset{z\in \mathbb{R}}{arg \min }。\ǞǞǞǞ \tau(z; \nu)。$$

$$\mathbb{E} \tau(z; \nu)=\underset{Z\sim \nu}{mathbb{E}}\left[\left|\mathbb{Y}_{Z<z}-tau\right| \times(Z-z)^2\right] 。$$

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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。

Posted on Categories:CS代写, Reinforcement learning, 强化学习, 计算机代写

## avatest™帮您通过考试

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avatest.™ 为您的留学生涯保驾护航 在计算机Computers代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的计算机Computers代写服务。我们的专家在强化学习Reinforcement learning代写方面经验极为丰富，各种强化学习Reinforcement learning相关的作业也就用不着 说。

## CS代写|强化学习代写Reinforcement learning代考|Challenges In Risk-Sensitive Control

Many convenient properties of the risk-neutral objective do not carry over to risk-sensitive control. As a consequence, finding an optimal policy is usually significantly more involved. This remains true even when the risk-sensitive objective (Equation 7.18) can be evaluated efficiently, for example by using distributional dynamic programming to approximate the return-distribution function $\eta^\pi$. In this section we illustrate some of these challenges by characterising optimal policies for the variance-constrained control problem.

The variance-constrained problem introduces risk sensitivity by forbidding policies whose return variance is too high. Given a parameter $C \geq 0$, the objective is to
$$\begin{array}{ll} \operatorname{maximise} & \mathbb{E}\pi\left[G^\pi\left(X_0\right)\right] \ \text { subject to } & \operatorname{Var}\pi\left(G^\pi\left(X_0\right)\right) \leq C . \end{array}$$
Equation $7.19$ can be shown to satisfy our definition of a risk-sensitive control problem if we express it in terms of a Lagrange multiplier:
$$J_{\mathrm{vC}}(\pi)=\min {\lambda \geq 0}\left(\mathbb{E}\pi\left[G^\pi\left(X_0\right)\right]-\lambda\left(\operatorname{Var}\pi\left(G^\pi\left(X_0\right)\right)-C\right)\right) .$$ The variance-penalised and variance-constrained problems are related in that they share the Pareto set $\boldsymbol{\pi}{\mathrm{PAR}} \subseteq \boldsymbol{\pi}{\mathrm{H}}$ of possibly optimal solutions. A policy $\pi$ is in the set $\pi{\mathrm{PAR}}$ if we have that for all $\pi^{\prime} \in \boldsymbol{\pi}_{\mathrm{H}}$,
(a) $\operatorname{Var}\left(G^\pi\left(X_0\right)\right)>\operatorname{Var}\left(G^{\pi^{\prime}}\left(X_0\right)\right) \Longrightarrow \mathbb{E}\left[G^\pi\left(X_0\right)\right]>\mathbb{E}\left[G^{\pi^{\prime}}\left(X_0\right)\right]$, and
(b) $\operatorname{Var}\left(G^\pi\left(X_0\right)\right)=\operatorname{Var}\left(G^{\pi^{\prime}}\left(X_0\right)\right) \Longrightarrow \mathbb{E}\left[G^\pi\left(X_0\right)\right] \geq \mathbb{E}\left[G^{\pi^{\prime}}\left(X_0\right)\right]$.

## CS代写|强化学习代写Reinforcement learning代考|Conditional Value-At-Risk

In the previous section, we saw that solutions to the variance-constrained control problem can take unintuitive forms, including the need to penalise better-thanexpected outcomes. One issue is that variance only coarsely measures what we mean by “risk” in the common sense of the word. To refine our meaning, we may identify two types of risk: downside risk, involving undesirable outcomes such as greater-than-expected losses, and upside risk, involving what we may informally call a stroke of luck. In some situations, it is possible and useful to separately account for these two types of risk.

To illustrate this point, we now present a distributional algorithm for optimising conditional value-at-risk (CVaR), based on work by Bäuerle and $\mathrm{Ott}$ [2011] and Chow et al. [2015]. One benefit of working with full return distributions is that the algorithmic template we present here can be reasonably adjusted to deal with other risk measures, including the entropic risk measure described in Example 7.17. For conciseness, in what follows we will state without proof a few technical facts about conditional value-at-risk which can be found in those sources and the work of Rockafellar and Uryasev [2002].

Conditional value-at-risk measures downside risk by focusing on the lower tail behaviour of the return distribution, specifically the expected value of this tail. This expected value quantifies the magnitude of losses in extreme scenarios. Let $Z$ be a random variable with cumulative and inverse cumulative distribution functions $F_Z$ and $F_Z^{-1}$, respectively. For a parameter $\tau \in(0,1)$, the $\mathrm{CVaR}$ of $Z$ is
$$\operatorname{CVAR}_\tau(Z)=\frac{1}{\tau} \int_0^\tau F_Z^{-1}(u) \mathrm{d} u .$$
When the inverse cumulative distribution $F_Z^{-1}$ is strictly increasing, the righthand side of Equation $7.20$ is equivalent to
$$\mathbb{E}\left[Z \mid Z \leq F_Z^{-1}(\tau)\right]$$

## CS代写|强化学习代写|强化学习代考|风险敏感控制中的挑战

$$\纹理 {最大化 } \mathbb{E} \pi\left[G^pi\left(X_0\right)\right] \text { subject to } \ooperatorname{Var} \pi\left(G^pi\left(X_0\right)\right) \leq C 。$$

(a) $operatorname{Var}\left(G^\pi\left(X_0\right)\right)>operatorname{Var}\left(G^{pi^{prime}}\left(X_0\right)\right) \Longrightarrow \mathbb{E}\left[G^\pi\left(X_0\right)\right]>\mathbb{E}\left[G^{pi^{prime}\left(X_0\right)\right]$。和
(b) $operatorname{Var}\left(G^\pi\left(X_0\right)\right)=operatorname{Var}\left(G^{pi^{prime}}\left(X_0\right)\right) \Longrightarrow \mathbb{E}\left[G^\pi\left(X_0\right)\right] \geq \mathbb{E}\left[G^{pi^{prime}\left(X_0\right)]$

## CS代写|强化学习代写|条件价值-风险代写

$$\operatorname{CVAR}_tau(Z)=\frac{1}{tau}\int_0^\tau F_Z^{-1}(u) \mathrm{d} u$$

$$\left.Q^{(} x, a\right)=\sup\pi\in \pi\operatorname{MSE}。\pi\left[sum t=0^{infty}\gamma^t R_t mid X=x, A=a\right] 。$$

$$\left.Q^{(} x, a\right)=\mathbb{E}\left[R+\gamma max d \in \mathcal{A}]。Q^{\left(X^{prime}, a^{prime}\right)}mid X=x, A=a\right］。$$

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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。