## 计算机代写|强化学习代写Reinforcement learning代考|COMP579

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|强化学习代写Reinforcement learning代考|Mathematics

Mathematical logic is another foundation of deep reinforcement learning. Discrete optimization and graph theory are of great importance for the formalization of reinforcement learning, as we will see in Sect. 2.2.2 on Markov decision processes. Mathematical formalizations have enabled the development of efficient planning and optimization algorithms that are at the core of current progress.

Planning and optimization are an important part of deep reinforcement learning. They are also related to the field of operations research, although there the emphasis is on (non-sequential) combinatorial optimization problems. In AI, planning and optimization are used as building blocks for creating learning systems for sequential, high-dimensional, problems that can include visual, textual, or auditory input.
The field of symbolic reasoning is based on logic, and it is one of the earliest success stories in artificial intelligence. Out of work in symbolic reasoning came heuristic search [34], expert systems, and theorem proving systems. Wellknown systems are the STRIPS planner [17], the Mathematica computer algebra system [13], the logic programming language PROLOG [14], and also systems such as SPARQL for semantic (web) reasoning [3, 7].

Symbolic AI focuses on reasoning in discrete domains, such as decision trees, planning, and games of strategy, such as chess and checkers. Symbolic AI has driven success in methods to search the web, to power online social networks, and to power online commerce. These highly successful technologies are the basis of much of our modern society and economy. In 2011 the highest recognition in computer science, the Turing award, was awarded to Judea Pearl for work in causal reasoning (Fig. 1.9). ${ }^2$ Pearl later published an influential book to popularize the field [35].
Another area of mathematics that has played a large role in deep reinforcement learning is the field of continuous (numerical) optimization. Continuous methods are important, for example, in efficient gradient descent and backpropagation methods that are at the heart of current deep learning algorithms.

## 计算机代写|强化学习代写Reinforcement learning代考|Engineering

In engineering, the field of reinforcement learning is better known as optimal control. The theory of optimal control of dynamical systems was developed by Richard Bellman and Lev Pontryagin [8]. Optimal control theory originally focused on dynamical systems, and the technology and methods relate to continuous optimization methods such as used in robotics (see Fig. $1.10$ for an illustration of optimal control at work in docking two space vehicles). Optimal control theory is of central importance to many problems in engineering.

To this day reinforcement learning and optimal control use a different terminology and notation. States and actions are denoted as $s$ and $a$ in state-oriented reinforcement learning, where the engineering world of optimal control uses $x$ and $u$. In this book the former notation is used.

Biology has a profound influence on computer science. Many nature-inspired optimization algorithms have been developed in artificial intelligence. An important nature-inspired school of thought is connectionist AI.

Mathematical logic and engineering approach intelligence as a top-down deductive process; observable effects in the real world follow from the application of theories and the laws of nature, and intelligence follows deductively from theory. In contrast, connectionism approaches intelligence in a bottom-up fashion. Connectionist intelligence emerges out of many low-level interactions. Intelligence follows inductively from practice. Intelligence is embodied: the bees in bee hives, the ants in ant colonies, and the neurons in the brain all interact, and out of the connections and interactions arises behavior that we recognize as intelligent [11].
Examples of the connectionist approach to intelligence are nature-inspired algorithms such as ant colony optimization [15], swarm intelligence [11, 26], evolutionary algorithms $[4,18,23]$, robotic intelligence [12], and, last but not least, artificial neural networks and deep learning [19, 21, 30].

It should be noted that both the symbolic and the connectionist school of AI have been very successful. After the enormous economic impact of search and symbolic AI (Google, Facebook, Amazon, Netflix), much of the interest in AI in the last two decades has been inspired by the success of the connectionist approach in computer language and vision. In 2018 the Turing award was awarded to three key researchers in deep learning: Bengio, Hinton, and LeCun (Fig. 1.11). Their most famous paper on deep learning may well be [30].

# 强化学习代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 计算机代写|强化学习代写Reinforcement learning代考|ST455

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|强化学习代写Reinforcement learning代考|Sequential Decision Problems

Learning to operate in the world is a high-level goal; we can be more specific. Reinforcement learning is about the agent’s behavior. Reinforcement learning can find solutions for sequential decision problems, or optimal control problems, as they are known in engineering. There are many situations in the real world where, in order to reach a goal, a sequence of decisions must be made. Whether it is baking a cake, building a house, or playing a card game; a sequence of decisions has to be made. Reinforcement learning provides efficient ways to learn solutions to sequential decision problems.

Many real-world problems can be modeled as a sequence of decisions [33]. For example, in autonomous driving, an agent is faced with questions of speed control, finding drivable areas, and, most importantly, avoiding collisions. In healthcare, treatment plans contain many sequential decisions, and factoring the effects of delayed treatment can be studied. In customer centers, natural language processing can help improve chatbot dialogue, question answering, and even machine translation. In marketing and communication, recommender systems recommend news, personalize suggestions, deliver notifications to user, or otherwise optimize the product experience. In trading and finance, systems decide to hold, buy, or sell financial titles, in order to optimize future reward. In politics and governance, the effects of policies can be simulated as a sequence of decisions before they are implemented. In mathematics and entertainment, playing board games, card games, and strategy games consists of a sequence of decisions. In computational creativity, making a painting requires a sequence of esthetic decisions. In industrial robotics and engineering, the grasping of items and the manipulation of materials consist of a sequence of decisions. In chemical manufacturing, the optimization of production processes consists of many decision steps that influence the yield and quality of the product. Finally, in energy grids, the efficient and safe distribution of energy can be modeled as a sequential decision problem.

In all these situations, we must make a sequence of decisions. In all these situations, taking the wrong decision can be very costly.

The algorithmic research on sequential decision making has focused on two types of applications: (1) robotic problems and (2) games. Let us have a closer look at these two domains, starting with robotics.

## 计算机代写|强化学习代写Reinforcement learning代考|Robotics

In principle, all actions that a robot should take can be pre-programmed step by step by a programmer in meticulous detail. In highly controlled environments, such as a welding robot in a car factory, this can conceivably work, although any small change or any new task requires reprogramming the robot.

It is surprisingly hard to manually program a robot to perform a complex task. Humans are not aware of their own operational knowledge, such as what “voltages” we put on which muscles when we pick up a cup. It is much easier to define a desired goal state, and let the system find the complicated solution by itself. Furthermore, in environments that are only slightly challenging, when the robot must be able to respond more flexibly to different conditions, an adaptive program is needed.

It will be no surprise that the application area of robotics is an important driver for machine learning research, and robotics researchers turned early on to finding methods by which the robots could teach themselves certain behavior.

The literature on robotics experiments is varied and rich. A robot can teach itself how to navigate a maze, how to perform manipulation tasks, and how to learn locomotion tasks.

Research into adaptive robotics has made quite some progress. For example, one of the recent achievements involves flipping pancakes [29] and flying an aerobatic model helicopter [1,2]; see Figs. 1.1 and 1.2. Frequently, learning tasks are combined with computer vision, where a robot has to learn by visually interpreting the consequences of its own actions.

# 强化学习代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 计算机代写|强化学习代写Reinforcement learning代考|CS7642

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|强化学习代写Reinforcement learning代考|What Is Deep Reinforcement Learning

Deep reinforcement learning is the combination of deep learning and reinforcement learning.

The goal of deep reinforcement learning is to learn optimal actions that maximize our reward for all states that our environment can be in (the bakery, the dance hall, the chess board). We do this by interacting with complex, high-dimensional environments, trying out actions, and learning from the feedback.

The field of deep learning is about approximating functions in high-dimensional problems, problems that are so complex that tabular methods cannot find exact solutions anymore. Deep learning uses deep neural networks to find approximations for large, complex, high-dimensional environments, such as in image and speech recognition. The field has made impressive progress; computers can now recognize pedestrians in a sequence of images (to avoid running over them) and can understand sentences such as: “What is the weather going to be like tomorrow?”

The field of reinforcement learning is about learning from feedback; it learns by trial and error. Reinforcement learning does not need a pre-existing dataset to train on: it chooses its own actions and learns from the feedback that an environment provides. It stands to reason that in this process of trial and error, our agent will

make mistakes (the fire extinguisher is essential to survive the process of learning to bake bread). The field of reinforcement learning is all about learning from success as well as from mistakes.

In recent years the two fields of deep and reinforcement learning have come together and have yielded new algorithms that are able to approximate highdimensional problems by feedback on their actions. Deep learning has brought new methods and new successes, with advances in policy-based methods, model-based approaches, transfer learning, hierarchical reinforcement learning, and multi-agent learning.

The fields also exist separately, as deep supervised learning and tabular reinforcement learning (see Table 1.1). The aim of deep supervised learning is to generalize and approximate complex, high-dimensional, functions from pre-existing datasets, without interaction; Appendix B discusses deep supervised learning. The aim of tabular reinforcement learning is to learn by interaction in simpler, low-dimensional, environments such as Grid worlds; Chap. 2 discusses tabular reinforcement learning.
Let us have a closer look at the two fields.

## 计算机代写|强化学习代写Reinforcement learning代考|Deep Learning

Classic machine learning algorithms learn a predictive model on data, using methods such as linear regression, decision trees, random forests, support vector machines, and artificial neural networks. The models aim to generalize, to make predictions. Mathematically speaking, machine learning aims to approximate a function from data.

In the past, when computers were slow, the neural networks that were used consisted of a few layers of fully connected neurons and did not perform exceptionally well on difficult problems. This changed with the advent of deep learning and faster computers. Deep neural networks now consist of many layers of neurons and use different types of connections. ${ }^1$ Deep networks and deep learning have taken the accuracy of certain important machine learning tasks to a new level and have allowed machine learning to be applied to complex, high-dimensional, problems, such as recognizing cats and dogs in high-resolution (mega-pixel) images.

Deep learning allows high-dimensional problems to be solved in real time; it has allowed machine learning to be applied to day-to-day tasks such as the face recognition and speech recognition that we use in our smartphones.

Let us look more deeply at reinforcement learning, to see what it means to learn from our own actions.

Reinforcement learning is a field in which an agent learns by interacting with an environment. In supervised learning we need pre-existing datasets of labeled examples to approximate a function; reinforcement learning only needs an environment that provides feedback signals for actions that the agent is trying out. This requirement is easier to fulfill, allowing reinforcement learning to be applicable to more situations than supervised learning.

Reinforcement learning agents generate, by their actions, their own on-the-fly data, through the environment’s rewards. Agents can choose which actions to learn from; reinforcement learning is a form of active learning. In this sense, our agents are like children, that, through playing and exploring, teach themselves a certain task. This level of autonomy is one of the aspects that attracts researchers to the field. The reinforcement learning agent chooses which action to perform-which hypothesis to test—and adjusts its knowledge of what works, building up a policy of actions that are to be performed in the different states of the world that it has encountered. (This freedom is also what makes reinforcement learning hard, because when you are allowed to choose your own examples, it is all too easy to stay in your comfort zone, stuck in a positive reinforcement bubble, believing you are doing great, but learning very little of the world around you.)

# 强化学习代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|强化学习project代写reinforence learning代考|Exploration Methods in Sparse Reward Environments

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|强化学习project代写reinforence learning代考|The Problem of Naive Exploration

In practice the “exploration-exploitation dilemma” is frequently addressed naively by dithering $[27,48,49]$. In continuous action spaces Gaussian noise is added to actions, while in discrete action spaces actions are chosen $\epsilon$-greedily, meaning that optimal actions are chosen with probability $1-\epsilon$ and random actions with probability $\epsilon$. These two approaches work in environments where random sequences of actions are likely to cause positive rewards or to “do the right thing”. Since rewards in sparse domains are infrequent, getting a random positive reward can become very unlikely, resulting in a worst case sample complexity exponential in the amount of states and actions $[20,33,34,56]$. For example, Fig. 1 shows a case where random exploration suffers from exponential sample complexity.

Empirically, this shortcoming can be observed in numerous benchmarks environments, such as the Arcade Learning Environment [5]. Games like Montezuma’s Revenge or Pitfall have sparse reward signals and consequently agents with ditheringbased exploration learn almost nothing [27, 48]. While Montezuma’s Revenge has become the standard benchmark for hard exploration problems, it is important to stress that successfully solving it may not always be a good indicator of intelligent exploration strategies.

This bad exploration behaviour is in partly due to the lack of a prior assumption about the world and its behaviour. As pointed out by [10], in a randomized version of Montezuma’s Revenge (Fig.3) humans perform significantly worse because their prior knowledge is diminished by the randomization, while for RL agents there is no difference due to the lack of prior in the first place. Augmenting an RL agent with prior knowledge could provide a more guided exploration. Yet, we can vastly improve over random exploration even without making use of prior.

A good exploration algorithm should be able to solve hard exploration problems with sparse rewards in large state-action spaces while remaining computationally tractable. According to [33] it is necessary that such an algorithm performs “deep exploration” rather than “myopic exploration”. An agent doing deep exploration will take several coherent actions to explore instead of just locally choosing the most interesting states independently. This is analogous to the general goal of the agent: maximizing the future expected reward rather than the reward of the next timestep.

## 机器学习代写|强化学习project代写reinforence learning代考|Optimism in the Face of Uncertainty

Many of the provably efficient algorithms are based on optimism in the face of uncertainty (OFU) [24] in which the agent acts greedily w.r.t. action values that are optimistic by including an exploration bonus. Either the agent then experiences a high reward and the action was indeed optimal or the agent experiences a low reward and learns that the action was not optimal. After visiting a state-action pair, the exploration bonus is reduced. This approach is superior to naive approaches in that it avoids actions where low value and low information gain are possible. Generally, under the assumption that the agent can visit every state-action pair infinitely many times, the overestimation will decrease and almost optimal behaviour is obtained. Optimal behaviour cannot be obtained due to the bias introduced by the exploration bonus. Most of the algorithms are optimal up to polynomial in the amount of states, actions or the horizon length. The literature provides many variations of these algorithms which use bounds with varying efficacy or different simplifying assumptions, e.g. $[3,6,9,19,20,22]$.

The bounds are often expressed in a framework called probably approximately correct $(\mathrm{PAC})$ learning. Formally, the PAC bound is expressed by a confidence parameter $\delta$ and an accuracy parameter $\epsilon$ w.r.t. which the algorithms are shown to be $\epsilon$ optimal with probability $1-\delta$ after a polynomial amount of timesteps in $\frac{1}{\sigma}, \frac{1}{\epsilon}$ and some factors depending on the MDP at hand.

## 机器学习代写|强化学习project代写reinforence learning代考|Intrinsic Rewards

A large body of work deals with efficient exploration through intrinsic motivation. This takes inspiration from the psychology literature [45] which divides human motivation into extrinsic and intrinsic. Extrinsic motivation describes doing an activity to attain a reward or avoid punishment, while intrinsic rewards describe doing an activity for the sake of curiosity or doing the activity itself. Analogously, we can define the environments reward signal $e_{t}$ at timestep $t$ to be extrinsic and augment it with an intrinsic reward signal $i_{t}$. The agent then tries to maximize $r_{t}=e_{t}+i_{t}$. In the context of a sparse reward problem, the intrinsic reward can fill the gaps between the sparse extrinsic rewards, possibly giving the agent quality feedback at every timestep. In non-tabular MDPs theoretical guarantees are not provided, though, and therefore there is no agreement on an optimal definition of the best intrinsic reward. Intuitively, the intrinsic reward should guide the agent towards optimal behaviour.
An upside of intrinsic reward methods are their straightforward implementation and application. Intrinsic rewards can be used in conjunction with any RL algorithm by just providing the modified reward signal to the learning algorithm $[4,7,57]$. When the calculation of the intrinsic reward and the learning algorithm itself both scale to high dimensional states and actions, the resulting combination is applicable to large state-action spaces as well. However, increased performance is not guaranteed [4]. In the following sections, we will present different formulations of intrinsic rewards.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|强化学习project代写reinforence learning代考|Intrinsic Motivation

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|强化学习project代写reinforence learning代考|Intrinsic Motivation

As stated before, another large branch of methods tackling the curse of sparse rewards is based on the idea of intrinsic motivation. These methods, similar to shaping, originate in behavioral science. Harlow [6] observed that even in the absence of

extrinsic rewards, monkeys have intrinsic drives, such as curiosity to solve complex puzzles. And these intrinsic drives can even be on par in strength with extrinsic incentives, such as food.

Singh et al. [12] transferred the notion of intrinsic motivation to reinforcement learning, illuminating it from an evolutionary perspective. Instead of postulating and hard-coding innate reward signals, an evolutionary-like process was run to optimize the reward function. The resulting reward functions turned out to incentivize exploration in addition to providing task-related guidance to the agent.

It is interesting to compare the computational discovery of Singh et al. [12] that the evolutionary optimal reward consists of two parts-one part responsible for providing motivation for solving a given task and the other part incentivizing exploration-with the way the reward signal is broken up in psychology into a primary reinforcer (basic needs) and a secondary reinforcer (abstract desires correlated with later satisfaction of basic needs). The primary reinforcer corresponds to the immediate physical reward defined by the environment the agent finds itself in. The secondary reinforcer corresponds to the evolutionary beneficial signal, which can be described as curiosity or desire for novelty/surprise, that helps the agent quickly adapt to variations in the environment.

Taking advantage of this two-part reward signal structure – task reward plus exploration bonus-Schmidhuber [11] proposed to design the exploration bonus directly, instead of performing costly evolutionary reward optimization. A variety of exploration bonuses have been described since then. Among the first ones were prediction error and improvement in the prediction error [11]. Recently, a large-scale study of curiosity-driven learning has been carried out [3], which showed that many problems, including Atari games and Mario, can be solved even without explicit task-specific rewards, by agents driven by pure curiosity.

However, curiosity is only one example of an intrinsic motivation signal. There is vast literature on intrinsic motivation, studying signals such as information gain, diversity, empowerment, and many more. We direct the interested reader to a comprehensive recent survey on intrinsic motivation in reinforcement learning [1] for further information.

## 机器学习代写|强化学习project代写reinforence learning代考|Introduction

Recent deep RL algorithms achieved impressive results, such as learning to play Atari games from pixels [27], how to walk [49] or reaching superhuman performance at chess, go and shogi [51]. However, a highly informative reward signal is typically necessary, and without it RL performs poorly, as shown in domains such as Montezuma’s Revenge [5].

The quality of the reward signal depends on multiple factors. First, the frequency at which rewards are emitted is crucial. Frequently emitted rewards are called “dense”, in contrast to infrequent emissions which are called “sparse”. Since improving the policy relies on getting feedback via rewards, the policy cannot be improved until a reward is obtained. In situations where this occurs very rarely, the agent can barely improve. Furthermore, even if the agent manages to obtain a reward, the feedback provided by it might still be less informative than the one of dense signals. In the case of infrequent rewards, in fact, it may be necessary to perform several action to achieve a reward. Hence, assigning credit to specific actions from a long sequence of actions is harder, since there are more actions to reason about.

One of the benchmarks for sparse rewards is the Arcade Learning Environment [5], which features several games with sparse rewards, such as Montezuma’s Revenge and Pitfall. The performance of most of RL algorithms in these games is poor, and

## 机器学习代写|强化学习project代写reinforence learning代考|Exploration Methods

Exploration methods aim to increase the agents knowledge about the environment. Since the agent starts off in an unknown environment, it is necessary to explore and gain knowledge about its dynamics and reward function. At any point the agent can exploit the current knowledge to gain the highest possible (to its current knowledge) cumulative reward. However, these two behaviours are conflicting ways of acting. Exploration is a long term endeavour where the agent tries to maximize the possibility of high rewards in the future, while exploitation is making use of the current knowledge and maximizing the expected rewards in the short term. The agent needs to strike a balance between these two contrasting behaviours, often referred to as “exploration-exploitation dilemma”.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|强化学习project代写reinforence learning代考| Sparse Rewards

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|强化学习project代写reinforence learning代考|Reward Shaping

Many interesting problems are most naturally characterized by a sparse reward signal. Evolution, discussed in 2.1, provides an extreme example. In many cases, it is more intuitive to describe a task completion requirement by a set of constraints rather than by a dense reward function. In such environments, the easiest way to construct a reward signal would be to give a reward on each transition that leads to all constraints being fulfilled.

Theoretically, a general purpose reinforcement learning algorithm should be able to deal with the sparse reward setting. For example, Q-learning [17] is one of the few algorithms that comes with a guarantee that it will eventually find the optimal policy provided all states and actions are experienced infinitely often. However, from a practical standpoint, finding a solution may be infeasible in many sparse reward environments. Stanton and Clune [15] point out that sparse environments require a “prohibitively large” number of training steps to solve with undirected exploration, such as, e.g., $\varepsilon$-greedy exploration employed in Q-learning and other RL algorithms.
A promising direction for tackling sparse reward problems is through curiositybased exploration. Curiosity is a form of intrinsic motivation, and there is a large body of literature devoted to this topic. We discuss curiosity-based learning in more detail in Sect. 5. Despite significant progress in the area of intrinsic motivation, no strong optimality guarantees, matching those for classical RL algorithms, have been established so far. Therefore, in the next section, we take a look at an alternative direction, aimed at modifying the reward signal in such a way that the optimal policy stays invariant but the learning process can be greatly facilitated.

The term “shaping” was originally introduced in behavioral science by Skinner [13]. Shaping is sometimes also called “successive approximation”. $\mathrm{Ng}$ et al. [8] formalized the term and popularized it under the name “reward shaping” in reinforcement learning. This section details the connection between the behavioral science definition and the reinforcement learning definition, highlighting the advantages and disadvantages of reward shaping for improving exploration in sparse reward settings.

## 机器学习代写|强化学习project代写reinforence learning代考|Shaping in Behavioral Science

Skinner laid down the groundwork on shaping in his book [13], where he aimed at establishing the “laws of behavior”. Initially, he found out by accident that learning of certain tasks can be sped up by providing intermediate rewards. Later, he figured out that it is the discontinuities in the operands (units of behavior) that are responsible for making tasks with sparse rewards harder to learn. By breaking down these discountinuities via successive approximation (a.k.a. shaping), he could show that a desired behavior can be taught much more efficiently.

It is remarkable that many of the experimental means described by Skinner [13] cohere to the method of potential fields introduced into reinforcement learning by Ng et al. [8] 46 years later. For example, Skinner employed angle- and position-based rewards to lead a pigeon into a goal region where a big final reward was awaiting. Interestingly, in addition to manipulating the reward, Skinner was able to further speed up pigeon training by manipulating the emvironment. By keeping a pigeon hungry prior to the experiment, he could make the pigeon peck at a switch with higher probability, getting to the reward attached to that behavior more quickly.
Since manipulating the environment is often outside engineer’s control in realworld applications of reinforcement learning, reward shaping techniques provide the main practical tool for modulating the learning process. The following section takes a closer look at the theory of reward shaping in reinforcement learning and highlights its similarities and distinctions from shaping in behavioral science.

## 机器学习代写|强化学习project代写reinforence learning代考|Reward Shaping in Reinforcement Learning

Although practitioners have always engaged in some form of reward shaping, the first theoretically grounded framework has been put forward by Ng et al. [8] under the name potential-based reward shaping $(P B R S)$. According to this theory, the shaping signal $F$ must be a function of the current state and the next state, i.e., $F: \mathcal{S} \times$ $\mathcal{S} \rightarrow \mathbb{R}$. The shaping signal is added to the original reward function to yield the

new reward signal $R^{\prime}\left(s, a, s^{\prime}\right)=R\left(s, a, s^{\prime}\right)+F\left(s, s^{\prime}\right)$. Crucially, the reward shaping term $F\left(s, s^{\prime}\right)$ must admit a representation through a potential function $\Phi(s)$ that only depends on one argument. The dependence takes the form of a difference of potentials
$$F\left(s, s^{\prime}\right)=\gamma \Phi\left(s^{\prime}\right)-\Phi(s) .$$
When condition (1) is violated, undesired effects should be expected. For example, in [9], a term rewarding closeness to the goal was added to a bicycle-riding task where the objective is to drive to a target location. As a result, the agent learned to drive in circles around the starting point. Since driving away from the goal is not punished, such policy is indeed optimal. A potential-based shaping term (1) would discourage such cycling solutions.

Comparing Skinner’s shaping approach from Sect. $4.1$ to PRBS, the reward provided to the pigeon for turning or moving in the right direction can be seen as arising from a potential field based on the direction and distance to the goal. From the description in [13], however, it is not clear whether also punishments (i.e., negative rewards) were given for moving in a wrong direction. If not, then Skinner’s pigeons must have suffered from the same problem as the cyclist in [9]. However, such behavior was not observed, which could be attributed to the animals having a low discount factor or receiving an internal punishment for energy expenditure.

Despite its appeal, some researchers view reward shaping quite negatively. As stated in [9], reward shaping goes against the “tabula rasa” ideal-demanding that the agent learns from scratch using a general (model-free RL) algorithm-by infusing prior knowledge into the problem. Sutton and Barto [16, p. 54] support this view, stating that
the reward signal is not the place to impart to the agent prior knowledge about how to achieve what we want it to do.

As an example in the following subsection shows, it is also often quite hard to come up with a good shaping reward. In particular, the shaping term $F\left(s, s^{\prime}\right)$ is problemspecific, and for each choice of the reward $R\left(s, a, s^{\prime}\right)$ needs to be devised anew. In Sect. 5, we consider more general shaping approaches that only depend on the environment dynamics and not the reward function.

## 机器学习代写|强化学习project代写reinforence learning代考|Reward Shaping

“塑造”一词最初是由 Skinner [13] 在行为科学中引入的。整形有时也称为“逐次逼近”。ñG等。[8] 将该术语形式化并在强化学习中以“奖励塑造”的名义推广。本节详细介绍了行为科学定义和强化学习定义之间的联系，强调了奖励塑造对于改善稀疏奖励设置中的探索的优缺点。

## 机器学习代写|强化学习project代写reinforence learning代考|Shaping in Behavioral Science

Skinner 在他的书 [13] 中奠定了塑造的基础，他的目标是建立“行为法则”。最初，他偶然发现通过提供中间奖励可以加快对某些任务的学习。后来，他发现正是操作数（行为单位）中的不连续性导致奖励稀疏的任务更难学习。通过逐次逼近（又名整形）分解这些折扣，他可以证明可以更有效地教授期望的行为。

## 机器学习代写|强化学习project代写reinforence learning代考|Reward Shaping in Reinforcement Learning

F(s,s′)=C披(s′)−披(s).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|强化学习project代写reinforence learning代考|Reinforcement Learning

Abstract The reward signal is responsible for determining the agent’s behavior, and therefore is a crucial element within the reinforcement learning paradigm. Nevertheless, the mainstream of RL research in recent years has been preoccupied with the development and analysis of learning algorithms, treating the reward signal as given and not subject to change. As the learning algorithms have matured, it is now time to revisit the questions of reward function design. Therefore, this chapter reviews the history of reward function design, highlighting the links to behavioral sciences and evolution, and surveys the most recent developments in RL. Reward shaping, sparse and dense rewards, intrinsic motivation, curiosity, and a number of other approaches are analyzed and compared in this chapter.

With the sharp increase of interest in machine learning in recent years, the field of reinforcement learning (RL) has also gained a lot of traction. Reinforcement learning is generally thought to be particularly promising, because it provides a constructive, optimization-based formalization of the behavior learning problem that is applicable to a large class of systems. Mathematically, the RL problem is represented by a Markov decision process (MDP) whose transition dynamics and/or the reward function are unknown to the agent.

The reward function, being an essential part of the MDP definition, can be thought of as ranking various proposal behaviors. The goal of a learning agent is then to find the behavior with the highest rank. However, there is often a discrepancy between a task and a reward function. For example, a task for a robot may be to open a door; the success in such a task can be evaluated by a binary function that returns one if the door is eventually open and zero otherwise. In practice, though, the reward function

can be made more informative, including such terms as the proximity to the door handle and the force applied to the door to open it. In the former case, we are dealing with a sparse reward scenario, and in the latter case, we have a dense reward scenario. Is the dense reward better for learning? If yes, how to design a dense reward with desired properties? Are there any requirements that the dense reward has to satisfy if what one really cares about is the sparse reward formulation? Such and related questions constitute the focus of this chapter.

At the end of the day, it is the engineer who has to decide on the reward function. Figure 1 shows a typical RL project structure, highlighting the key interactions between its parts. A feedback loop passing through the engineer is especially emphasized, showing that the reward function and the learning algorithm are typically adjusted by the engineer in an iterative fashion based on the given task. The environment, on the other hand, which is identified with the system dynamics in this chapter, is depicted as being outside of engineer’s control, reflecting the situation in real-world applications of reinforcement learning. This chapter reviews and systematizes techniques of reward function design to provide practical guidance to the engineer.

## 机器学习代写|强化学习project代写reinforence learning代考|Evolutionary Reward Signals: Survival and Fitness

Biological evolution is an example of a process where the reward signal is hard to quantify. At the same time, it is perhaps the oldest learning algorithm and therefore has been studied very thoroughly. As one of the first computational modeling approaches, Smith [14] builds a connection between mathematical optimization and biological evolution. He mainly tries to explain the outcome of evolution by identifying the main characteristics of an optimization problem: a set of constraints, an optimization criterion, and heredity. He focuses very much on the individual and identifies the reproduction rate, gait(s), and the foraging strategy as major constraints. These constraints are supposed to cover the control distribution and what would be the dynamics equations in classical control. For the optimization criterion, he chooses the inclusive fitness, which again is a measure of reproduction capabilities. Thus, he takes a very fine-grained view that does not account for long-term behavior but rather falls back to a “greedy” description of the individual.

Reiss [10] criticizes this very simplistic understanding of fitness and acknowledges that the measurement of fitness is virtually impossible in reality. More recently, Grafen [5] attempts to formalize the inclusive notion of the fitness definition. He states that inclusive fitness is only understood in a narrow set of simple situations and even questions whether it is maximized by natural selection at all. To circumvent the direct specification of fitness, another, more abstract, view can be taken. Here, the process is treated as not being fully observable. It is sound to assume that just the rules of physics – which induce, among other things, the concept of survival-form a strict framework, where the survival of an individual is extremely noisy but its fitness is a consistent (probabilistic) latent variable.

From this perspective, survival can be seen as an extremely sparse reward signal. When viewing a human population as an agent, it becomes apparent that the agent not only learned to model its environment (e.g., using science) and to improve itself (e.g., via sexual selection), but also to invent and inherit cultural traditions (e.g., via intergenerational knowledge transfer). In reinforcement learning terms, it is hard to determine the horizon/discounting rate on the population and even on the individual scale. Even considering only a small set of particular choices of an individuum, different studies come to extremely different results, as shown in [4].

So there is no definitive answer on how to specify the reward function and discounting scheme of the natural evolution in terms of a (multi-agent) reinforcement learning setup.

## 机器学习代写|强化学习project代写reinforence learning代考|Monetary Reward in Economics

In contrast to the biological evolution discussed in Sect. 2.1, the reward function arises quite naturally in economics. Simply put, the reward can be identified with the amount of money. As stated by Hughes [7], the learning aspect is really important a

in the economic setup, because albeit many different models exist for financial markets, these are in most cases based on coarse-grained macroeconomic or technical indicators [2]. Since only an extremely small fraction of a market can be captured by direct observation, the agent should learn the mechanics of a particular environment implicitly by taking actions and receiving the resulting reward.

An agent trading in a market and receiving the increase/decrease in value of its assets as the reward at each time-step is also an example for a setup with a dense (as opposed to sparse) reward signal. At every time-step, there is some (arguably unbiased) signal of its performance. In this case, the density of the reward signal increases with the liquidity of the particular market. This example still leaves the question of discounting open. But in economic problems, the discounting rate has the interpretation of an interest-/inflation-rate and should be viewed as dictated by the environment rather than chosen as a learning parameter in most cases. This is also implied by the usage of the term ‘discounting’ in economics where, e.g., the discounted cash flow analysis is based on essentially the same interpretation.

## 机器学习代写|强化学习project代写reinforence learning代考|Evolutionary Reward Signals: Survival and Fitness

Reiss [10] 批评了这种对适应度非常简单的理解，并承认在现实中测量适应度几乎是不可能的。最近，Grafen [5] 试图将适应度定义的包容性概念正式化。他指出，仅在一组狭窄的简单情况下才能理解包容性适应度，甚至质疑它是否完全通过自然选择而最大化。为了规避适应度的直接说明，可以采用另一种更抽象的观点。在这里，该过程被视为不可完全观察。假设只有物理学规则——其中包括生存的概念——形成一个严格的框架是合理的，其中个体的生存是非常嘈杂的，但它的适应度是一个一致的（概率）潜在变量。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|强化学习project代写reinforence learning代考|Further Comparison

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|强化学习project代写reinforence learning代考|Further Comparison

$\mathrm{TD}(0)$ and RG both perform SGD on the $\overline{\mathrm{BE}}$, with $\mathrm{TD}(0)$ simply ignoring the fact, that the one-step TD error also depends on the parameters $\theta$. When assuming linear function approximation, this comparison can also be shown using objective function formulations. Since $\Pi$ represents a orthogonal projection, the relation
$$|\overline{\mathrm{BE}}(\theta)|^{2}=|\overline{\mathrm{PBE}}(\theta)|^{2}+\left|B_{\pi} \hat{v}{\theta}-\Pi B{\pi} \hat{v}{\theta}\right|^{2}$$ is valid (compare Fig. 1). Since the TD-fix-point is congruent with the fix-point of the $\overline{\mathrm{PBE}}, \mathrm{TD}(0)$ only minimizes the $\overline{\mathrm{PBE}}$ and ignores the term $\left|B{\pi} \hat{v}{\theta}-\Pi B{\pi} \hat{v}_{\theta}\right|^{2}$, that is crucial for guaranteed convergence. In contrast RG minimizes both parts of the $\overline{\mathrm{BE}}$ objective. Furthermore the relation shows, that the $\overline{\mathrm{BE}}$, minimized by $\mathrm{RG}$, is an upper bound for the $\overline{\mathrm{PBE}}$, minimized by TD-learning. So minimizing the $\overline{\mathrm{BE}}$ ensures small TD errors. Since optimizing the $\overline{\mathrm{BE}}$ objective using $\mathrm{RG}$ in a way includes the optimization of the $\overline{\mathrm{PBE}}$ objective done by $\mathrm{TD}(0), \mathrm{RG}$ appears to be more difficult from a numerical point of view [8]. The $\overline{\mathrm{BE}}$ objective also suffers from higher variance in its estimates and is therefore harder to optimize [8].

Assuming linear function approximation Li [6] compared $\mathrm{TD}(0)$ and RG with respect to prediction errors $(\overline{\mathrm{VE}})$. The derived bounds for TD $(0)$ are tighter than those for RG, i.e. performance of $\mathrm{TD}(0)$ seems to result in a smaller $\overline{\mathrm{VE}}$. With respect to RG Scherrer [8] also derived an upper bound for the $\overline{\mathrm{VE}}$ using the $\overline{\mathrm{BE}}$. However Dann, Neumann and Peters [3] observed this bound to be too loose for many MDPs in real applications. Sun and Bagnell [9] managed to tighten the bounds for the prediction error of RG even more, even with less strict assumptions than all previous attempts and even for nonlinear function approximation. Although the bounds for $\mathrm{TD}(0)$ are still tighter than those for RG, Sun and Bagnell [9] find in experiments, that residual gradient methods have the potential to achieve smaller prediction errors than temporal-difference methods. Those results are contradictory to the derived bounds

and to the work of Scherrer [8], that finds, that approximation functions derived using the fix-point of the $\overline{\mathrm{PBE}}$ often achieve a lower $\overline{\mathrm{VE}}$ than functions congruent with the fix-point of the $\overline{\mathrm{BE}}$.

Nevertheless, the main point affecting RG is the double sampling problem. Also the $\overline{\text { TDE }}$ objective, that is optimized by RG when simply sampling just one successor state for each state, has not been investigated much in research [3]. In addition Lagoudakis and Parr [5] found, that policy iteration making use of the $\overline{\text { PBE }}$ objective results in control policies of higher quality. Furthermore, congruent to Baird [1], Scherrer [8] and Dann, Neumann and Peters [3] also find TD $(0)$ to converge much faster than RG. Finally Sutton and Barto [11] question the learnability of the Bellman Error and therefore the $\overline{\mathrm{BE}}$ as an objective in general. Altogether TD $(0)$ seems to be preferable, as long as it does not diverge. In the next section, more recent approaches are stated, which combine the advantages of temporal-differences methods with guaranteed convergence.

## 机器学习代写|强化学习project代写reinforence learning代考|Recent Methods and Approaches

In 2009 Sutton, Maei and Szepesvári [13] introduced a stable off-policy temporaldifference algorithm called gradient temporal-difference (GTD). GTD was the first algorithm achieving guaranteed off-policy convergence and linear complexity in memory and per-time-step computation using temporal differences and linear function approximation. GTD performs SGD on a new objective, called norm of the expected TD update (NEU). When optimizing the NEU objective, there are two estimates $\theta$ and $\omega$ of the parameters of the approximation function. First the approximation value function $\hat{v}{\theta}$ is mapped against the one-step TD estimations of the true values of the states (the targets), which are calculated using $\hat{v}{\omega}$. Second $\hat{v}{\omega}$ is mapped against $\hat{v}{\theta}$. Maintaining two individual approximation functions, one for estimating the targets and one for the actual value function approximation, was also one of two key ideas by Mnih et al. [7] to achieve greater success with deep QLearning. Q-Learning is closely related to the problem of non-linear critic-learning. (The second key idea was the introduction of an experience replay memory.) Like Q-Learning, GTD was also extended by Bhatnagar et al. [2] to non-linear function approximation. As all non-linear optimization approaches, it also suffers from potential failures caused by the non-convexity of the optimization objective. GTD, though achieving a lot desirable properties, still converges much slower than conventional $\mathrm{TD}(0)$. Therefore Sutton et al. [12] introduced two new non-linear approximation algorithms, gradient temporal-difference 2 (GTD2) and linear TD with gradient correction (TDC), which converge both faster than GTD. They both perform SGD directly on the $\overline{\mathrm{PBE}}$ objective and TDC even seems to achieve the same (sometimes even better) convergence speed as $\mathrm{TD}(0)$.

## 机器学习代写|强化学习project代写reinforence learning代考|Conclusion

We have reviewed the fundamental contents to understand critic learning. We explained all basic objective functions and compared Temporal-difference learning and the Residual-Gradient algorithm. Thereby Temporal-difference learning was found to be the preferable choice. Also some more recent approaches based on Temporal-difference learning have been reviewed. Like the Residual-Gradient algorithm those approaches are also stable in the off-policy case, but possess better properties.

Nevertheless several aspects have not been considered in this paper, like other optimization techniques to solve the discussed objective functions (e.g. least-squares or probabilistic approaches), extensions like eligibility-traces and further comparison and investigation concerning to the related topic of Q-Learning (and its achievements like $\mathrm{DQN}$ and dueling networks).

## 机器学习代写|强化学习project代写reinforence learning代考|Further Comparison

|乙和¯(θ)|2=|磷乙和¯(θ)|2+|乙圆周率在^θ−圆周率乙圆周率在^θ|2是有效的（比较图 1）。由于 TD-fix-point 与磷乙和¯,吨D(0)只会最小化磷乙和¯并忽略该术语|乙圆周率在^θ−圆周率乙圆周率在^θ|2，这对于保证收敛至关重要。相比之下，RG 最小化了乙和¯客观的。此外，该关系表明，乙和¯, 最小化RG, 是上界磷乙和¯，通过 TD 学习最小化。所以最小化乙和¯确保小的 TD 误差。由于优化乙和¯客观使用RG在某种程度上包括优化磷乙和¯目标完成吨D(0),RG从数值的角度来看似乎更困难[8]。这乙和¯Objective 的估计也存在较大的方差，因此更难优化 [8]。

Scherrer [8] 的工作发现，使用磷乙和¯经常达到较低的在和¯比与固定点一致的函数乙和¯.

## 机器学习代写|强化学习project代写reinforence learning代考|Recent Methods and Approaches

2009 年，Sutton、Maei 和 Szepesvári [13] 引入了一种稳定的离策略时间差算法，称为梯度时间差 (GTD)。GTD 是第一个使用时间差异和线性函数逼近在内存和每时间步计算中实现有保证的非策略收敛和线性复杂性的算法。GTD 对一个新目标执行 SGD，称为预期 TD 更新 (NEU) 的范数。在优化 NEU 目标时，有两个估计θ和ω的近似函数的参数。首先是近似值函数在^θ映射到状态（目标）的真实值的一步 TD 估计，这些估计是使用在^ω. 第二在^ω映射到在^θ. 维护两个单独的近似函数，一个用于估计目标，一个用于实际值函数近似，这也是 Mnih 等人的两个关键思想之一。[7] 通过深度 QLearning 取得更大的成功。Q-Learning 与非线性批评学习问题密切相关。（第二个关键思想是引入经验回放记忆。）与 Q-Learning 一样，GTD 也被 Bhatnagar 等人扩展。[2] 到非线性函数逼近。与所有非线性优化方法一样，它也存在由优化目标的非凸性引起的潜在故障。GTD 虽然实现了很多理想的特性，但仍然比传统的收敛速度慢得多吨D(0). 因此萨顿等人。[12] 引入了两种新的非线性逼近算法，梯度时间差 2（GTD2）和带梯度校正的线性 TD（TDC），它们的收敛速度都比 GTD 快。他们都直接在磷乙和¯目标和 TDC 甚至似乎达到了相同（有时甚至更好）的收敛速度吨D(0).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|强化学习project代写reinforence learning代考|Bellman Equation and Temporal Differences

statistics-lab™ 为您的留学生涯保驾护航 在代写强化学习reinforence learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写强化学习reinforence learning代写方面经验极为丰富，各种代写强化学习reinforence learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|强化学习project代写reinforence learning代考|Bellman Equation and Temporal Differences

As an alternative to MC estimates, we can make use of the Bellman equation, that expresses a value function in a recursive way

$$v_{\pi}(s)=\mathbb{E}{\mathcal{P}, \pi}\left[r\left(s{t}, a_{t}\right)+\gamma v_{\pi}\left(s_{t+1}\right) \mid s_{t}=s .\right]$$
For any arbitrary value function, the mean squared error can be reformulated using the Bellman equation. That results in the mean squared Bellman error objective
$$\overline{\mathrm{BE}}(\theta)=\mathbb{E}{\mu}\left[\left(\hat{v}{\theta}(s)-\mathbb{E}{\mathcal{P}, \pi}\left[r\left(s{t}, a_{t}\right)+\gamma \hat{v}{\theta}\left(s{t+1}\right) \mid \pi, s_{t}=s\right]\right)^{2}\right] .$$
Again no parametric value function can achieve $\overline{\mathrm{BE}}(\theta)=0$, because then it would be identical to $v_{\pi}$, what is not possible for non-trivial value functions. The mean squared Bellman error can be simplified to $\overline{\mathrm{BE}}(\theta)=\mathbb{E}{\mu}\left[\left(\mathbb{E}{\mathcal{P}{, \pi}}\left[\delta{t} \mid s_{t}\right]\right)^{2}\right]$, where $\delta_{t}$ refers to the temporal-difference (TD) error
$$\delta_{t}=r\left(s_{t}, a_{t}\right)+\gamma \hat{v}{\theta}\left(s{t+1}\right)-\hat{v}{\theta}\left(s{t}\right) .$$
Taking a closer look at the simplified mean squared Bellman error points out the so called double sampling problem. The outer expectation value is taken concerning the multiplication of a random variable with itself. To get an unbiased estimator for the product of two random variables, two independently generated sample from the corresponding distribution are necessary. In case of the mean squared Bellman error that means, that for one state $s_{t}$, two successor states $s_{t+1}$ needs to be sampled independently. In most Reinforcement Learning settings, sampling such two successor states independently is not possible. Special cases overcoming the double sampling problem, e.g. cases, in which a model of the MDP is available or in which the MDP is deterministic, are usually less relevant in practice $[1,11]$.

In practice we often want to learn from experience, collected during single trajectories. Consequently only one successor state per state is available. When only using a single successor state for calculating the estimation value, the square of the mean squared Bellman error moves into the inner expectation value. The resulting formula is referred to as the mean squared temporal-difference error
\begin{aligned} \overline{\operatorname{TDE}}(\theta) &=\mathbb{E}{\mu}\left[\mathbb{E}{\mathcal{P}, \pi}\left[\delta_{t}^{2} \mid s_{t}\right]\right] \ &=\mathbb{E}{\mu}\left[\hat{v}{\theta}(s)-\mathbb{E}{\mathcal{P}, \pi}\left[\left(r\left(s{t}, a_{t}\right)+\gamma \hat{v}{\theta}\left(s{t+1}\right)\right)^{2} \mid \pi, s_{t}=s\right]\right] . \end{aligned}
The objectives of the mean squared temporal-difference error and the mean squared Bellman error differ and result in different approximate parametric value functions. Furthermore a parametric value function can now achieve $\overline{\mathrm{TDE}}(\theta)=0[3,11]$.
One last alternative to the stated objective functions is the mean squared projected Bellman error. It is related to the mean squared Bellman error. When constructing the mean squared Bellman error objective, first the Bellman operator is applied to the approximation function. In a second step the weighted estimation value of the difference between the resulting function and the approximation function is constructed. When defining the Bellman operator as $\left(B_{\pi} v_{\pi}\right)\left(s_{t}\right)=$

$\mathbb{E}{\mathcal{P}, \pi}\left[r\left(s{t}, a_{t}\right)+\gamma v_{\pi}\left(s_{t+1}\right) \mid \pi, s_{t}=s\right]$, the mean squared Bellman error can be rewritten as $\overline{\mathrm{BE}}(\theta)=\mathbb{E}{\mu}\left[\left(\hat{v}{\theta}(s)-\leftB_{\pi} \hat{v}{\theta}\right\right)^{2}\right]$. However often $\left(B{\pi} v_{\pi}\right)(s) \notin \mathcal{H}{\theta}$. But using the projection operator $\Pi,\left(B{\pi} v_{\pi}\right)(s)$ can be projected back into $\mathcal{H}{\theta}$. That results in the mean squared projected Bellman error $$\overline{\operatorname{PBE}}(\theta)=\mathbb{E}{\mu}\left[\left(\hat{v}{\theta}(s)-\left\Pi\left(B{\pi} \hat{v}{\theta}\right)\right\right)^{2}\right]$$ Analogous to the mean squared temporal-difference error, approximate value functions can achieve $\overline{\mathrm{PBE}}(\theta)=0$. It is important to mention, that the optimization of all mentioned objective functions in general results in different approximation functions, i.e. $$\begin{gathered} \arg \min {\theta} \overline{\mathrm{VE}}(\theta) \neq \arg \min {\theta} \overline{\mathrm{BE}}(\theta) \ \neq \arg \min {\theta} \overline{\mathrm{TDE}}(\theta) \neq \arg \min {\theta} \overline{\mathrm{PBE}}(\theta) . \end{gathered}$$ Only when $v{\pi} \in \mathcal{H}{\theta}$, then methods optimizing the $\overline{\mathrm{BE}}$ and the $\overline{\mathrm{PBE}}$ as an objective converge to the same and true value function $v{\pi}$, i.e. $\arg \min {\theta} \overline{\mathrm{VE}}(\theta)=\arg \min {\theta}$ $\overline{\mathrm{BE}}(\theta)=\operatorname{arg~min}_{\theta} \overline{\mathrm{PBE}}(\theta)[3,8,11]$.

## 机器学习代写|强化学习project代写reinforence learning代考|Error Sources of Policy Evaluation Methods

Three general, conceptual error sources of Policy Evaluation methods result from the previous explanations [3]:

• Objective bias: The minimum of the objective function often does not correspond with the minimum of the mean squared error, e.g. arg $\min {\theta} \overline{\mathrm{VE}} \neq$ $\arg {\min }^{\theta}$
• Sampling error: Since it is impossible to collect samples over the whole state set $\mathcal{S}$, learning the approximation function has to be done using only a limited number of samples.
• Optimization error: Optimization errors occur, when the chosen optimization methods does not find the (global) optimum, e.g. due to non-convexity of the objective function.

When trying to learn the value function of a target policy $\pi$ using samples collected by a behavior policy $b$, commonly referred to as off-policy learning, two main problems occur. First, the probability of a trajectory occurring after visiting a certain state might be different for $b$ and $\pi$. As a result the probability for the observed cumulative discounted reward might be different and more or less relevant for the

estimation of the true value of the state. This problem can easily be solved using importance sampling. As the stated objectives in this paper all make use of temporal differences, importance sampling simplifies to weighting only as many steps as used for bootstrapping.

The second problem occurs, because the stationary distributions for behavior policy $b$ and target policy $\pi$ differ, i.e. $d^{b}(s) \neq \mu(s)$. This disparity causes the order and frequency of updates for states to change in such a way, that some weights might diverge. There are very simple examples, e.g. the “star problem” introduced by Baird [1], which causes fundamental critic learning methods to diverge. In the next section some more details concerning to the off-policy case are discussed [11].

## 机器学习代写|强化学习project代写reinforence learning代考|Temporal Differences and Bellman Residuals

In the following, two basic fundamental critic-learning approaches are discussed, which aim to find the best possible parametric approximation function. They both use Stochastic Gradient Descent (SGD) to minimize an objective, thus the may suffer from optimization error, especially in the case of nonlinear function approximation.

Temporal-difference learning (TD-learning) was introduced by Sutton [10]. The simplest version of TD-learning, called TD $(0)$, tries to minimize the mean squared error. But instead of using MC estimates to approximate to true value function, it uses onestep temporal-differences estimates. The resulting parameter update function is
$$\theta_{t+1}=\theta_{t}+\alpha_{t}\left[R_{t}+\gamma \hat{v}{\theta{\mathrm{r}}}\left(s_{t+1}\right)-\hat{v}{\theta{t}}\left(s_{t}\right)\right] \frac{\delta \hat{v}{\theta{\mathrm{r}}}\left(s_{t}\right)}{\delta \theta}$$
where $\alpha$ is the learning rate of SGD. So a dependency on the quality of the function approximation is introduced. Since $R_{t}+\gamma \hat{v}{\theta{t}}\left(s_{t+1}\right)$ and $v_{\pi}(s)$ differ, Sutton and Barto [11] describe this procedure to be “semi-gradient” as the objective introduces a bias. Since TD $(0)$ converges to the fix-point of the $\overline{\mathrm{PBE}}$ objective, the often used term “TD-fix-point” simply refers to this fix-point [3]. The main problem with TDlearning is, that there are very simple examples, for which TD $(0)$ diverges, e.g. the already mentioned “star problem” introduced by Baird [1]. So TD-learning suffers from $d^{b}(s) \neq \mu(s)$ in the off-policy case and can diverge.

Due to the instability of TD-learning, Baird [1] introduced the Residual-Gradient algorithm (RG) with guaranteed off-policy convergence. RG directly performs SGD on the $\overline{\mathrm{BE}}$ objective. The resulting parameter update function is
$$\theta_{t+1}=\theta_{t}+\alpha_{t}\left[R_{t}+\gamma \hat{v}{\theta{t}}\left(s_{t+1}\right)-\hat{v}{\theta{t}}\left(s_{t}\right)\right]\left(\frac{\delta \hat{v}{\theta{t}}\left(s_{t}\right)}{\delta \theta}-\gamma \frac{\delta \hat{v}{\theta{\mathrm{r}}}\left(s_{t+1}\right)}{\delta \theta}\right)$$
The only difference between the updates of $\mathrm{TD}(0)$ and RG is a correction of the multiplicative term. A drawback of RG is, that it converges very slow and hence requires extensive interaction between actor and environment [1].

## 机器学习代写|强化学习project代写reinforence learning代考|Bellman Equation and Temporal Differences

d吨=r(s吨,一种吨)+C在^θ(s吨+1)−在^θ(s吨).

TDE¯(θ)=和μ[和磷,圆周率[d吨2∣s吨]] =和μ[在^θ(s)−和磷,圆周率[(r(s吨,一种吨)+C在^θ(s吨+1))2∣圆周率,s吨=s]].

J(\pi)=\mathbb{E}{\mathcal{P}, \pi}\left[\sum{t=0}^{\infty} \gamma^{t} R_{t}\right]
$$where \gamma \in[0,1] is the discount factor. The discount factor can be used to determine how much importance is given to future rewards. Assuming ergodicity also allows to define a stationary distribution \mu(s) over \mathcal{S}, that determines the probability for an agent to be in state s at any time step [3,4]. ## 机器学习代写|强化学习project代写reinforence learning代考|Critic Learning To maximize future rewards an estimation of the accumulated discounted reward is required. This accumulated reward is referred to as the value v_{\pi} of a state s. The corresponding value function$$
v_{\pi}(s)=\mathbb{E}{\mathcal{P}, \pi}\left[\sum{t=0}^{\infty} \gamma^{t} R_{t} \mid S_{0}=s\right]
$$returns the value we can expect after starting in a state s and following a policy \pi. Its estimation plays a fundamental role in Reinforcement Learning, because based on the values we can select the actions. For example, the important concept of policy iteration alternates between evaluating a policy, i.e. estimating the value of each state following a given policy, and improving the policy, e.g. making it greedy concerning the estimated values. When the state set is small and discrete, estimating the value function can be realized by tabular methods. Those methods simply try to learn and remember the true value for each state individually. However, tabular methods are not feasible, when the state space is large or continuous. One of the most common approaches in this case is learning a parametric function, that estimates the value of a given state as precise as possible. In this context, the idea of policy iteration is also called Actor-Critic Learning, where the term actor refers to the deduced policy and the term critic refers to the learned value function. So critic learning is the problem of learning a parametric value function given an MDP and a policy [11]. ## 机器学习代写|强化学习project代写reinforence learning代考|Objective Functions and Temporal Differences To assess the quality of a parametric value function, first we review the mean squared error between the approximate and the true values of the states as an objective function. When approximating the true value function, it is more important to estimate those states correctly, that have a higher frequency of occurrence, than those, that only occur infrequently. Therefore the mean squared errors are weighted using the stationary distribution \mu(s). This weighted mean squared error, or simply mean squared error, is thus given by$$
\overline{\mathrm{VE}}(\theta)=\mathbb{E}{\mu}\left[\left(\hat{v}{\theta}(s)-v_{\pi}(s)\right)^{2}\right],
$$which is identical to \sum_{s \in \mathcal{S}} \mu(s)\left[\hat{v}{\theta}(s)-v{\pi}(s)\right]^{2}, assuming a finite state set. The \theta refers to the parameters of the parametric function. { }^{1} There is one central insight when discussing critic learning. That is, that there is no parametric value function, that can achieve \overline{\mathrm{VE}}(\theta)=0, as long as the true value function is non-trivial and the number of parameters is less than the number of states [11]. Hence all parametric value functions only form a subspace inside the total space of all possible value functions, that map states s \in \mathcal{S} to real numbers \mathbb{R}. This subspace is referred to as \mathcal{H}{\theta}. As already mentioned, usually v{\pi} \notin \mathcal{H}{\theta}. Nevertheless there is a value function \hat{v}{\theta} \in \mathcal{H}{\theta}, that is closest to the true value function in terms of the mean squared error, i.e. \theta=\arg \min {\theta^{\prime}} \overline{\mathrm{VE}}\left(\theta^{\prime}\right). This function can be obtained by applying the projection operator \Pi onto the true value function. This operator projects the true value function from outside to inside of \mathcal{H}{\theta}, i.e.$$ \left(\Pi v{\pi}\right)(s) \doteq \hat{v}{\theta}(s) \quad \text { with } \quad \theta=\arg \min {\theta^{\prime}} \overline{\mathrm{VE}}\left(\theta^{\prime}\right) .

The most straightforward way to learn the approximation value function is to get an estimator for the true value of each state $v_{\pi}(s)$ and then use a standard optimization technique to obtain the parameters $\theta$, that minimize the mean squared error. Monte Carlo (MC) estimates of the true values can be used for that. That means, that the actor starts interaction with the environment and retrospectively calculates the discounted average reward for each state visited after finishing the interaction and observing the rewards. This kind of estimation is unbiased and thus the optimization procedure, assuming convexity, will eventually result in $\Pi v_{\pi}$. But learning the critic using MC estimates is not preferable due to two main reasons. First, we have to wait until the end of the interaction between actor and environment before being able to update and improve the approximation value function. Second, the estimates of the state values, although being unbiased, suffers from a high variance. Thus the learning process is very slow and requires extensive interaction between actor and environment $[3,11]$.

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