### 统计代写|强化学习作业代写Reinforcement Learning代考|Eligibility Traces and TD

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• Foundations of Data Science 数据科学基础

## 统计代写|强化学习作业代写Reinforcement Learning代考|Eligibility Traces and TD

Eligibility traces unify the MC and TD methods in an algorithmically efficient way. TD methods when combined with eligibility trace produce $\operatorname{TD}(\lambda)$ where $\lambda=0$, making it equivalent to the one-step TD that we have studied so far. That’s the reason why one-step TD is also known as $\operatorname{TD}(0)$. The value of $\lambda=1$ makes it similar to the regular $\infty$-step TD or in other words an MC method. Eligibility trace makes it possible to apply MC methods on nonepisodic tasks. We will cover only high-level concepts of eligibility trace and $\operatorname{TD}(\lambda)$.
In the previous section, we looked at n-step returns with $\mathrm{n}=1$ taking us to the regular TD method and $n=\infty$ taking us to MC. We also touched upon the fact that neither extreme is good. An algorithm performs best with some intermediate value of

n. n-step offered a view on how to unify TD and MC. What eligibility does is to offer an efficient way to combine them without keeping track of the n-step transitions at each step. Until now we have looked at an approach of updating a state value based on the next $n$ transitions in the future. This is called the forward view. However, you could also look backward, i.e., at each time step $t$, and see the impact that the reward at time step $t$ would have on the preceding $n$ states in past. This is known as backward view and forms the core of $\operatorname{TD}(\lambda)$. The approach allows an efficient implementation of integrating n-step returns in TD learning.
Look back at Figure 4-20. What if instead of choosing different values of $n$, we combined all the n-step returns with some weight? This is known as $\lambda$-return, and the equation is as follows:
$$G_{t}^{\lambda}=(1-\lambda) \sum_{n=1}^{T-t-1} \lambda^{n-1} G_{t: t+n}+\lambda^{T-t-1} G_{t}$$
Here, $G_{t: t+n}$ is the n-step return which uses bootstrapped value of remaining steps at the end of the $n^{\text {th }}$ step. It is defined as follows:
$$G_{t: t+n}=R_{t+1}+\gamma R_{t+2}+\ldots+\gamma^{n-1} R_{t+n}+\gamma^{n} V\left(S_{t+n}\right)$$
If we put $\lambda=0$ in (4.13), we get the following:
$$G_{t}^{0}=G_{t: t+1}=R_{t+1}+\gamma V\left(S_{t+1}\right)$$

## 统计代写|强化学习作业代写Reinforcement Learning代考|Summary

In this chapter, we looked at the model-free approach to reinforcement learning. We started by estimating the state value using the Monte Carlo approach. We looked at the “first visit” and “every visit” approaches. We then looked at the bias and variance tradeoff in general and specifically in the context of the $\mathrm{MC}$ approaches. With the foundation of MC estimation in place, we looked at MC control methods connecting it with the GPI framework for policy improvement that was introduced in Chapter 3 . We saw how GPI could be applied by swapping the estimation step of the approach from DP-based to an MC-based approach. We looked in detail at the exploration exploitation dilemma that needs to be balanced, especially in the model-free world where the transition probabilities are not known. We then briefly talked about the off-policy approach in the context of the MC methods.

TD was the next approach we looked into with respect to model-free learning. We started off by establishing the basics of TD learning, starting with TD-based value estimation. This was followed by a deep dive into SARSA, an on-policy TD control method. We then looked into Q-learning, a powerful off-policy TD learning approach, and some of its variants like expected SARSA.
In the context of TD learning, we also introduced the concept of state approximation to convert continuous state spaces into approximate discrete state values. The concept of state approximation will form the bulk of the next chapter and will allow us to combine deep learning with reinforcement learning.

Before concluding the chapter, we finally looked at n-step returns, eligibility traces, and $\operatorname{TD}(\lambda)$ as ways to combine TD and MC into a single framework.

## 统计代写|强化学习作业代写Reinforcement Learning代考|Function Approximation

In the previous three chapters, we looked at various approaches to planning and control, first using dynamic programming (DP), then using the Monte Carlo approach (MC), and finally using the temporal difference (TD) approach. In all these approaches, we always looked at problems where the state space and actions were both discrete. Only in the previous chapter toward the end did we talk about Q-learning in a continuous state space. We discretized the state values using an arbitrary approach and trained a learning model. In this chapter, we are going to extend that approach by talking about the theoretical foundations of approximation and how it impacts the setup for reinforcement learning. We will then look at the various approaches to approximating values, first with a linear approach that has a good theoretical foundation and then with a nonlinear approach specifically with neural networks. This aspect of combining deep learning with reinforcement learning is the most exciting development that has moved reinforcement learning algorithms to scale.

As usual, the approach will be to look at everything in the context of the prediction/ estimation setup where the agent tries to follow a given policy to learn the state value and/or action values. This will be followed by talking about control, i.e., to find the optimal policy. We will continue to be in a model-free world where we do not know the transition dynamics. We will then talk about the issues of convergence and stability in the world of function approximation. So far, the convergence has not been a big issue in the context of the exact and discrete state spaces. However, function approximation brings about new issues that need to be considered for theoretical guarantees and practical best practices. We will also touch upon batch methods and compare them with the incremental learning approach discussed in the first part of this chapter.

We will close the chapter with a quick overview of deep learning, basic theory, and the basics of building/training models using PyTorch and TensorFlow.

## 统计代写|强化学习作业代写Reinforcement Learning代考|Eligibility Traces and TD

n. n-step 提供了一个关于如何统一 TD 和 MC 的观点。资格的作用是提供一种有效的方法来组合它们，而无需跟踪每一步的 n 步转换。到目前为止，我们已经研究了一种基于下一个更新状态值的方法n未来的过渡。这称为前视。但是，您也可以向后看，即在每个时间步吨，并查看奖励在时间步的影响吨会在前面n过去的状态。这被称为后视，构成了运输署⁡(λ). 该方法允许在 TD 学习中有效地集成 n 步回报。

G吨λ=(1−λ)∑n=1吨−吨−1λn−1G吨:吨+n+λ吨−吨−1G吨

G吨:吨+n=R吨+1+CR吨+2+…+Cn−1R吨+n+Cn在(小号吨+n)

G吨0=G吨:吨+1=R吨+1+C在(小号吨+1)

## 统计代写|强化学习作业代写Reinforcement Learning代考|Summary

TD 是我们研究的下一个关于无模型学习的方法。我们从建立 TD 学习的基础开始，从基于 TD 的价值估计开始。随后深入研究了 SARSA，一种基于策略的 TD 控制方法。然后，我们研究了 Q-learning，一种强大的离策略 TD 学习方法，以及它的一些变体，如预期的 SARSA。

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