统计代写|强化学习作业代写Reinforcement Learning代考|Challenges in Approximation

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• Advanced Mathematical Statistics 高等数理统计学
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统计代写|强化学习作业代写Reinforcement Learning代考|Challenges in Approximation

While we leverage the knowledge of supervised learning-based methods like gradient descent explained earlier, we have to keep two things in mind that make gradient-based methods harder to work in reinforcement learning as compared to the supervised learning.
First, in supervised learning, the training data is held constant. The data is generated from the model, and while we do, the model does not change. It is a ground truth that is given to us and that we are trying to approximate by using the data to learn about the way inputs are mapped to outputs. The data provided to the training algorithm is external to the algorithm, and it does not depend on the algorithm in any way. It is given as constant and independent of the learning algorithm. Unfortunately, in RL, especially in a model-free setup, such is not the case. The data used to generate training samples are based on the policy the agent is following, and it is not a complete picture of the underlying model. As we explore the environment, we learn more, and a new set of training data is generated. We either use the MC-based approach of observing an actual trajectory or bootstrap under TD to form an estimate of the target value, the $y(t)$. As we explore and learn more, the target $y(t)$ changes, which is not the case in supervised learning. This is known as the problem of nonstationary targets.

Second, supervised learning is based on the theoretical premise of samples being uncorrelated to each other, mathematically known as i.i.d. (for “independent identically distributed”) data. However, in RL, the data we see depends on the policy that the agent followed to generate the data. In a given episode, the states we see are dependent on the policy the agent is following at that instant. States that come in later time steps depend

on the action (decisions) the agent took earlier. In other words, the data is correlated. The next state $s_{t+1}$ we see depends on the current state $s_{t}$ and the action $a_{t}$ agent takes in that state.
These two issues make function approximation harder in an RL setup. As we go along, we will see the various approaches that have been taken to address these challenges.
With a broad understanding of the approach, it is time now to start with our usual course of first looking at value prediction/estimate to learn a function that can represent the value functions. We will then look at the control aspect, i.e., the process of the agent trying to optimize the policy. It will follow the usual pattern of using Generalized Policy Iteration (GPI), just like the approach in the previous chapter.

统计代写|强化学习作业代写Reinforcement Learning代考|Incremental Prediction

In this section, we will look at the prediction problem, i.e., how to estimate the state values using function approximation.

Following along, let’s try to extend the supervised training process of finding a model using training data consisting of inputs and targets to function approximation under RL using the loss function in (5.4) and weight update in (5.5). If you compare the loss function in (5.4) and MC/TD updates in (5.2) and (5.3), you can draw a parallel by thinking of MC and TD updates as operations, which are trying to minimize the error between the actual target $v_{\pi}(s)$ and the current estimate $v(s)$. We can represent the loss function as follows:
$$J(w)=E_{\pi}\left[V_{\pi}(s)-V_{t}(s)\right]^{2}$$
Following the same derivation as in (5.5) and using stochastic gradient descent (i.e., replacing expectation with update at each sample), we can write the update equation for weight vector $w$ as follows:
$$\begin{gathered} w_{t+1}=w_{t}-\alpha \cdot \nabla_{w} J(w) \ w_{t+1}=w_{t}+\alpha \cdot\left[V_{\pi}(s)-V_{t}(s ; w)\right] \cdot \nabla_{w} V_{t}(s ; w) \end{gathered}$$

However, unlike supervised learning, we do not have the actual/target output values $V_{\pi}(s)$; rather, we use estimates of these targets. With $\mathrm{MC}$, the estimate/target of $V_{\pi}(s)$ is $G_{\mathrm{r}}(s)$, while the estimate/target under $\mathrm{TD}(0)$ is $R_{t+1}+\gamma * V_{t}\left(s^{\prime}\right)$. Accordingly, the updates under $\mathrm{MC}$ and $\mathrm{TD}(0)$ with functional approximation can be written as follows.
Here is the MC update:
$$w_{t+1}=w_{t}+\alpha \cdot\left[G_{t}(s)-V_{t}(s ; w)\right] \cdot \nabla_{w} V_{t}(s ; w)$$
Here is the $\operatorname{TD}(0)$ update:
$$w_{t+1}=w_{t}+\alpha \cdot\left[R_{t+1}+\gamma * V_{t}\left(s^{\prime} ; w\right)-V_{t}(s ; w)\right] \cdot \nabla_{w} V_{t}(s ; w)$$
A similar set of equations can be written for q-values. We will see that in the next section. This is along the same lines of what we did for the MC and TD control sections in the previous chapter.

Let’s first consider the setup of linear approximation where the state value $\hat{v}(s ; w)$ can be expressed as a dot product of state vector $x(s)$ and weight vector $w$ :
$$\hat{v}(s ; w)=x(s)^{T} \cdot w=\sum_{i} x_{i}(s) * w_{i}$$
The derivative of $\hat{v}(s ; w)$ with respect to $w$ will now be simply state vector $x(s)$.
$$\Delta_{w} V_{t}(s ; w)=x(s)$$
Combining (5.11) with equation (5.7) gives us the following:
$$w_{t+1}=w_{t}+\alpha \cdot\left[V_{x}(s)-V_{t}(s ; w)\right] \cdot x(s)$$

统计代写|强化学习作业代写Reinforcement Learning代考|Incremental Control

Just like in the previous chapter, we will follow a similar approach. We start with function approximation to estimate the q-values.
$$\hat{q}(s, a ; w) \approx q_{\pi}(s, a)$$

Like before, we form a loss function between the target and current value.
$$J(w)=E_{\pi}\left[\left(q_{\pi}(s, a)-\hat{q}(s, a ; w)\right)^{2}\right]$$
Loss is minimized with respect to $w$ to carry out stochastic gradient descent:
$$w_{t+1}=w_{t}-\alpha \cdot \nabla_{u} J(w)$$
where,
$$\nabla_{w} J(w)=\left(q_{n}(s, a)-\hat{q}(s, a ; w)\right) . \nabla_{w} \hat{q}(s, a ; w)$$
Like before, we can simplify the equation when $\hat{q}(s, a ; w)$ uses linear approximation with $\hat{q}(s, a ; w)=x(s, a)^{T} . w$. The derivative $\nabla_{w} \hat{q}(s, a ; w)$, in a linear case as shown previously, will become $\nabla_{w} \hat{q}(s, a ; w)=x(s, a)$.

Next, as we do not know the true q-value $q_{n}(s, a)$, we replace it with the estimates using either MC or TD, giving us a set of equations.
Here is the MC update:
$$w_{t+1}=w_{t}+\alpha \cdot\left[G_{t}(s)-q_{t}(s, a ; w)\right] \cdot \nabla_{w} q_{t}(s, a)$$
Here is the $\operatorname{TD}(0)$ update:
$$w_{t+1}=w_{t}+\alpha \cdot\left[R_{t+1}+\gamma * q_{t}\left(s^{\prime}, a^{\prime} ; w\right)-q_{t}(s, a ; w)\right] \cdot \nabla_{w} q_{t}(s ; a ; w)$$
These equations allow us to carry out q-value estimation/prediction. This is the evaluation step of Generalized Policy Iteration where we carry out multiple rounds of gradient descent to improve on the q-value estimates for a given policy and get them close to the actual target values.
Evaluation is followed by greedy policy maximization to improve the policy. Figure $5-4$ shows the process of iteration under GPI with function approximation.

统计代写|强化学习作业代写Reinforcement Learning代考|Incremental Prediction

Ĵ(在)=和圆周率[在圆周率(s)−在吨(s)]2

Δ在在吨(s;在)=X(s)

统计代写|强化学习作业代写Reinforcement Learning代考|Incremental Control

q^(s,一种;在)≈q圆周率(s,一种)

Ĵ(在)=和圆周率[(q圆周率(s,一种)−q^(s,一种;在))2]

∇在Ĵ(在)=(qn(s,一种)−q^(s,一种;在)).∇在q^(s,一种;在)

有限元方法代写

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MATLAB代写

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