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

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• 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

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

Reinforcement learning can be used to solve very big problems with many discrete state configurations or problems with continuous state space. Consider the game of backgammon, which has close to $10^{20}$ discrete states, or consider the game of Go, which has close to $10^{170}$ discrete states. Also consider environments like self-driving cars, drones, or robots: these have a continuous state space.
Up to now we saw problems where the state space was discrete and also small in size, such as the grid world with $\sim 100$ states or the taxi world with 500 states. How do we scale the algorithms we have learned so far to bigger environments or environments with continuous state spaces? All along we have been representing the state values $V(s)$ or the action values $Q(s, a)$ with a table, with one entry for each value of state $s$ or a combination of state $s$ and action $a$. As the numbers increase, the table size is going to become huge, making it infeasible to be able to store state or action values in a table. Further, there will be too many combinations, which can slow down the learning of a policy. The algorithm may spend too much time in states that are very low probability in a real run of the environment.

We will take a different approach now. Let’s represent the state value (or state-action value) with the following function:
$$\begin{gathered} \hat{v}(s ; w) \approx v_{\pi}(s) \ \hat{q}(s, a ; w) \approx q_{\pi}(s, a) \end{gathered}$$
Instead of representing values in a table, they are now being represented by the function $\hat{v}(s ; w)$ or $\hat{q}(s, a ; w)$ where the parameter $w$ is dependent on the policy being followed by the agent, and where $s$ or $(s, a)$ are the inputs to the state or state-value functions. We choose the number of parameters $|w|$ which is lot smaller than the number of states $|s|$ or the number of state-action pairs $(|s| x|a|)$. The consequence of this approach is that there is a generalization of representation of state of the stateaction values. When we update the weight vector $w$ based on some update equation for a given state $s$, it not only updates the value for that specific $s$ or $(s, a)$, but also updates the values for many other states or state actions that are close to the original $s$ or $(s, a)$ for which the update has been carried out. This depends on the geometry of the function. The other values of states near $s$ will also be impacted by such an update as shown previously. We are approximating the values with a function that is a lot more restricted than the number of states. Just to be specific, instead of updating $v(s)$ or $q(s, a)$ directly, we now update the parameter set $w$ of the function, which in turn impacts the value estimates $\hat{v}(s ; w)$ or $\hat{q}(s, a ; w)$. Of course, like before, we carry out the $w$ update using the MC or TD approach. There are various approaches to function approximation. We could feed the state vector (the values of all the variables that signify the state, e.g., position, speed, location, etc.) and get $\hat{v}(s ; w)$, or we could feed state and action vectors and get $\hat{q}(s, a ; w)$ as an output. An alternate approach that is very dominant in the case of actions being discrete and coming from a small set is to feed state vector $s$ and get $|A|$ number of $\hat{q}(s, a ; w)$, one for each action possible $(|A|$ denotes the number of possible actions). Figure 5-1 shows the schematic.

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

Function approximation is a topic studied extensively in the field of supervised learning wherein based on training data we build a generalization of the underlying model. Most of the theory from supervised learning can be applied to reinforcement learning with functional approximation. However, RL with functional approximation brings to fore new issues such as how to bootstrap as well as its impact on nonstationarity. In supervised learning, while the algorithm is learning, the problem/model from which the training data was generated does not change. However, when it comes to RL with function approximation, the way the target (labeled output in supervised learning) is formed, it induces nonstationarity, and we need to come up with new ways to handle it. What we mean by nonstationarity is that we do not know the actual target values of $v(s)$ or $q(s, a)$. We use either the MC or TD approach to form estimates and then use these estimates as “targets.” And as we improve our estimates of target values, we used the revised estimates as new targets. In supervised learning it is different; the targets are given and fixed during training. The learning algorithm has no impact on the targets. In reinforcement learning, we do not have actual targets, and we are using estimates of the target values. As these estimates change, the targets being used in the learning algorithm change; i.e., they are not fixed or stationary during the learning.
Let’s revisit the update equations for $\mathrm{MC}$ (equation 4.2) and TD (equation 4.4), reproduced here. We have modified the equations to make both MC and TD use the same notations of subscript $t$ for the current time and $t+1$ for the next instant. Both equations carry out the same update to move $V_{t}(s)$ closer to its target, which is $G_{t}(s)$ in the case of the $\mathrm{MC}$ update and $R_{t+1}+\gamma * V_{t}(s)$ for the $\operatorname{TD}(0)$ update.
$$\begin{gathered} V_{t+1}(s)=V_{t}(s)+\alpha\left[G_{t}(s)-V_{t}(s)\right] \ V_{t+1}(s)=V_{t}(s)+\alpha\left[R_{t+1}+\gamma * V_{t}\left(s^{\prime}\right)-V_{t}(s)\right] \end{gathered}$$
This is similar to what we do in supervised learning, especially in linear least square regression. We have the output values/targets $y(t)$, and we have the input features $x(t)$, together called training data. We can choose a model Model $_{w}[x(t)]$ like the polynomial linear model, decision tree, or support vectors, or even other nonlinear models like neural nets. The training data is used to minimize the error between what the model is predicting and what the actual output values are from the training set. The is called the minimizing loss function and is represented as follows.

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

Let’s look at the mountain car problem that was discussed in Figure 2-2. The car has a two-dimensional state, a position, and a velocity. Suppose we divide the twodimensional state space into overlapping circles with each circle representing a feature. If state $S$ lies inside a circle, that particular feature is present and has a value of 1 ; otherwise, the feature is absent and has a value of 0 . The number of features is the number of circles. Let’s say we have $p$ circles; then we have converted a two-dimensional continuous state space to a p-dimensional state space where each dimension can be 0 or 1. In other words, each dimension can belong to ${0,1}$.ellipses, the generalization will be more in the direction of the elongation. We could also choose shapes other than circles to control the amount of generalization.

Now consider the case with large, densely packed circles. A large circle makes the initial generalization wide where two faraway states are connected because they fall inside at least one common circle. However, the density (i.e., number of circles) allows us to control the fine-grained generalization. By having many circles, we ensure that even nearby states have at least one feature that is different between two states. This will hold even when each of the individual circles is big. With the help of experiments with varying configurations of the circle size and number of circles, one can fine-tune the size and number of circles to control the generalization appropriate for the problem/domain in question.

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