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

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

To calculate the value function, let us look again at the tree in Fig. $2.5$ on page 33 , and imagine that it is many times larger, with subtrees that extend to fully cover the state space. Our task is to compute the value of the root, based on the reward values at the real leaves, using the transition function $T_a$. One way to calculate the value $V(s)$ is to traverse this full state space tree, computing the value of a parent node by taking the reward value and the sum of the children, discounting this value by $\gamma$.
This intuitive approach was first formalized by Richard Bellman in 1957. Bellman showed that discrete optimization problems can be described as a recursive backward induction problem [7]. He introduced the term dynamic programming to recursively traverse the states and actions. The so-called Bellman equation shows the relationship between the value function in state $s$ and the future child state $s^{\prime}$, when we follow the transition function.

The discrete Bellman equation of the value of state $s$ after following policy $\pi$ is $^3$
$$V^\pi(s)=\sum_{a \in A} \pi(a \mid s)\left[\sum_{s^{\prime} \in S} T_a\left(s, s^{\prime}\right)\left[R_a\left(s, s^{\prime}\right)+\gamma \cdot V^\pi\left(s^{\prime}\right)\right]\right],$$
where $\pi$ is the probability of action $a$ in state $s, T$ is the stochastic transition function, $R$ is the reward function, and $\gamma$ is the discount rate. Note the recursion on the value function and that for the Bellman equation, the transition and reward functions must be known for all states by the agent.

Together, the transition and reward models are referred to as the dynamics model of the environment. The dynamics model is often not known by the agent, and model-free methods have been developed to compute the value function and policy function without them.

The recursive Bellman equation is the basis of algorithms to compute the value function, and other relevant functions to solve reinforcement learning problems. In the next section we will study these solution methods.

## 计算机代写|强化学习代写Reinforcement learning代考|MDP Solution Methods

The Bellman equation is a recursive equation: it shows how to calculate the value of a state, out of the values of applying the function specification again on the successor states. Figure $2.7$ shows a recursive picture, of a picture in a picture, in a picture, etc. In algorithmic form, dynamic programming calls its own code on states that are closer and closer to the leaves, until the leaves are reached, and the recursion cannot go further.

Dynamic programming uses the principle of divide and conquer: it begins with a start state whose value is to be determined by searching a large subtree, which it does by going down into the recursion, finding the value of sub-states that are closer to terminals. At terminals the reward values are known, and these are then used in the construction of the parent values, as it goes up, back out of the recursion, and ultimately arrives at the root value itself.

A simple dynamic programming method to iteratively traverse the state space to calculate Bellman’s equation is value iteration (VI). Pseudocode for a basic version of VI is shown in Listing 2.1, based on [2]. Value iteration converges to the optimal value function by iteratively improving the estimate of $V(s)$. The value function $V(s)$ is first initialized to random values. Value iteration repeatedly updates $Q(s, a)$ and $V(s)$ values, looping over the states and their actions, until convergence occurs (when the values of $V(s)$ stop changing much).

Value iteration works with a finite set of actions. It has been proven to converge to the optimal values, but, as we can see in the pseudocode in Listing 2.1, it does so quite inefficiently by essentially repeatedly enumerating the entire state space in a triply nested loop, traversing the state space many times. Soon we will see more efficient methods.

# 强化学习代考

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

$$V^\pi(s)=\sum_{a \in A} \pi(a \mid s)\left[\sum_{s^{\prime} \in S} T_a\left(s, s^{\prime}\right)\left[R_a\left(s, s^{\prime}\right)+\gamma \cdot V^\pi\left(s^{\prime}\right)\right]\right]$$

## 有限元方法代写

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## MATLAB代写

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