### 机器学习代写|强化学习project代写reinforence learning代考|Reviewing On-Policy Critic Learning in the Context

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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

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

Abstract Critic learning is a fundamental problem in Reinforcement Learning. This paper aims to review some of the basic contents, that are essential to understand critic learning. We review the most important objective functions in the context of critic learning, state some general error sources of policy evaluation methods and explain problems occurring for the off-policy case. Using this knowledge we then compare the fundamental approaches for critic learning, Temporal Differences and Residual Learning. In the end we give a short overview about some more recent critic-learning methods.

In the setting of Reinforcement Learning an agent interacts with an environment by performing actions and receiving reward. This interaction can be formulated as a Markov Decision Process (MDP), defined by a state set $\mathcal{S}$, an action set $\mathcal{A}$, a transition function $\mathcal{P}: \mathcal{S} \times \mathcal{A} \times \mathcal{S} \rightarrow \mathbb{R}$ and a reward function $R: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$. At each discrete time step $t \in \mathbb{N}{\geq 0}$ the agent chooses an action $A{t}$ dependent on the current state $S_{t}$ of the environment and its current policy $\pi$. After performing the chosen action, the environment changes its state according to the transition function and the agent receives a reward $R_{t}$ according to the reward function [3].

The goal of Reinforcement Learning is to find the so-called optimal policy $\pi^{*}$, that maximizes the future expected discounted reward

$$J(\pi)=\mathbb{E}{\mathcal{P}, \pi}\left[\sum{t=0}^{\infty} \gamma^{t} R_{t}\right]$$
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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