強化學習基礎：基本概念和動態規劃

問題描述

目標：最大化指望累加獎勵(expected cumulative reward)

• Episodic Task: 在某個時間步結束，$S_0,A_0,R_1,S_1,A_1,R_2,S_2,\cdots,A_{T-1},R_T,S_T(i.e., terminal\text{ }states)$
• Continuing Task: 沒有明確的結束信號，$S_0,A_0,R_1,S_1,A_1,R_2,S_2,\cdots$
• The discounted return(cumulative reward) at time step t: $G_{t}=R_{t+1}+\gamma R_{t+2}+\gamma^{2} R_{t+3}+\ldots,\text{ }0\leq\gamma\leq{1}$

馬爾可夫決策過程MDP

• $\mathcal{S}$: the set of all (nonterminal) states; $\mathcal{S}^+$: S+{terminal states} (only for episodic task)
• $\mathcal{A}$: the set of possible actions; $\mathcal{A}(s)$:  the set of possible actions available in state $s\in\mathcal{S}$
• $\mathcal{R}$: the set of rewards; $\gamma$: discount rate, $0\leq\gamma\leq{1}$
• the one-step dynamics: $p\left(s^{\prime}, r | s, a\right) = \mathbb{P}\left(S_{t+1}=s^{\prime}, R_{t+1}=r | S_{t}=s, A_{t}=a\right)\text{ for each possible }s^{\prime}, r, s, \text { and } a$

問題求解

• Deterministic Policy $\pi$：a mapping $\mathcal{S} \rightarrow \mathcal{A}$;  Stochastic Policy $\pi$: a mapping $\mathcal{S} \times \mathcal{A} \rightarrow[0,1], \text{ i.e., }\pi(a | s)=\mathbb{P}\left(A_{t}=a | S_{t}=s\right)$
• State-Value Function以及Action-Value Function

• Bellman Equation: $v_{\pi}(s)=\mathbb{E}_{\pi}\left[R_{t+1}+\gamma v_{\pi}\left(S_{t+1}\right) | S_{t}=s\right]$

• Policy $\pi^{\prime}\geq\pi \text{ } \Longleftrightarrow \text{ } v_{\pi^{\prime}}(s) \geq v_{\pi}(s)$ for all $s \in \mathcal{S}$; Optimal policy(may not be unique) $\pi_*\geq\pi \text{ for all possible }\pi$
• All optimal policies have the same state-value function $v_*$​ and the same action-value function $q_*$
• $\pi_{*}(s)=\arg \max _{a \in \mathcal{A}(s)} q_{*}(s, a)$

求解方法：Dynamic Programming

假設條件：智能體(Agent)事先知道馬爾科夫決策過程(MDP)的信息，不須要從與環境(Environment)的交互中逐漸學習

方法1：Policy Iteration

• 問題一（左圖）：estimate the state-value function $v_{\pi}$​ corresponding to a policy $\pi$
• 問題二（右圖）：obtain the action-value function $q_{\pi}$​ from the state-value function $v_{\pi}$

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• 問題三（左圖）：take an estimate of the state-value function $v_{\pi}$​ corresponding to a policy $\pi$, returns a new policy $\pi^{\prime}\geq\pi$
• 問題四（右圖）：solve an MDP in the dynamic programming setting 

方法2：Truncated Policy Iteration

In this approach, the evaluation step is stopped after a fixed number of sweeps through the state space

方法3：Value Iteration

In this approach, each sweep over the state space simultaneously performs policy evaluation and policy improvement