强化学习（五）：时间差分学习

Temporal-Difference Learning

TD在强化学习中处于中心位置，它结合了DP与MC两种思想。如MC, TD可以直接从原始经验中学习，且不需要对环境有整体的认知。也如DP一样，它不需要等到最终结果才开始学习，它Bootstrap，即它的每步估计会部分地基于之前的估计。

最简单的TD形式：

$$V(S_t) \leftarrow V(S_t) + \alpha [R_{t+1} + \gamma V(S_{t+1} ) - V(S_t)]$$

这个可被称为TD(0)或一步TD(one-step TD)。

# Tabular TD(0) for estimating v_pi
Input: the policy pi to be evaluated
Algorithm parameter: step size alpha in (0,1]
Initialize V(s), for all s in S_plus, arbitrarily except that V(terminal) = 0

Loop for each episode
    Initialize S
    for step in episode:
        A = action given by pi for S
        Take action A, observe R, S'
        V(S) = V(S) + alpha *[ R+gamma V(S') - V(S)]
        S = S'
        if S == terminal:
            break

TD error:

$$\delta_t \dot = R_{t+1} + \gamma V(S_{t+1}) - V(S_t)$$

在每一时刻，TD error是因为估计所产生的误差。

Advantage of TD Prediction Methods

Sarsa: On-policy TD Control

$$Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha[R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) - Q(S_t,A_t)]$$

Sarsa (State, Action, Reward, State, Action) 表达是这个五元组元素之间的关系。TD error 可表示

$$\delta_t = R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) - Q(S_t,A_t)$$

# Sarsa (on-policy TD control) for estimating Q = q
Algorithm parameters: step size  alpha in (0,1], small epsilon > 0
Initialize Q(s,a), for all s in S_plus, a in A(s), arbitrarily except that Q(terminal,.) = 0

Loop for each episode:
     Initialize S
     Choose A from S using policy derived from Q (e.g., epsilon-greedy)
     Loop for each step of episode:
          Take action A, observe R, S'
          Choose A' from S' using policy derived from Q (e.g., epsilon-greedy)
          Q(S,A) = Q(S,A) + alpha[R + gamma Q(S',A') - Q(S,A)]
          S = S',A=A'
          if S = terminal:
              break

Q-learning: Off-policy TD Control

$$Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha[R_{t+1} + \gamma \max_a Q(S_{t+1}, a) - Q(S_t,A_t)]$$

# Sarsa (on-policy TD control) for estimating Q = q
Algorithm parameters: step size  alpha in (0,1], small epsilon > 0
Initialize Q(s,a), for all s in S_plus, a in A(s), arbitrarily except that Q(terminal,.) = 0

Loop for each episode:
     Initialize S
     
     Loop for each step of episode:
        
          Choose A from S using policy derived from Q (e.g., epsilon-greedy)
          Take action A, observe R, S'
          Q(S,A) = Q(S,A) + alpha[R + gamma max_a Q(S',a) - Q(S,A)]
          S = S'
          if S = terminal:
              break

Q-learning　直接逼近q＊, 最优的action-value 函数独立于行为策略。

Expected Sarsa

$$Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha[R_{t+1} + \gamma E[ Q(S_{t+1}, A_{t+1})|S_{t+1}] - Q(S_t,A_t)]\\ \leftarrow Q(S_t,A_t) +\alpha [R_{t+1} + \gamma \sum_{a}\pi(a|S_{t+1})Q(S_{t+1},a) - Q(S_t,A_t)]$$

Double Q-learning

$$Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha[R_{t+1} + \gamma Q_2(S_{t+1}, \arg\max_a Q_1(S_{t+1},a)) - Q(S_t,A_t)]$$

# Double Q-learning, for estimating Q1 = Q2 = q*

Algorithm parameters: step size alpha in (0,1],small epsilon >0
Initialize Q1(s,a) and Q2(s,a), for all s in S_plus, a in A(s), such that Q(terminal,.) = 0

Loop for each episode:
     Initialize S
     Loop for each step of episode:
          Choose A from S using the policy epsilon-greedy in Q1+Q2
          Take action A, observe R, S'
          with 0.5 probability:
               Q1(S,A) = Q1(S,A) + alpha(R + gamma Q2(S',arg max_a Q1(S',a)) - Q1(S,A))
    	 else:
               Q2(S,A) = Q2(S,A) + alpha(R + gamma Q1(S',arg max_a Q2(S',a)) - Q2(S,A))
     	 S = S'
         if S = terminal:
                break