1. Learner is not told what to do
2. Trial and error search
3. Delayed reward
4. We need to explore round exploit

• Policy what to do
• Reward what is good
• Value what is good because it predicts reward
• Model what follows what

• Evaluating actions vs. inst???
• Example: n-armed bandit
  • evaluate feedback
• after each play $a_t$ we got a reward r_t where $ E\{r_t \mid a_t \} = Q^*(a_t)$
• optimize reward ??? 1000 plays
• Exploration/
  • ??? $Q_t(a) = Q^*(a)$ action value estimate
  • ???
• $a_t = a_t^*$ ⇒ exploitation
• $a_t \ne a_t^*$ ⇒ exploration

Suppose by the ??? play actions $a$ had been choosen $k_a$ times, producing rewards $r_1, r_2, \dots r_{k,a}$ $$Q_t(a) = \frac{r_1, r_2, \dots r_{k,a}}{k_a}$$ $$\lim_{k \rightarrow \infty} Q_t(a) = Q^*(a)$$

• feeding: $a_t = a_t^* = avg_a max Q_t(a)$
• $\epsilon$-feeding = $a_t^*$????

• $n=10$ possible actions
• Each Q^*(a) is chosen rounding from N(0,1)
• 1000 plays, avergage our 2000 experiments

• Softmax grade action probabilities by estimated values $Q_t$
• Bolzmann-distribution: $$\pi(a_t) = \frac{e^{Q_t(a)/\tau}}{\sum^n_{b=1} e^{Q_t(b)/\tau}}$$

$$ Q_k = \frac{r_1, r_2, \ldots, r_k}{k}$$

$$Q_{k+1} = Q_k+\frac{1}{k+1}\[r_{k+1} - Q_k\]$$ Common form: NewEstimate == OldEstimate + StepSize[Target - OldEstimate] ==== Agent-Estimation??? ==== Learn a policy: Policy at step t, $\pi_t$ is a mapping from states to action probabilities $\pi_t(s,a)$=probability that $a_k=a$ when $S_k = S$

Return: $$r_{t+1}, r_{t+2}, \ldots$$

We want to maximize the expected reward, $E\{R_t\}$ for each t

$$R_t = r_{t+1} + r_{t+2} + \ldots + r_T$$

Discounted reward $$R_t = r_{t+1} + \gamma r_{t+2} + \gamma^2 r_{t+3} + \ldots = \sum_{k=0}^\infty \gamma^k r_{t+k+1} \text{where} 0 \le \gamma \le 1$$

$$\text{shortsited} 0 \leftarrow \gamma \rightarrow 1 \text{farsited}$$

$$Pr\{s_{t+1} = s', r_{t+1} = r \mid s_t, a_t, r_t, s_{t-1}, a_{t-1}, r_{t-1}, \ldots s_0, a_0, r_0 \} = Pr \{s_{t+1} = s', r_{t+1} = 1 \mid s_t, a_t \}