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--- uni:8:ml:start [2015-06-11 09:29] – skrupellos
+++ uni:8:ml:start [2020-11-18 18:11] (current) – external edit 127.0.0.1
@@ Line 9: / Line 9: @@
   - Delayed reward
   - 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 feedback ====
+  * 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
+== Action Value Methods ==
+Suppose by the ??? play actions $a$ had been choosen $k_a$ times, producing rewards $r_1, r_2, \dots r_{k,a}$
+$$Q_t(a) = \frac{r_1, r_2, \dots r_{k,a}}{k_a}$$
+$$\lim_{k \rightarrow \infty} Q_t(a) = Q^*(a)$$
+== \epsilon-feeding action selection ==
+  * feeding: $a_t = a_t^* = avg_a max Q_t(a)$
+  * $\epsilon$-feeding = $a_t^*$????
+== In the 10-Armed test bed ==
+  * $n=10$ possible actions
+  * Each Q^*(a) is chosen rounding from N(0,1)
+  * 1000 plays, avergage our 2000 experiments
+== Softmax action selection ==
+  * Softmax grade action probabilities by estimated values $Q_t$
+  * Bolzmann-distribution: $$\pi(a_t) = \frac{e^{Q_t(a)/\tau}}{\sum^n_{b=1} e^{Q_t(b)/\tau}}$$
+===== Something else =====
+$$ Q_k = \frac{r_1, r_2, \ldots, r_k}{k}$$
+==== Incremental implementation ====
+$$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}$$
+==== Markov Property ====
+$$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 \}