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content/CSE510/CSE510_L15.md

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 # CSE510 Deep Reinforcement Learning (Lecture 15)
 
+## Motivation
+
+For policy gradient methods over stochastic policies
+
+$$
+\pi_\theta(a|s) = P[a|s,\theta]
+$$
+
+Advantages
+
+- Potentially learning optimal solutions for multi-agent settings
+- Dealing with partial observable settings
+- Sufficient exploration
+
+Disadvantages
+
+- Can not learning a deterministic policy
+- Extension to continuous action space is not straightforward
+
+### On-Policy vs. Off-Policy Policy Gradients
+
+On-Policy Policy Gradients:
+
+- Training samples are collected according to the current policy.
+
+Off-Policy Algorithms:
+
+- Enable the reuse of past experience.
+- Samples can be collected by an exploratory behavior policy.
+
+How to design off-policy policy gradient?
+
+- Using importance sampling
+
 ## Off-Policy Actor-Critic (OffPAC)
 
+Stochastic Behavior Policy for exploration.
+
+- For collecting data. Labelled as $\beta(a|s)$
+
+The objective function is:
+
+$$
+\begin{aligned}
+J(\theta)=\mathbb{E}_{s\sim d^\beta}[V^{\pi}(s)]
+&= \sum_{s\in S} d^\beta(s) \sum_{a\in A} \pi_\theta(a|s) Q^{\pi}(s,a)\\
+\end{aligned}
+$$
+
+$d^\beta(s)$ is the stationary distribution under the behavior policy $\beta(a|s)$.
+
+### Solving the Off-Policy Policy Gradient
+
+$$
+\begin{aligned}
+\nabla_\theta J(\theta) &= \nabla_\theta \mathbb{E}_{s\sim d^\beta}\left[\sum_{a\in A} \pi_\theta(a|s) Q^{\pi}(s,a)\right]\\
+&= \mathbb{E}_{s\sim d^\beta}\left[\sum_{a\in A} \nabla_\theta \pi_\theta(a|s) Q^{\pi}(s,a)+\pi_\theta(a|s) \nabla_\theta Q^{\pi}(s,a)\right]\\
+&= \mathbb{E}_{s\sim d^\beta}\left[\sum_{a\in A} \nabla_\theta \pi_\theta(a|s) Q^{\pi}(s,a)\right]\\
+&= \mathbb{E}_{s\sim d^\beta}\left[\sum_{a\in A} \beta(a|s) \frac{1}{\beta(a|s)} \nabla_\theta \pi_\theta(a|s) Q^{\pi}(s,a)\right]\\
+&= \mathbb{E}_{s\sim d^\beta}\left[\sum_{a\in A} \beta(a|s) \nabla_\theta \log \beta(a|s) Q^{\pi}(s,a)\right]\\
+&= \mathbb{E}_{\beta}\left[\frac{1}{\beta(a|s)} \nabla_\theta \pi_\theta(a|s) Q^{\pi}(s,a)\right]\\
+&= \mathbb{E}_{\beta}\left[\frac{\pi_\theta(a|s)}{\beta(a|s)} Q^{\pi}(s,a)\nabla_\theta \log \pi_\theta(a|s)\right]\\
+\end{aligned}
+$$
+
+To compute the off-policy policy gradient, $Q^{\pi}(s,a)$ is estimated given data collected by $\beta$.
+
+Common solution:
+
+- Importance sampling
+- Tree backup
+- Gradient temporal-difference learning
+- Retrace [Munos et al., 2016] [IMPALA](https://arxiv.org/abs/1802.01561)
+
+### Importance Sampling
+
+Assume that samples come in the form of episodes.
+
+$M$ is the number of episodes containing $(s,a), t_m$ be the first time when $(s,a)$ appears in episode $m$.
+
+The first-visit importance sampling estimator of $Q^{\pi}(s,a)$ is:
+
+$$
+Q^{IS}(s,a)\coloneqq \frac{1}{M}\sum_{m=1}^M R_m w_m
+$$
+
+$R_m$ is the return following $(s,a)$ in episode $m$.
+
+$$
+R_m\coloneqq r_{t_m +1}+\gamma r_{t_m +2}+\cdots+\gamma^{T_m-t_m -1} r_{T_m}
+$$
+
+$w_m$ is the importance sampling weight:
+
+$$
+w_m\coloneqq \frac{\pi(a_{t_m}|s_{t_m})}{\beta(a_{t_m}|s_{t_m})}\frac{\pi(a_{t_m+1}|s_{t_m+1})}{\beta(a_{t_m+1}|s_{t_m+1})}\cdots\frac{\pi(a_{T_m}|s_{T_m})}{\beta(a_{T_m}|s_{T_m})}
+$$
+
+### Per-decision algorithm
+
+Consider the parts we used in importance sampling:
+
+$$
+R_m w_m=\sum_{i=t_m+1}^{T_m}\gamma^{i-t_m-1} r_i \frac{\pi(a_{t_m}|s_{t_m})}{\beta(a_{t_m}|s_{t_m})}\cdots \frac{\pi(a_{t_{i-1}}|s_{t_{i-1}})}{\beta(a_{t_{i-1}}|s_{t_{i-1}})}\frac{\pi(a_{t_i}|s_{t_i})}{\beta(a_{t_i}|s_{t_i})}\cdots \frac{\pi(a_{T_m-1}|s_{T_m-1})}{\beta(a_{T_m-1}|s_{T_m-1})}
+$$
+
+Intuitively, $r_i$ should not depend on the actions taken after $t_i$.
+
+This gives the per-decision importance sampling estimator:
+
+$$
+Q^{PD}(s,a)\coloneqq \frac{1}{M}\sum_{m=1}^M \sum_{k=1}^{T_m-t_m} \gamma^{k-1} r_{t_m+k}\prod_{i=t_m}^{t_m+k-1} \frac{\pi(a_{t_i}|s_{t_i})}{\beta(a_{t_i}|s_{t_i})}
+$$
+
+The per-decision importance sampling estimator is consistence and unbiased estimator of $Q^{\pi}(s,a)$.
+
+Proof as exercise.
+
+<details>
+<summary>Hints</summary>
+- Show the expectation of $Q^{PD}(s,a)$ is the same as $Q^{IS}(s,a)$.
+- $Q^{IS}(s,a)$ is a consistence and unbiased estimator of $Q^{\pi}(s,a)$.
+</details>
+
 ## Deterministic Policy Gradient (DPG)
 
-## Deep Deterministic Policy Gradient (DDPG)
+The objective function is:
-§ Extensions of DDPG
+
+$$
+J(\theta)=\int_{s\in S} \pho^{\mu}(s) r(s,\mu_\theta(s)) ds
+$$
+
+where $\pho^{\mu}(s)$ is the stationary distribution under the behavior policy $\mu_\theta(s)$.
+
+Proof along the same lines of the standard policy gradient theorem.
+
+$$
+\nabla_\theta J(\theta) = \mathbb{E}_{\mu_\theta}[\nabla_\theta Q^{\mu_\theta}(s,a)]=\mathbb{E}_{s\sim \pho^{\mu}}[\nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)}]
+$$
+
+### Issues for DPG
+
+The formulations up to now can only use on-policy data.
+
+
+## Deep Deterministic Policy Gradient (DDPG)

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