# CSE510 Deep Reinforcement Learning (Lecture 13)

> Recap from last lecture
>
> For any differentiable policy $\pi_\theta(s,a)$, for any o the policy objective functions $J=J_1, J_{avR}$ or $\frac{1}{1-\gamma} J_{avV}$
> 
> The policy gradient is
>
> $\nabla_{\theta}J(\theta)=\mathbb{E}_{\pi_{\theta}}\left[\nabla_\theta \log \pi_\theta(s,a)Q^{\pi_\theta}(s,a)\right]$

## Problem for policy gradient method

Data Inefficiency

- On-policy method: for each new policy, we need to generate a completely new trajectory
- The data is thrown out after just one gradient update
- As complex neural networks need many updates, this makes the training process very slow

Unstable update： step size is very important

- If step size is too large:
  - Large step -> bad policy
  - Next batch is generated from current bad policy → collect bad samples
  - Bad samples -> worse policy (compare to supervised learning: the correct label and data in the following batches may correct it)
- If step size is too small: the learning process is slow

## Deriving the optimization objection function of Trusted Region Policy Optimization (TRPO)

### Objective of Policy Gradient Methods

Policy Objective

$$
J(\pi_\theta)=\mathbb{E}_{\tau\sim \pi_theta}\sum_{t=0}^{\infty} \gamma^t r^t
$$

here $\tau$ is the trajectory for the policy $\pi_\theta$.

Policy objective can be written in terms of old one:

$$
J(\pi_{\theta'})-J(\pi_{\theta})=\mathbb{E}_{\tau \sim \pi_{\theta'}}\sum_{t=0}^{\infty}\gamma^t A^{\pi_\theta}(s_t,a_t)
$$

Equivalently for succinctness:

$$
J(\pi')-J(\pi)=\mathbb{E}_{\tau\in \pi'}\sum_{t=0}^{\infty} A^{\pi}(s_t,a_t)
$$

<details>
<summary> Proof</summary>

$$
\begin{aligned}
&\mathbb{E}_{\tau\sim \pi'}\left[\gamma^t  A^{\pi_\theta}(s_t,a_t)\right]\\
&=E_{\tau\sim \pi'}\left[\sum_{t=0}^{\infty}\gamma^t R(s_0)+\sum_{t=0}^{\infty} \gamma^{t+1}V^{\pi}(s_{t+1})-\sum_{t=0}^{\infty} \gamma^{t}V^\pi(s_t)\right]\\
&=J(\pi')+\sum_{t=1}^{\infty} \gamma^{t}V^{\pi}(s_t)-\sum_{t=0}^{\infty} \gamma^{t}V^\pi(s_t)\\
&=J(\pi')-\mathbb{E}_{\tau\sim\pi'}V^{\pi}(s_0)\\
&=J(\pi')-J(\pi)
\end{aligned}
$$

</details>

### Importance Sampling

Estimate one distribution by sampling form anther distribution

$$
\begin{aligned}
\mathbb{E}_{x\sim p}[f(x)]&=\int f(x)p(x)dx
&=\int f(x)\frac{p(x)}{q(x)}q(x)dx\\
&=\mathbb{E}_{x\sim q}\left[f(x)\frac{p(x)}{q(x)}\right]\\
&\approx \frac{1}{N}\sum_{i=1,x^i\in q}^N f(x^i)\frac{p(x^i)}{q(x^i)}
\end{aligned}
$$

### Estimating objective with importance sampling

Discounted state visit distribution:

$$
d^\pi(s)=(1-\gamma)\sum_{t=0}^{\infty}\gamma^t P(s_t=s|\pi)
$$

$$
\begin{aligned}
J(\pi')-J(\pi)&=\mathbb{E}_{\tau\sim\pi'}\sum_{t=0}^{\infty} \gamma^t A^{\pi}(s_t,a_t)\\
&=\mathbb{E}_{s\sim d^{\pi'}, a\sim \pi'} A^{\pi}(s,a)\\
&=\mathbb{E}_{s\sim d^{\pi'}, a\sim \pi}\left[A^{\pi'}(s,a)\frac{\pi'(a|s)}{\pi(a|s)}\right]\\
\end{aligned}
$$

Using the old policy to sample states form a policy that we are trying to optimize.

$$
L_\pi(\pi')=\mathbb{E}_{s\sim d^{\pi'}, a\sim \pi}\left[A^{\pi'}(s,a)\frac{\pi'(a|s)}{\pi(a|s)}\right]
$$

### Lower bound of Optimization

> [!NOTE]
>
> (Kullback-Leibler) KL divergence is a measure of the difference between two probability distributions.
>
> $D_{KL}(\pi(\dot|s)||\pi'(\dot|s))=\int_s \pi(a|s)\log \frac{\pi(a|s)}{\pi'(a|s)}da$

$$
J(\pi')-J(\pi)\geq L_\pi(\pi')-C\max_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))
$$

where $C$ is a constant.

Optimizing the objective function:

$$
\max_{\pi'} J(\pi')-J(\pi)
$$

By maximizing the lower bound

$$
\max_{\pi'} L_\pi(\pi')-C\max_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))
$$

### Monotonic Improvement Theorem

Proof of improvement guarantee: Suppose $\pi_{k+1}$ and $\pi_k$ are related by

$$
\pi_{k+1}=\max_{\pi'} L_\pi(\pi')-C\max_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))
$$

$\pi_{k}$ is a feasible point, and the objective at $\pi_k$ is equal to 0.

$$
L_{\pi_k}(\pi_{k})\propto \mathbb{E}_{s,a\sim d^{\pi_k}}[A^{\pi_k}(s,a)]=0
$$
$$
D_{KL}(\pi_k||\pi_k)[s]=0
$$

Optimal value $\geq 0$.

By the performance bound, $J_{pi_{k+1}}-J_{pi_k}\geq 0$.

### Final objective function

$$
\max_{\pi'}\mathbb{E}_{s\sim d^{\pi}, a\sim \pi}[A^{\pi'}(s,a)\frac{\pi'(a|s)}{\pi(a|s)}]-C\max_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))
$$

by approximation

$$
\max_{\pi'}\mathbb{E}_{s\sim d^{\pi}, a\sim \pi}[A^{\pi'}(s,a)\frac{\pi'(a|s)}{\pi(a|s)}]-C\mathbb{E}_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))
$$

With the Lagrangian Duality, the objective is mathematically the same as following using a trust region constraint:

$$
\max_{\pi'} L_\pi(\pi')
$$

such that

$$
\mathbb{E}_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))\leq \delta
$$

$C$ gets very high when $\gamma$ is close to one and the corresponding gradient step size becomes too small.

$$
C\propto \frac{\epsilon \gamma}{(1-\gamma)^2}
$$

- Empirical results show that it needs to more adaptive
- But Tuning $C$ is hard (need some trick just like PPO)
- TRPO uses trust region constraint and make $\delta$ a tunable hyperparameter.

## Trust Region Policy Optimization (TRPO)

$$
\max_{\pi'} L_\pi(\pi')
$$

such that

$$
\mathbb{E}_{s\in S}D_{KL}(\pi(\dot|s)||\pi'(\dot|s))\leq \delta
$$

Make linear approximation to $L_{\pi_{\theta_{old}}}$ and quadratic approximation to KL term.

Maximize $g\cdot(\theta-\theta_{old})-\frac{\beta}{2}(\theta-\theta_{old})^T F(\theta-\theta_{old})$

where $g=\frac{\partial}{\partial \theta}L_{\pi_{\theta_{old}}}(\pi_{\theta})\vert_{\theta=\theta_{old}}$ and $F=\frac{\partial^2}{\partial \theta^2}\overline{KL}_{\pi_{\theta_{old}}}(\pi_{\theta})\vert_{\theta=\theta_{old}}$

<details>
<summary>Taylor Expansion of KL Term</summary>

$$
D_{KL}(\pi_{\theta_{old}}|\pi_{\theta})\approx D_{KL}(\pi_{\theta_{old}}|\pi_{\theta_{old}})+d^T \nabla_\theta D_{KL}(\pi_{\theta_{old}}|\pi_{\theta})\vert_{\theta=\theta_{old}}+\frac{1}{2}d^T \nabla_\theta^2 D_{KL}(\pi_{\theta_{old}}|\pi_{\theta})\vert_{\theta=\theta_{old}}d
$$

$$
\begin{aligned}
\nabla_\theta D_{KL}(\pi_{\theta_{old}}|\pi_{\theta})&=-\nabla_\theta \mathbb{E}_{x\sim \pi_{\theta_{old}}}\log P_\theta(x)\vert_{\theta=\theta_{old}}\\
&=-\mathbb{E}_{x\sim \pi_{\theta_{old}}}\nabla_\theta \log P_\theta(x)\vert_{\theta=\theta_{old}}\\
&=-\mathbb{E}_{x\sim \pi_{\theta_{old}}}\frac{1}{\pi_{\theta_{old}}(x)}\nabla_\theta P_\theta(x)\vert_{\theta=\theta_{old}}\\
&=\int_x P_{\theta_{old}}(x)\frac{1}{P_{\theta_{old}}(x)}\nabla_\theta P_\theta(x)\vert_{\theta=\theta_{old}} dx\\
&=\int_x P_{\theta_{old}}(x)\nabla_\theta P_\theta(x)\vert_{\theta=\theta_{old}} dx\\
&=\nabla_\theta \int_x P_{\theta_{old}}(x) \vert_{\theta=\theta_{old}} dx\\
&=0
\end{aligned}
$$

$$
\begin{aligned}
\nabla_\theta^2 D_{KL}(\pi_{\theta_{old}}|\pi_{\theta})\vert_{\theta=\theta_{old}}&=-\mathbb{E}_{x\sim \pi_{\theta_{old}}}\nabla_\theta^2 \log P_\theta(x)\vert_{\theta=\theta_{old}}\\
&=-\mathbb{E}_{x\sim \pi_{\theta_{old}}}\nabla_\theta \left(\frac{\nabla_\theta P_\theta(x)}{P_\theta(x)}\right)\vert_{\theta=\theta_{old}}\\
&=-\mathbb{E}_{x\sim \pi_{\theta_{old}}}\left(\frac{\nabla_\theta^2 P_\theta(x)-\nabla_\theta P_\theta(x)\nabla_\theta P_\theta(x)^T}{P_\theta(x)^2}\right)\vert_{\theta=\theta_{old}}\\
&=-\mathbb{E}_{x\sim \pi_{\theta_{old}}}\left(\frac{\nabla_\theta^2 P_\theta(x)\vert_{\theta=\theta_{old}}}P_{\theta_{old}}(x)\right)+\mathbb{E}_{x\sim \pi_{\theta_{old}}}\left(\nabla_\theta \log P_\theta(x)\nabla_\theta \log P_\theta(x)^T\right)\vert_{\theta=\theta_{old}}\\
&=\mathbb{E}_{x\sim \pi_{\theta_{old}}}\nabla_\theta\log P_\theta(x)\nabla_\theta\log P_\theta(x)^T\vert_{\theta=\theta_{old}}\\
\end{aligned}
$$

</details>
