NoteNextra-origin/content/CSE510/CSE510_L14.md

# CSE510 Deep Reinforcement Learning (Lecture 14)

## Advanced Policy Gradient Methods

### Trust Region Policy Optimization (TRPO)

"Recall" from last lecture

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

such that

$$
\mathbb{E}_{s\sim d^{\pi}} D_{KL}(\pi(\dot|s)||\pi'(\dot|s))\leq \delta
$$

Unconstrained penalized objective:

$$
d^*=\arg\max_{d} J(\theta+d)-\lambda(D_{KL}\left[\pi_\theta||\pi_{\theta+d}\right]-\delta)
$$

$\theta_{new}=\theta_{old}+d$

First order Taylor expansion for the loss and second order for the KL:

$$
\approx \arg\max_{d} J(\theta_{old})+\nabla_\theta J(\theta)\mid_{\theta=\theta_{old}}d-\frac{1}{2}\lambda(d^\top\nabla_\theta^2 D_{KL}\left[\pi_{\theta_{old}}||\pi_{\theta}\right]\mid_{\theta=\theta_{old}}d)+\lambda \delta
$$

If you are really interested, try to fill the solving the KL Constrained Problem section.

#### Natural Gradient Descent

Setting the gradient to zero:

$$
\begin{aligned}
0&=\frac{\partial}{\partial d}\left(-\nabla_\theta J(\theta)\mid_{\theta=\theta_{old}}d+\frac{1}{2}\lambda(d^\top F(\theta_{old})d\right)\\
&=-\nabla_\theta J(\theta)\mid_{\theta=\theta_{old}}+\frac{1}{2}\lambda F(\theta_{old})d
\end{aligned}
$$

$$
d=\frac{2}{\lambda} F^{-1}(\theta_{old})\nabla_\theta J(\theta)\mid_{\theta=\theta_{old}}
$$

The natural gradient is

$$
\tilde{\nabla}J(\theta)=F^{-1}(\theta_{old})\nabla_\theta J(\theta)
$$

$$
\theta_{new}=\theta_{old}+\alpha F^{-1}(\theta_{old})\hat{g}
$$

$$
D_{KL}(\pi_{\theta_{old}}||\pi_{\theta})\approx \frac{1}{2}(\theta-\theta_{old})^\top F(\theta_{old})(\theta-\theta_{old})
$$

$$
\frac{1}{2}(\alpha g_N)^\top F(\alpha g_N)=\delta
$$

$$
\alpha=\sqrt{\frac{2\delta}{g_N^\top F g_N}}
$$

However, due to the quadratic approximation, the KL constrains may be violated.

#### Linear search

We do Linear search for the best step size by making sure that

- Improving the objective
- Satisfying the KL constraint

TRPO = NPG + Linesearch + monotonic improvement theorem

#### Summary of TRPO

Pros

- Proper learning step
- [Monotonic improvement guarantee](./CSE510_L13.md#Monotonic-Improvement-Theorem)

Cons

- Poor scalability
  - Second-order optimization: computing Fisher Information Matrix and its inverse every time for the current policy model is expensive
- Not quite sample efficient
  - Requiring a large batch of rollouts to approximate accurately

### Proximal Policy Optimization (PPO)

> Proximal Policy Optimization (PPO), which perform comparably or better than state-of-the-art approaches while being much simpler to implement and tune. -- OpenAI

[link to paper](https://arxiv.org/pdf/1707.06347)

Idea:

- The constraint helps in the training process. However, maybe the constraint is not a strict constraint:
- Does it matter if we only break the constraint just a few times?

What if we treat it as a “soft” constraint? Add proximal value to the objective function?

#### PPO with Adaptive KL Penalty

$$
\max_{\theta} \hat{\mathbb{E}_t}\left[\frac{\pi_\theta(a_t|s_t)}{\pi_{\theta_{old}}(a_t|s_t)}\hat{A}_t\right]-\beta \hat{\mathbb{E}_t}\left[KL[\pi_{\theta_{old}}(\dot|s_t),\pi_{\theta}(\dot|s_t)]\right]
$$

Use adaptive $\beta$ value.

$$
L^{KLPEN}(\theta)=\hat{\mathbb{E}_t}\left[\frac{\pi_\theta(a_t|s_t)}{\pi_{\theta_{old}}(a_t|s_t)}\hat{A}_t\right]-\beta \hat{\mathbb{E}_t}\left[KL[\pi_{\theta_{old}}(\dot|s_t),\pi_{\theta}(\dot|s_t)]\right]
$$

Compute $d=\hat{\mathbb{E}_t}\left[KL[\pi_{\theta_{old}}(\dot|s_t),\pi_{\theta}(\dot|s_t)]\right]$

- If $d<d_{target}/1.5$, $\beta\gets \beta/2$
- If $d>d_{target}\times 1.5$, $\beta\gets \beta\times 2$

#### PPO with Clipped Objective

$$
\max_{\theta} \hat{\mathbb{E}_t}\left[\frac{\pi_\theta(a_t|s_t)}{\pi_{\theta_{old}}(a_t|s_t)}\hat{A}_t\right]
$$

$$
r_t(\theta)=\frac{\pi_\theta(a_t|s_t)}{\pi_{\theta_{old}}(a_t|s_t)}
$$

- Here, $r_t(\theta)$ measures how much the new policy changes the probability of taking action $a_t$ in state $s_t$:
  - If $r_t > 1$ :the action becomes more likely under the new policy.
  - If $r_t < 1$ :the action becomes less likely.
- We'd like $r_tA_t$ to increase if $A_t > 0$ (good actions become more probable) and decrease if $A_t < 0$.
- But if $r_t$ changes too much, the update becomes **unstable**, just like in vanilla PG.

We limit $r_t(\theta)$ to be in a range:

$$
L^{CLIP}(\theta)=\hat{\mathbb{E}_t}\left[\min(r_t(\theta)\hat{A}_t, clip(r_t(\theta), 1-\epsilon, 1+\epsilon)\hat{A}_t)\right]
$$

> Trusted region Policy Optimization (TRPO): Don't move further than $\delta$ in KL.
> Proximal Policy Optimization (PPO): Don't let $r_t(\theta)$ drift further than $\epsilon$.

#### PPO in Practice

$$
L_t^{CLIP+VF+S}(\theta)=\hat{\mathbb{E}_t}\left[L_t^{CLIP}(\theta)+c_1L_t^{VF}(\theta)+c_2S[\pi_\theta](s_t)\right]
$$

Here $L_t^{CLIP}(\theta)$ is the surrogate objective function.

$L_t^{VF}(\theta)$ is a squared-error loss for "critic" $(V_\theta(s_t)-V_t^{target})^2$.

$S[\pi_\theta](s_t)$ is the entropy bonus to ensure sufficient exploration. Encourage diversity of actions.

$c_1$ and $c_2$ are trade-off parameters, in paper $c_1=1$ and $c_2=0.01$.

### Summary for Policy Gradient Methods

Trust region policy optimization (TRPO)

- Optimization problem formulation
- Natural gradient ascent + monotonic improvement + line search
- But require second-order optimization

Proximal policy optimization (PPO)

- Clipped objective
- Simple yet effective

Take-away:

- Proper step size is critical for policy gradient methods
- Sample efficiency can be improved by using important sampling