# CSE510 Deep Reinforcement Learning (Lecture 12)

## Policy Gradient Theorem

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

$$
\nabla_{\theta}J(\theta)=\mathbb{E}_{\pi_{\theta}}\left[\nabla_\theta \log \pi_\theta(s,a)Q^{\pi_\theta}(s,a)\right]
$$

## Policy Gradient Methods

Advantages of Policy-Based RL

Advantages:

- Better convergence properties
- Effective in high-dimensional or continuous action spaces
- Can learn stochastic policies

Disadvantages:

- Typically converge to a local rather than global optimum
- Evaluating a policy is typically inefficient and high variance

### Anchor-Critic Methods

#### Q Actor-Critic

Reducing Variance Using a Critic

Monte-Carlo Policy Gradient still has high variance.

We use a critic to estimate the action-value function $Q_w(s,a)\approx Q^{\pi_\theta}(s,a)$.

Anchor-critic algorithms maintain two sets of parameters:

Critic: updates action-value function parameters $w$

Actor: updates policy parameters $\theta$, in direction suggested by the critic.

Actor-critic algorithms follow an approximate policy gradient:

$$
\nabla_\theta J(\theta) \approx \mathbb{E}_{\pi_{\theta}}\left[\nabla_\theta \log \pi_\theta(s,a)Q_w(s,a)\right]
$$
$$
\Delta \theta = \alpha \nabla_\theta \log \pi_\theta(s,a)Q_w(s,a)
$$

Action-Value Actor-Critic

- Simple actor-critic algorithm based on action-value critic
- Using linear value function approximation $Q_w(s,a)=\phi(s,a)^\top w$

Critic: updates $w$ by linear $TD(0)$
Actor: updates $\theta$ by policy gradient

```python
def Q_actor-critic(states,theta):
    actions=sample_actions(a,pi_theta)
    for i in range(num_steps):
        reward=sample_rewards(actions,states)
        transition=sample_transition(actions,states)
        new_actions=sample_action(transition,theta)
        delta=sample_reward+gamma*Q_w(transition, new_actions)-Q_w(states, actions)
        theta=theta+alpha*nabla_theta*log(pi_theta(states, actions))*Q_w(states, actions)
        w=w+beta*delta*phi(states, actions)
        a=new_actions
        s=transition
```

#### Advantage Actor-Critic

Reducing variance using a baseline

- We subtract a baseline function $B(s)$ form the policy gradient
- This can reduce the variance without changing expectation

$$
\begin{aligned}
\mathbb{E}_{\pi_\theta}\left[\nabla_\theta\log \pi_\theta(s,a)B(s)\right]&=\sum_{s\in S}d^{\pi_\theta}(s)\sum_{a\in A}\nabla_{\theta}\pi_\theta(s,a)B(s)\\
&=\sum_{s\in S}d^{\pi_\theta}B(s)\nabla_\theta\sum_{a\in A}\pi_\theta(s,a)\\
&=0
\end{aligned}
$$

A good baseline is the state value function $B(s)=V^{\pi_\theta}(s)$

So we can rewrite the policy gradient using the advantage function $A^{\pi_\theta}(s,a)=Q^{\pi_\theta}(s,a)-V^{\pi_theta}(s)$

$$
\nabla_\theta J(\theta)=\mathbb{E}\left[\nabla_\theta \log \pi_\theta(s,a) A^{\pi_theta}(s,a)\right]
$$

##### Estimating the Advantage function

**Method 1:** direct estimation

> May increase the variance

The advantage function can significantly reduce variance of policy gradient

So the critic should really estimate the advantage function

For example, by estimating both $V^{\pi_theta}(s)$ and $Q^{\pi_theta}(s,a)$

Using two function approximators and two parameter vectors,

$$
V_v(s)\approx V^{\pi_\theta}(s)\\
Q_w(s,a)\approx Q^{\pi_\theta}(s,a)\\
A(s,a)=Q_w(s,a)-V_v(s)
$$

And updating both value functions by e.g. TD learning

**Method 2:** using the TD error

> We can prove that TD error is an unbiased estimation of the advantage function

For the true value function $V^{\pi_\theta}(s)$, the TD error $\delta^{\pi_\theta}$

$$
\delta^{\pi_\theta} = r + \gamma V^{\pi_\theta}(s) - V^{\pi_\theta}(s)
$$

is an unbiased estimate of the advantage function

$$
\begin{aligned}
\mathbb{E}_{\pi_\theta}[\delta^{\pi_\theta}| s,a]&=\mathbb{E}_{\pi_\theta}[r + \gamma V^{\pi_\theta}(s') |s,a]-V^{\pi_\theta}(s)\\
&=Q^{\pi_\theta}(s,a)-V^{\pi_\theta}(s)\\
&=A^{\pi_\theta}(s,a)
\end{aligned}
$$

So we can use the TD error to compute the policy gradient

$$
\Delta \theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) \delta^{\pi_\theta}]
$$

In practice, we can use an approximate TD error $\delta_v=r+\gamma V_v(s')-V_v(s)$ to compute the policy gradient

### Summary of policy gradient algorithms

THe policy gradient has many equivalent forms.

$$
\begin{aligned}
\nabla_\theta J(\theta) &= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) v_t] \text{  REINFORCE} \\
&= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) Q_w(s,a)] \text{  Q Actor-Critic} \\
&= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) A^{\pi_\theta}(s,a)] \text{  Advantage Actor-Critic} \\
&= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) \delta^{\pi_\theta}] \text{  TD Actor-Critic}
\end{aligned}
$$

Each leads s stochastic gradient ascent algorithm.

Critic use policy evaluation to estimate the $Q^\pi(s,a)$ or $A^\pi(s,a)$ or $V^\pi(s)$.

## Compatible Function Approximation

If the following two conditions are satisfied:

1. Value function approximation is a compatible with the policy
    $$
    \nabla_w Q_w(s,a) = \nabla_\theta \log \pi_\theta(s,a)
    $$
2. Value function parameters $w$ minimize the MSE
    $$
    \epsilon = \mathbb{E}_{\pi_\theta}[(Q^{\pi_\theta}(s,a)-Q_w(s,a))^2]
    $$
    Note $\epsilon$ need not be zero, just need to be minimized.

Then the policy gradient is exact

$$
\nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) Q_w(s,a)]
$$

Remember:

$$
\nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) Q^{\pi_\theta}(s,a)]
$$

### Challenges with Policy Gradient Methods

- 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

