205 lines
6.3 KiB
Markdown
205 lines
6.3 KiB
Markdown
# CSE510 Deep Reinforcement Learning (Lecture 12)
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## Policy Gradient Theorem
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For any differentiable policy $\pi_\theta(s,a)$, for any o the policy objective functions $J=J_1, J_{avR}$ or $\frac{1}{1-\gamma} J_{avV}$
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The policy gradient is
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$$
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\nabla_{\theta}J(\theta)=\mathbb{E}_{\pi_{\theta}}\left[\nabla_\theta \log \pi_\theta(s,a)Q^{\pi_\theta}(s,a)\right]
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$$
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## Policy Gradient Methods
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Advantages of Policy-Based RL
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Advantages:
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- Better convergence properties
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- Effective in high-dimensional or continuous action spaces
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- Can learn stochastic policies
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Disadvantages:
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- Typically converge to a local rather than global optimum
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- Evaluating a policy is typically inefficient and high variance
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### Anchor-Critic Methods
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#### Q Actor-Critic
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Reducing Variance Using a Critic
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Monte-Carlo Policy Gradient still has high variance.
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We use a critic to estimate the action-value function $Q_w(s,a)\approx Q^{\pi_\theta}(s,a)$.
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Anchor-critic algorithms maintain two sets of parameters:
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Critic: updates action-value function parameters $w$
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Actor: updates policy parameters $\theta$, in direction suggested by the critic.
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Actor-critic algorithms follow an approximate policy gradient:
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$$
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\nabla_\theta J(\theta) \approx \mathbb{E}_{\pi_{\theta}}\left[\nabla_\theta \log \pi_\theta(s,a)Q_w(s,a)\right]
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$$
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$$
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\Delta \theta = \alpha \nabla_\theta \log \pi_\theta(s,a)Q_w(s,a)
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$$
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Action-Value Actor-Critic
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- Simple actor-critic algorithm based on action-value critic
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- Using linear value function approximation $Q_w(s,a)=\phi(s,a)^\top w$
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Critic: updates $w$ by linear $TD(0)$
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Actor: updates $\theta$ by policy gradient
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```python
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def Q_actor-critic(states,theta):
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actions=sample_actions(a,pi_theta)
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for i in range(num_steps):
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reward=sample_rewards(actions,states)
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transition=sample_transition(actions,states)
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new_actions=sample_action(transition,theta)
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delta=sample_reward+gamma*Q_w(transition, new_actions)-Q_w(states, actions)
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theta=theta+alpha*nabla_theta*log(pi_theta(states, actions))*Q_w(states, actions)
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w=w+beta*delta*phi(states, actions)
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a=new_actions
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s=transition
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```
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#### Advantage Actor-Critic
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Reducing variance using a baseline
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- We subtract a baseline function $B(s)$ form the policy gradient
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- This can reduce the variance without changing expectation
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$$
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\begin{aligned}
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\mathbb{E}_{\pi_\theta}\left[\nabla_\theta\log \pi_\theta(s,a)B(s)\right]&=\sum_{s\in S}d^{\pi_\theta}(s)\sum_{a\in A}\nabla_{\theta}\pi_\theta(s,a)B(s)\\
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&=\sum_{s\in S}d^{\pi_\theta}B(s)\nabla_\theta\sum_{a\in A}\pi_\theta(s,a)\\
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&=0
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\end{aligned}
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$$
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A good baseline is the state value function $B(s)=V^{\pi_\theta}(s)$
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So we can rewrite the policy gradient using the advantage function $A^{\pi_\theta}(s,a)=Q^{\pi_\theta}(s,a)-V^{\pi_theta}(s)$
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$$
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\nabla_\theta J(\theta)=\mathbb{E}\left[\nabla_\theta \log \pi_\theta(s,a) A^{\pi_theta}(s,a)\right]
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$$
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##### Estimating the Advantage function
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**Method 1:** direct estimation
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> May increase the variance
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The advantage function can significantly reduce variance of policy gradient
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So the critic should really estimate the advantage function
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For example, by estimating both $V^{\pi_theta}(s)$ and $Q^{\pi_theta}(s,a)$
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Using two function approximators and two parameter vectors,
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$$
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V_v(s)\approx V^{\pi_\theta}(s)\\
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Q_w(s,a)\approx Q^{\pi_\theta}(s,a)\\
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A(s,a)=Q_w(s,a)-V_v(s)
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$$
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And updating both value functions by e.g. TD learning
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**Method 2:** using the TD error
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> We can prove that TD error is an unbiased estimation of the advantage function
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For the true value function $V^{\pi_\theta}(s)$, the TD error $\delta^{\pi_\theta}$
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$$
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\delta^{\pi_\theta} = r + \gamma V^{\pi_\theta}(s) - V^{\pi_\theta}(s)
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$$
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is an unbiased estimate of the advantage function
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$$
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\begin{aligned}
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\mathbb{E}_{\pi_\theta}[\delta^{\pi_\theta}| s,a]&=\mathbb{E}_{\pi_\theta}[r + \gamma V^{\pi_\theta}(s') |s,a]-V^{\pi_\theta}(s)\\
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&=Q^{\pi_\theta}(s,a)-V^{\pi_\theta}(s)\\
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&=A^{\pi_\theta}(s,a)
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\end{aligned}
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$$
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So we can use the TD error to compute the policy gradient
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$$
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\Delta \theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) \delta^{\pi_\theta}]
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$$
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In practice, we can use an approximate TD error $\delta_v=r+\gamma V_v(s')-V_v(s)$ to compute the policy gradient
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### Summary of policy gradient algorithms
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THe policy gradient has many equivalent forms.
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$$
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\begin{aligned}
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\nabla_\theta J(\theta) &= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) v_t] \text{ REINFORCE} \\
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&= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) Q_w(s,a)] \text{ Q Actor-Critic} \\
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&= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) A^{\pi_\theta}(s,a)] \text{ Advantage Actor-Critic} \\
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&= \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) \delta^{\pi_\theta}] \text{ TD Actor-Critic}
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\end{aligned}
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$$
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Each leads s stochastic gradient ascent algorithm.
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Critic use policy evaluation to estimate the $Q^\pi(s,a)$ or $A^\pi(s,a)$ or $V^\pi(s)$.
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## Compatible Function Approximation
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If the following two conditions are satisfied:
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1. Value function approximation is a compatible with the policy
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$$
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\nabla_w Q_w(s,a) = \nabla_\theta \log \pi_\theta(s,a)
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$$
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2. Value function parameters $w$ minimize the MSE
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$$
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\epsilon = \mathbb{E}_{\pi_\theta}[(Q^{\pi_\theta}(s,a)-Q_w(s,a))^2]
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$$
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Note $\epsilon$ need not be zero, just need to be minimized.
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Then the policy gradient is exact
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$$
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\nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) Q_w(s,a)]
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$$
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Remember:
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$$
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\nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a) Q^{\pi_\theta}(s,a)]
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$$
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### Challenges with Policy Gradient Methods
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- Data Inefficiency
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- On-policy method: for each new policy, we need to generate a completely new
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- trajectory
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- The data is thrown out after just one gradient update
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- As complex neural networks need many updates, this makes the training process very slow
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- Unstable update： step size is very important
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- If step size is too large:
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- Large step -> bad policy
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- Next batch is generated from current bad policy -> collect bad samples
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- Bad samples -> worse policy (compare to supervised learning: the correct label and data in the following batches may correct it)
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- If step size is too small: the learning process is slow
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