# CSE510 Deep Reinforcement Learning (Lecture 16)

## Deterministic Policy Gradient (DPG)

### Learning Deterministic Policies

- Deterministic policy gradients [Silver et al., ICML 2014]
  - Explicitly learn a deterministic policy.
  - $a = \mu_\theta(s)$
- Advantages
  - Existing optimal deterministic policy for MDPs
  - Naturally dealing with a continuous action space
  - Expected to be more efficient than learning stochastic policies
    - Computing stochastic gradient requires more samples, as it integrates over both state and action space.
    - Deterministic gradient is preferable as it integrates over state space only.

### Deterministic Policy Gradient

The objective function is:

$$
J(\theta)=\int_{s\in S} \rho^{\mu}(s) r(s,\mu_\theta(s)) ds
$$

where $\rho^{\mu}(s)$ is the stationary distribution under the behavior policy $\mu_\theta(s)$.

The policy gradient from the standard policy gradient theorem is:

$$
\nabla_\theta J(\theta) = \mathbb{E}_{s\sim \rho^{\mu}}[\nabla_\theta Q^{\mu_\theta}(s,a)]=\mathbb{E}_{s\sim \rho^{\mu}}[\nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)}]
$$

#### Issues for DPG

The formulations up to now can only use on-policy data.

Deterministic policy can hardly guarantee sufficient
exploration.

- Solution: Off-policy training using a stochastic behavior policy.

#### Off-Policy Deterministic Policy Gradient (Off-DPG)

Use a stochastic behavior policy $\beta(a|s)$. The modified objective function is:

$$
J(\mu_\theta)=\int_{s\in S} \rho^{\beta}(s) Q^{\mu_\theta}(s,\beta(s)) ds
$$

The gradients are:

$$
\begin{aligned}
\nabla_\theta J(\mu_\theta) &\approx \int_{s\in S} \rho^{\beta}(s) \nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)} ds\\
&= \mathbb{E}_{s\sim \rho^{\beta}}[\nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)}]
\end{aligned}
$$

Importance sampling is avoided in the actor due to the absence of integral over actions.

#### Policy Evaluation in DPG

Importance sampling can also be avoided in the critic.

Gradient TD-like algorithm can be directly applied to the critic.

$$
\mathcal{L}_{critic}(w) = \mathbb{E}[r_t+\gamma Q^w(s_{t+1},a_{t+1})-Q^w(s_t,a_t)]^2
$$

#### Off-Policy Deterministic Actor-Critic

$$
\delta_t=r_t+\gamma Q^w(s_{t+1},a_{t+1})-Q^w(s_t,a_t)
$$

$$
w_{t+1} = w_t + \alpha_w \delta_t \nabla_w Q^w(s_t,a_t)
$$

$$
\theta_{t+1} = \theta_t + \alpha_\theta \nabla_\theta \mu_\theta(s_t) \nabla_a Q^{\mu_\theta}(s_t,a_t)\vert_{a=\mu_\theta(s_t)}
$$

### Deep Deterministic Policy Gradient (DDPG)

Insights from DQN + Deterministic Policy Gradients

    - Use a replay buffer
- Critic is updated every timestep (Sample from buffer, minibatch):

$$
\mathcal{L}_{critic}(w) = \mathbb{E}[r_t+\gamma Q^w(s_{t+1},a_{t+1})-Q^w(s_t,a_t)]^2
$$

Actor is updated every timestep:

$$
\nabla_a Q(s_t,a;w)|_{a=\mu_\theta(s_t)} \nabla_\theta \mu_\theta(s_t)
$$

Smoothing target updated at every timestep:

$$
w_{t+1} = \tau w_t + (1-\tau) w_{t+1}
$$

$$
\theta_{t+1} = \tau \theta_t + (1-\tau) \theta_{t+1}
$$

Exploration: add noise to the action selection: $a_t = \mu_\theta(s_t) + \mathcal{N}_t$

Batch normalization used for training networks

### Extension of DDPG

Overestimation bias is an issue of Q-learning in which the maximization of a noisy value estimate

$$
DDPG:\nabla_\theta J(\theta) = \mathbb{E}_{s\sim \rho^{\mu}}[\nabla_\theta \mu_\theta(s) \nabla_a Q^{\mu_\theta}(s,a)\vert_{a=\mu_\theta(s)}]
$$

#### Double DQN is not enough

Because the slow-changing policy in an actor-critic setting

- the current and target value estimates remain too similar to avoid maximization bias.
- Target value of Double DQN: $r_t + \gamma Q^w'(s_{t+1},\mu_\theta(s_{t+1}))$

#### TD3: Twin Delayed Deep Deterministic policy gradient

Address overestimation bias:

- Double Q-learning is unbiased in tabular settings, but still slight overestimation with function approximation.

$$
y_1 = r + \gamma Q^{\theta_2'}(s', \pi_{\phi_1}(s'))
$$
$$
y_2 = r + \gamma Q^{\theta_1'}(s', \pi_{\phi_2}(s'))
$$
 It is possible that $Q^{\theta_2}(s, \pi_{\phi_1}(s)) > Q^{\theta_1}(s, \pi_{\phi_1}(s))$

Clipped double Q-learning:

$$
y_1 = r + \gamma \min_{i=1,2} Q^{\theta_i'}(s', \pi_{\phi_i}(s'))
$$

High-variance estimates provide a noisy gradient.

Techniques in TD3 to reduce the variance:

- Update the policy at a lower frequency than the value network.
- Smoothing the value estimate:
  $$
  y=r+\gamma \mathbb{E}_{\epsilon}[Q^{\theta'}(s', \pi_{\phi'}(s')+\epsilon)]
  $$

Update target:

$$
y=r+\gamma \mathbb{E}_{\epsilon}[Q^{\theta'}(s', \pi_{\phi'}(s')+\epsilon)]
$$

where $\epsilon\sim clip(\mathcal{N}(0, \sigma), -c, c)$

#### Other methods

- Generalizable Episode Memory for Deep Reinforcement Learning
- Distributed Distributional Deep Deterministic Policy Gradient
  - Distributional critic
  - N-step returns are used to update the critic
  - Multiple distributed parallel actors
  - Prioritized experience replay
- 