# Deep Deterministic Policy Gradient (DDPG)

## Overview

DDPG is a popular DRL algorithm for continuous control. It extends DQN to work with the continuous action space by introducing a deterministic actor that directly outputs continuous actions. DDPG also combines techniques from DQN, such as the replay buffer and target network.

Original paper:

Reference resources:

## Implemented Variants

Variants Implemented Description
ddpg_continuous_action.py, docs For continuous action space

Below is our single-file implementation of DDPG:

## ddpg_continuous_action.py

The ddpg_continuous_action.py has the following features:

• For continuous action space
• Works with the Box observation space of low-level features
• Works with the Box (continuous) action space

### Usage

poetry install
poetry install -E pybullet
python cleanrl/ddpg_continuous_action.py --help
python cleanrl/ddpg_continuous_action.py --env-id HopperBulletEnv-v0
poetry install -E mujoco # only works in Linux
python cleanrl/ddpg_continuous_action.py --env-id Hopper-v3


### Explanation of the logged metrics

Running python cleanrl/ddpg_continuous_action.py will automatically record various metrics such as actor or value losses in Tensorboard. Below is the documentation for these metrics:

• charts/episodic_return: episodic return of the game
• charts/SPS: number of steps per second
• losses/qf1_loss: the mean squared error (MSE) between the Q values at timestep $$t$$ and the Bellman update target estimated using the reward $$r_t$$ and the Q values at timestep $$t+1$$, thus minimizing the one-step temporal difference. Formally, it can be expressed by the equation below. $$J(\theta^{Q}) = \mathbb{E}_{(s,a,r,s') \sim \mathcal{D}} \big[ (Q(s, a) - y)^2 \big],$$ with the Bellman update target $$y = r + \gamma \, Q^{'}(s', a')$$, where $$a' \sim \mu^{'}(s')$$, and the replay buffer $$\mathcal{D}$$.

• losses/actor_loss: implemented as -qf1(data.observations, actor(data.observations)).mean(); it is the negative average Q values calculated based on the 1) observations and the 2) actions computed by the actor based on these observations. By minimizing actor_loss, the optimizer updates the actors parameter using the following gradient (Lillicrap et al., 2016, Algorithm 1)1:

$\nabla_{\theta^{\mu}} J \approx \frac{1}{N}\sum_i\left.\left.\nabla_{a} Q\left(s, a \mid \theta^{Q}\right)\right|_{s=s_{i}, a=\mu\left(s_{i}\right)} \nabla_{\theta^{\mu}} \mu\left(s \mid \theta^{\mu}\right)\right|_{s_{i}}$
• losses/qf1_values: implemented as qf1(data.observations, data.actions).view(-1), it is the average Q values of the sampled data in the replay buffer; useful when gauging if under or over estimation happens.

### Implementation details

Our ddpg_continuous_action.py is based on the OurDDPG.py from sfujim/TD3, which presents the the following implementation difference from (Lillicrap et al., 2016)1:

1. ddpg_continuous_action.py uses a gaussian exploration noise $$\mathcal{N}(0, 0.1)$$, while (Lillicrap et al., 2016)1 uses Ornstein-Uhlenbeck process with $$\theta=0.15$$ and $$\sigma=0.2$$.

2. ddpg_continuous_action.py runs the experiments using the openai/gym MuJoCo environments, while (Lillicrap et al., 2016)1 uses their proprietary MuJoCo environments.

3. ddpg_continuous_action.py uses the following architecture:

class QNetwork(nn.Module):
def __init__(self, env):
super(QNetwork, self).__init__()
self.fc1 = nn.Linear(np.array(env.single_observation_space.shape).prod() + np.prod(env.single_action_space.shape), 256)
self.fc2 = nn.Linear(256, 256)
self.fc3 = nn.Linear(256, 1)

def forward(self, x, a):
x = torch.cat([x, a], 1)
x = F.relu(self.fc1(x))
x = F.relu(self.fc2(x))
x = self.fc3(x)
return x

class Actor(nn.Module):
def __init__(self, env):
super(Actor, self).__init__()
self.fc1 = nn.Linear(np.array(env.single_observation_space.shape).prod(), 256)
self.fc2 = nn.Linear(256, 256)
self.fc_mu = nn.Linear(256, np.prod(env.single_action_space.shape))

def forward(self, x):
x = F.relu(self.fc1(x))
x = F.relu(self.fc2(x))

while (Lillicrap et al., 2016, see Appendix 7 EXPERIMENT DETAILS)1 uses the following architecture (difference highlighted):

class QNetwork(nn.Module):
def __init__(self, env):
super(QNetwork, self).__init__()
self.fc1 = nn.Linear(np.array(env.single_observation_space.shape).prod(), 400)
self.fc2 = nn.Linear(400 + np.prod(env.single_action_space.shape), 300)
self.fc3 = nn.Linear(300, 1)

def forward(self, x, a):
x = F.relu(self.fc1(x))
x = torch.cat([x, a], 1)
x = F.relu(self.fc2(x))
x = self.fc3(x)
return x

class Actor(nn.Module):
def __init__(self, env):
super(Actor, self).__init__()
self.fc1 = nn.Linear(np.array(env.single_observation_space.shape).prod(), 400)
self.fc2 = nn.Linear(400, 300)
self.fc_mu = nn.Linear(300, np.prod(env.single_action_space.shape))

def forward(self, x):
x = F.relu(self.fc1(x))
x = F.relu(self.fc2(x))

4. ddpg_continuous_action.py uses the following learning rates:

q_optimizer = optim.Adam(list(qf1.parameters()), lr=3e-4)
actor_optimizer = optim.Adam(list(actor.parameters()), lr=3e-4)

while (Lillicrap et al., 2016, see Appendix 7 EXPERIMENT DETAILS)1 uses the following learning rates:

q_optimizer = optim.Adam(list(qf1.parameters()), lr=1e-4)
actor_optimizer = optim.Adam(list(actor.parameters()), lr=1e-3)

5. ddpg_continuous_action.py uses --batch-size=256 --tau=0.005, while (Lillicrap et al., 2016, see Appendix 7 EXPERIMENT DETAILS)1 uses --batch-size=64 --tau=0.001

6. ddpg_continuous_action.py also adds support for handling continuous environments where the lower and higher bounds of the action space are not $$[-1,1]$$, or are asymmetric. The case where the bounds are not $$[-1,1]$$ is handled in DDPG.py (Fujimoto et al., 2018)2 as follows:

class Actor(nn.Module):

...

def forward(self, state):
a = F.relu(self.l1(state))
a = F.relu(self.l2(a))
return self.max_action * torch.tanh(self.l3(a)) # Scale from [-1,1] to [-action_high, action_high]

On the other hand, in CleanRL's ddpg_continuous_action.py, the mean and the scale of the the action space are computed as action_bias and action_scale respectively. Those scalars are in turn used to scale the output of a tanh activation function in the actor to the original action space range:
class Actor(nn.Module):
def __init__(self, env):
...
# action rescaling
self.register_buffer("action_scale", torch.FloatTensor((env.action_space.high - env.action_space.low) / 2.0))
self.register_buffer("action_bias", torch.FloatTensor((env.action_space.high + env.action_space.low) / 2.0))

def forward(self, x):
x = F.relu(self.fc1(x))
x = F.relu(self.fc2(x))
x = torch.tanh(self.fc_mu(x))
return x * self.action_scale + self.action_bias # Scale from [-1,1] to [-action_low, action_high]


Additionally, when drawing exploration noise that is added to the actions produced by the actor, CleanRL's ddpg_continuous_action.py centers the distribution the sampled from at action_bias, and the scale of the distribution is set to action_scale * exploration_noise.

Info

Note that Humanoid-v2, InvertedPendulum-v2, Pusher-v2 have action space bounds that are not the standard [-1, 1]. See below.

Ant-v2 Observation space: Box(-inf, inf, (111,), float64) Action space: Box(-1.0, 1.0, (8,), float32)
HalfCheetah-v2 Observation space: Box(-inf, inf, (17,), float64) Action space: Box(-1.0, 1.0, (6,), float32)
Hopper-v2 Observation space: Box(-inf, inf, (11,), float64) Action space: Box(-1.0, 1.0, (3,), float32)
Humanoid-v2 Observation space: Box(-inf, inf, (376,), float64) Action space: Box(-0.4, 0.4, (17,), float32)
InvertedDoublePendulum-v2 Observation space: Box(-inf, inf, (11,), float64) Action space: Box(-1.0, 1.0, (1,), float32)
InvertedPendulum-v2 Observation space: Box(-inf, inf, (4,), float64) Action space: Box(-3.0, 3.0, (1,), float32)
Pusher-v2 Observation space: Box(-inf, inf, (23,), float64) Action space: Box(-2.0, 2.0, (7,), float32)
Reacher-v2 Observation space: Box(-inf, inf, (11,), float64) Action space: Box(-1.0, 1.0, (2,), float32)
Swimmer-v2 Observation space: Box(-inf, inf, (8,), float64) Action space: Box(-1.0, 1.0, (2,), float32)
Walker2d-v2 Observation space: Box(-inf, inf, (17,), float64) Action space: Box(-1.0, 1.0, (6,), float32)


### Experiment results

To run benchmark experiments, see benchmark/ddpg.sh. Specifically, execute the following command:

Below are the average episodic returns for ddpg_continuous_action.py (3 random seeds). To ensure the quality of the implementation, we compared the results against (Fujimoto et al., 2018)2.

Environment ddpg_continuous_action.py OurDDPG.py (Fujimoto et al., 2018, Table 1)2 DDPG.py using settings from (Lillicrap et al., 2016)1 in (Fujimoto et al., 2018, Table 1)2
HalfCheetah 9382.32 ± 1395.52 8577.29 3305.60
Walker2d 1598.35 ± 862.66 3098.11 1843.85
Hopper 1313.43 ± 684.46 1860.02 2020.46
Humanoid 897.74 ± 281.87 not available
Pusher -34.45 ± 4.47 not available
InvertedPendulum 645.67 ± 270.31 1000.00 ± 0.00
Info

Note that ddpg_continuous_action.py uses gym MuJoCo v2 environments while OurDDPG.py (Fujimoto et al., 2018)2 uses the gym MuJoCo v1 environments. According to the openai/gym#834, gym MuJoCo v2 environments should be equivalent to the gym MuJoCo v1 environments.

Also note the performance of our ddpg_continuous_action.py seems to be worse than the reference implementation on Walker2d and Hopper. This is likely due to openai/gym#938. We would have a hard time reproducing gym MuJoCo v1 environments because they have been long deprecated.

One other thing could cause the performance difference: the original code reported the average episodic return using determinisitc evaluation (i.e., without exploration noise), see sfujim/TD3/main.py#L15-L32, whereas we reported the episodic return during training and the policy gets updated between environments steps.

Learning curves:      ## ddpg_continuous_action_jax.py

The ddpg_continuous_action_jax.py has the following features:

### Usage

poetry install -E "mujoco jax"
poetry run pip install --upgrade "jax[cuda]==0.3.14" -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html
poetry run python -c "import mujoco_py"
python cleanrl/ddpg_continuous_action_jax.py --help
poetry install -E mujoco # only works in Linux
python cleanrl/ddpg_continuous_action_jax.py --env-id Hopper-v3

Warning

Note that JAX does not work in Windows . The official docs recommends using Windows Subsystem for Linux (WSL) to install JAX.

### Explanation of the logged metrics

See related docs for ddpg_continuous_action.py.

### Implementation details

See related docs for ddpg_continuous_action.py.

### Experiment results

To run benchmark experiments, see benchmark/ddpg.sh. Specifically, execute the following command:

Below are the average episodic returns for ddpg_continuous_action_jax.py (3 random seeds). To ensure the quality of the implementation, we compared the results against (Fujimoto et al., 2018)2.

Environment ddpg_continuous_action_jax.py (RTX 3060 TI) ddpg_continuous_action_jax.py (VM w/ TPU) ddpg_continuous_action.py (RTX 2060) OurDDPG.py (Fujimoto et al., 2018, Table 1)2
HalfCheetah 9910.53 ± 673.49 9790.72 ± 1494.85 9382.32 ± 1395.52 8577.29
Walker2d 1397.60 ± 677.12 1314.83 ± 689.71 1598.35 ± 862 3098.11
Hopper 1603.5 ± 727.281 1602.20 ± 696.11 1313.43 ± 684.46 1860.02
Info

Note that the experiments were conducted on different hardwares, so your mileage might vary. This inconsistency is because 1) re-running expeirments on the same hardware is computationally expensive and 2) requiring the same hardware is not inclusive nor feasible to other contributors who might have different hardwares.

That said, we roughly expect to see a 2-4x speed improvement from using ddpg_continuous_action_jax.py under the same hardware. And if you disable the --capture-video overhead, the speed improvement will be even higher.

Learning curves:      Tracked experiments and game play videos:

1. Lillicrap, T.P., Hunt, J.J., Pritzel, A., Heess, N.M., Erez, T., Tassa, Y., Silver, D., & Wierstra, D. (2016). Continuous control with deep reinforcement learning. CoRR, abs/1509.02971. https://arxiv.org/abs/1509.02971

2. Fujimoto, S., Hoof, H.V., & Meger, D. (2018). Addressing Function Approximation Error in Actor-Critic Methods. ArXiv, abs/1802.09477. https://arxiv.org/abs/1802.09477