# 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: 

* [Continuous control with deep reinforcement learning](https://arxiv.org/abs/1509.02971)

Reference resources:

* :material-github: [sfujim/TD3](https://github.com/sfujim/TD3)
* [Deep Deterministic Policy Gradient | Spinning Up in Deep RL](https://spinningup.openai.com/en/latest/algorithms/ddpg.html)
* :material-github: [ikostrikov/jaxrl](https://github.com/ikostrikov/jaxrl) (helpful reference when implemented [`ddpg_continuous_action_jax.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py))

## Implemented Variants


| Variants Implemented      | Description |
| ----------- | ----------- |
| :material-github: [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py), :material-file-document: [docs](/rl-algorithms/ddpg/#ddpg_continuous_actionpy) | For continuous action space |
| :material-github: [`ddpg_continuous_action_jax.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py), :material-file-document: [docs](/rl-algorithms/ddpg/#ddpg_continuous_action_jaxpy) | For continuous action space |

Below is our single-file implementation of DDPG:

## `ddpg_continuous_action.py`

The [ddpg_continuous_action.py](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/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"

    ```bash
    uv pip install .
    uv run python cleanrl/ddpg_continuous_action.py --help
    uv pip install ".[mujoco]"
    uv run python cleanrl/ddpg_continuous_action.py --env-id Hopper-v4
    ```

=== "pip"

    ```bash
    python cleanrl/ddpg_continuous_action.py --help
    pip install -r requirements/requirements-mujoco.txt
    python cleanrl/ddpg_continuous_actions.py --env-id Hopper-v4
    ```

### 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`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) is based on the [`OurDDPG.py`](https://github.com/sfujim/TD3/blob/master/OurDDPG.py) from :material-github: [sfujim/TD3](https://github.com/sfujim/TD3), which presents the the following implementation difference from (Lillicrap et al., 2016)[^1]:

1. [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/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 $\sigma=0.2$.

1. [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) runs the experiments using the `openai/gym` MuJoCo environments, while (Lillicrap et al., 2016)[^1] uses their proprietary MuJoCo environments.

1. [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) uses the following architecture:
    ```python
    class QNetwork(nn.Module):
        def __init__(self, env):
            super().__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().__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))
            # action rescaling
            self.register_buffer(
                "action_scale", torch.tensor((env.action_space.high - env.action_space.low) / 2.0, dtype=torch.float32)
            )
            self.register_buffer(
                "action_bias", torch.tensor((env.action_space.high + env.action_space.low) / 2.0, dtype=torch.float32)
            )

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

    ```python hl_lines="4-6 9-11 19-21"
    class QNetwork(nn.Module):
        def __init__(self, env):
            super(QNetwork, self).__init__()
            self.fc1 = nn.Linear(p.array(env.single_observation_space.shape).prod() + np.prod(env.single_action_space.shape), 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))
            # action rescaling
            self.register_buffer(
                "action_scale", torch.tensor((env.action_space.high - env.action_space.low) / 2.0, dtype=torch.float32)
            )
            self.register_buffer(
                "action_bias", torch.tensor((env.action_space.high + env.action_space.low) / 2.0, dtype=torch.float32)
            )

        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
    ```

1. [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) uses the following learning rates:

    ```python
    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:

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

1. [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/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`

1. [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/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`](https://github.com/sfujim/TD3/blob/385b33ac7de4767bab17eb02ade4a268d3e4e24f/DDPG.py#L15) (Fujimoto et al., 2018)[^2] as follows:
```python
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`](https://github.com/dosssman/cleanrl/blob/10b606e7bd9bd1b06e455e8ef542df2b7699a20c/cleanrl/ddpg_continuous_action.py#L98), 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:
```python
class Actor(nn.Module):
    def __init__(self, env):
        ...
        # action rescaling
        self.register_buffer(
            "action_scale", torch.tensor((env.action_space.high - env.action_space.low) / 2.0, dtype=torch.float32)
        )
        self.register_buffer(
            "action_bias", torch.tensor((env.action_space.high + env.action_space.low) / 2.0, dtype=torch.float32)
        )

    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`](https://github.com/dosssman/cleanrl/blob/10b606e7bd9bd1b06e455e8ef542df2b7699a20c/cleanrl/ddpg_continuous_action.py#L175) 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 :material-github: [benchmark/ddpg.sh](https://github.com/vwxyzjn/cleanrl/blob/master/benchmark/ddpg.sh). Specifically, execute the following command:


``` title="benchmark/ddpg.sh" linenums="1"
--8<-- "benchmark/ddpg.sh::7"
```


Below are the average episodic returns for [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/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`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) | [`OurDDPG.py`](https://github.com/sfujim/TD3/blob/master/OurDDPG.py) (Fujimoto et al., 2018, Table 1)[^2]  | [`DDPG.py`](https://github.com/sfujim/TD3/blob/master/DDPG.py) using settings from (Lillicrap et al., 2016)[^1] in (Fujimoto et al., 2018, Table 1)[^2]    |
| ----------- | ----------- | ----------- | ----------- |
| HalfCheetah-v4       | 10374.07 ± 157.37      |8577.29  | 3305.60|
| Walker2d-v4    | 1240.16 ± 390.10     |  3098.11 | 1843.85 |
| Hopper-v4    | 1576.78 ± 818.98         |  1860.02 | 2020.46 |
| InvertedPendulum-v4  |    642.68 ± 69.56    | 1000.00 ± 0.00  | 
| Humanoid-v4  |  1699.56 ± 694.22      |  not available | 
| Pusher-v4  |  -77.30 ± 38.78      |  not available | 



???+ info

    Note that [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) uses gym MuJoCo v4 environments while [`OurDDPG.py`](https://github.com/sfujim/TD3/blob/master/OurDDPG.py) (Fujimoto et al., 2018)[^2] uses 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 :material-github: [openai/gym#938](https://github.com/openai/baselines/issues/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`](https://github.com/sfujim/TD3/blob/385b33ac7de4767bab17eb02ade4a268d3e4e24f/main.py#L15-L32), whereas we reported the episodic return during training and the policy gets updated between environments steps.

Learning curves:

``` title="benchmark/ddpg_plot.sh" linenums="1"
--8<-- "benchmark/ddpg_plot.sh::9"
```

<img loading="lazy" src="https://huggingface.co/datasets/cleanrl/benchmark/resolve/main/benchmark/pr-424/ddpg.png">
<img loading="lazy" src="https://huggingface.co/datasets/cleanrl/benchmark/resolve/main/benchmark/pr-424/ddpg-time.png">

<iframe loading="lazy" src="https://wandb.ai/openrlbenchmark/openrlbenchmark/reports/MuJoCo-CleanRL-s-DDPG--VmlldzoxNjkyMjc1" style="width:100%; height:500px" title="MuJoCo: CleanRL's DDPG"></iframe>



## `ddpg_continuous_action_jax.py`

The [ddpg_continuous_action_jax.py](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py) has the following features:

* Uses [Jax](https://github.com/google/jax), [Flax](https://github.com/google/flax), and [Optax](https://github.com/deepmind/optax) instead of `torch`.  [ddpg_continuous_action_jax.py](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py) is roughly 2.5-4x faster than  [ddpg_continuous_action.py](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py)
* For continuous action space
* Works with the `Box` observation space of low-level features
* Works with the `Box` (continuous) action space

### Usage

=== "poetry"

    ```bash
    uv pip install ".[mujoco, jax]"
    uv run python cleanrl/ddpg_continuous_action_jax.py --help
    uv run python cleanrl/ddpg_continuous_action_jax.py --env-id Hopper-v4
    ```

=== "pip"

    ```bash
    pip install -r requirements/requirements-mujoco.txt
    pip install -r requirements/requirements-jax.txt
    python cleanrl/ddpg_continuous_action_jax.py --help
    python cleanrl/ddpg_continuous_action_jax.py --env-id Hopper-v4
    ```


???+ warning

    Note that JAX does not work in Windows :fontawesome-brands-windows:. The official [docs](https://github.com/google/jax#installation) recommends using Windows Subsystem for Linux (WSL) to install JAX.

### Explanation of the logged metrics

See [related docs](/rl-algorithms/ddpg/#explanation-of-the-logged-metrics) for `ddpg_continuous_action.py`.


### Implementation details

See [related docs](/rl-algorithms/ddpg/#implementation-details) for `ddpg_continuous_action.py`.


### Experiment results

To run benchmark experiments, see :material-github: [benchmark/ddpg.sh](https://github.com/vwxyzjn/cleanrl/blob/master/benchmark/ddpg.sh). Specifically, execute the following command:


``` title="benchmark/ddpg.sh" linenums="1"
--8<-- "benchmark/ddpg.sh:12:19"
```

Below are the average episodic returns for [`ddpg_continuous_action_jax.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py) (3 random seeds).


{!benchmark/ddpg.md!}

Learning curves:


``` title="benchmark/ddpg_plot.sh" linenums="1"
--8<-- "benchmark/ddpg_plot.sh:11:20"
```

<img loading="lazy" src="https://huggingface.co/datasets/cleanrl/benchmark/resolve/main/benchmark/pr-424/ddpg_jax.png">
<img loading="lazy" src="https://huggingface.co/datasets/cleanrl/benchmark/resolve/main/benchmark/pr-424/ddpg_jax-time.png">


???+ info

    These are some previous experiments with TPUs. Note the results are very similar to the ones above, but the runtime can be different due to different hardware used.


    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`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py) under the same hardware. And if you disable the `--capture_video` overhead, the speed improvement will be even higher.


    | Environment      | [`ddpg_continuous_action_jax.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py) (RTX 3060 TI) | [`ddpg_continuous_action_jax.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action_jax.py) (VM w/ TPU) | [`ddpg_continuous_action.py`](https://github.com/vwxyzjn/cleanrl/blob/master/cleanrl/ddpg_continuous_action.py) (RTX 3060 TI) | [`OurDDPG.py`](https://github.com/sfujim/TD3/blob/master/OurDDPG.py) (Fujimoto et al., 2018, Table 1)[^2]    |
    | ----------- | ----------- | ----------- | ----------- | ----------- |
    | HalfCheetah |  9592.25 ± 135.10 | 9125.06 ± 1477.58  | 10210.57 ± 196.22      |8577.29  |
    | Walker2d |  1083.15 ± 567.65  | 1303.82 ± 448.41 | 1661.14 ± 250.01     |  3098.11 | 
    | Hopper |  1275.28 ± 209.60  | 1145.05 ± 41.95 | 1007.44 ± 148.29         |  1860.02 |    

    Learning curves:

    <div class="grid-container">
    <img loading="lazy" src="../ddpg-jax/HalfCheetah-v2.png">
    <img loading="lazy" src="../ddpg-jax/HalfCheetah-v2-time.png">

    <img loading="lazy" src="../ddpg-jax/Walker2d-v2.png">
    <img loading="lazy" src="../ddpg-jax/Walker2d-v2-time.png">

    <img loading="lazy" src="../ddpg-jax/Hopper-v2.png">
    <img loading="lazy" src="../ddpg-jax/Hopper-v2-time.png">
    </div> 

    Tracked experiments and game play videos:

    <iframe loading="lazy" src="https://wandb.ai/openrlbenchmark/openrlbenchmark/reports/MuJoCo-CleanRL-s-DDPG-JAX--VmlldzoyMjQxMjE2" style="width:100%; height:500px" title="MuJoCo: CleanRL's DDPG + JAX"></iframe>


[^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