# Soft Actor-Critic (SAC)

## Overview

The Soft Actor-Critic (SAC) algorithm extends the DDPG algorithm by 1) using a stochastic policy, which in theory can express multi-modal optimal policies. This also enables the use of 2) entropy regularization based on the stochastic policy's entropy. It serves as a built-in, state-dependent exploration heuristic for the agent, instead of relying on non-correlated noise processes as in DDPG, or TD3 Additionally, it incorporates the 3) usage of two Soft Q-network to reduce the overestimation bias issue in Q-network-based methods.

Original papers: The SAC algorithm's initial proposal, and later updates and improvements can be chronologically traced through the following publications:

Reference resources:

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

Below is our single-file implementations of SAC:

## sac_continuous_action.py

The sac_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.
• Numerically stable stochastic policy based on openai/spinningup and pranz24/pytorch-soft-actor-critic implementations.
• Supports automatic entropy coefficient $$\alpha$$ tuning, enabled by default.

### Usage

poetry install

# Pybullet
poetry install -E pybullet

## Default
python cleanrl/sac_continuous_action.py --env-id HopperBulletEnv-v0

## Without Automatic entropy coef. tuning
python cleanrl/sac_continuous_action.py --env-id HopperBulletEnv-v0 --autotune False --alpha 0.2


### 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: the episodic return of the game during training

• charts/SPS: number of steps per second

• losses/qf1_loss, losses/qf2_loss: for each Soft Q-value network $$Q_{\theta_i}$$, $$i \in \{1,2\}$$, this metric holds the mean squared error (MSE) between the soft Q-value estimate $$Q_{\theta_i}(s, a)$$ and the entropy regularized Bellman update target estimated as $$r_t + \gamma \, Q_{\theta_{i}^{'}}(s', a') + \alpha \, \mathcal{H} \big[ \pi(a' \vert s') \big]$$.

More formally, the Soft Q-value loss for the $$i$$-th network is obtained by:

$J(\theta^{Q}_{i}) = \mathbb{E}_{(s,a,r,s') \sim \mathcal{D}} \big[ (Q_{\theta_i}(s, a) - y)^2 \big]$

with the entropy regularized, Soft Bellman update target: $$y = r(s, a) + \gamma ({\color{orange} \min_{\theta_{1,2}}Q_{\theta_i^{'}}(s',a')} - \alpha \, \text{log} \pi( \cdot \vert s'))$$ where $$a' \sim \pi( \cdot \vert s')$$, $$\text{log} \pi( \cdot \vert s')$$ approximates the entropy of the policy, and $$\mathcal{D}$$ is the replay buffer storing samples of the agent during training.

Here, $$\min_{\theta_{1,2}}Q_{\theta_i^{'}}(s',a')$$ takes the minimum Soft Q-value network estimate between the two target Q-value networks $$Q_{\theta_1^{'}}$$ and $$Q_{\theta_2^{'}}$$ for the next state and action pair, so as to reduce over-estimation bias.

• losses/qf_loss: averages losses/qf1_loss and losses/qf2_loss for comparison with algorithms using a single Q-value network.

• losses/actor_loss: Given the stochastic nature of the policy in SAC, the actor (or policy) objective is formulated so as to maximize the likelihood of actions $$a \sim \pi( \cdot \vert s)$$ that would result in high Q-value estimate $$Q(s, a)$$. Additionally, the policy objective encourages the policy to maintain its entropy high enough to help explore, discover, and capture multi-modal optimal policies.

The policy's objective function can thus be defined as:

$\text{max}_{\phi} \, J_{\pi}(\phi) = \mathbb{E}_{s \sim \mathcal{D}} \Big[ \text{min}_{i=1,2} Q_{\theta_i}(s, a) - \alpha \, \text{log}\pi_{\phi}(a \vert s) \Big]$

where the action is sampled using the reparameterization trick1: $$a = \mu_{\phi}(s) + \epsilon \, \sigma_{\phi}(s)$$ with $$\epsilon \sim \mathcal{N}(0, 1)$$, $$\text{log} \pi_{\phi}( \cdot \vert s')$$ approximates the entropy of the policy, and $$\mathcal{D}$$ is the replay buffer storing samples of the agent during training.

• losses/alpha: $$\alpha$$ coefficient for entropy regularization of the policy.

• losses/alpha_loss: In the policy's objective defined above, the coefficient of the entropy bonus $$\alpha$$ is kept fixed all across the training. As suggested by the authors in Section 5 of the Soft Actor-Critic And Applications paper, the original purpose of augmenting the standard reward with the entropy of the policy is to encourage exploration of not well enough explored states (thus high entropy). Conversely, for states where the policy has already learned a near-optimal policy, it would be preferable to reduce the entropy bonus of the policy, so that it does not become sub-optimal due to the entropy maximization incentive.

Therefore, having a fixed value for $$\alpha$$ does not fit this desideratum of matching the entropy bonus with the knowledge of the policy at an arbitrary state during its training.

To mitigate this, the authors proposed a method to dynamically adjust $$\alpha$$ as the policy is trained, which is as follows:

$\alpha^{*}_t = \text{argmin}_{\alpha_t} \mathbb{E}_{a_t \sim \pi^{*}_t} \big[ -\alpha_t \, \text{log}\pi^{*}_t(a_t \vert s_t; \alpha_t) - \alpha_t \mathcal{H} \big],$

where $$\mathcal{H}$$ represents the target entropy, the desired lower bound for the expected entropy of the policy over the trajectory distribution induced by the latter. As a heuristic for the target entropy, the authors use the dimension of the action space of the task.

### Implementation details

CleanRL's sac_continuous_action.py implementation is based on openai/spinningup.

1. sac_continuous_action.py uses a numerically stable estimation method for the standard deviation $$\sigma$$ of the policy, which squashes it into a range of reasonable values for a standard deviation:

LOG_STD_MAX = 2
LOG_STD_MIN = -5

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_mean = nn.Linear(256, np.prod(env.single_action_space.shape))
self.fc_logstd = nn.Linear(256, np.prod(env.single_action_space.shape))
# action rescaling
self.action_scale = torch.FloatTensor((env.action_space.high - env.action_space.low) / 2.0)
self.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))
mean = self.fc_mean(x)
log_std = self.fc_logstd(x)
log_std = torch.tanh(log_std)
log_std = LOG_STD_MIN + 0.5 * (LOG_STD_MAX - LOG_STD_MIN) * (log_std + 1)  # From SpinUp / Denis Yarats

return mean, log_std

def get_action(self, x):
mean, log_std = self(x)
std = log_std.exp()
normal = torch.distributions.Normal(mean, std)
x_t = normal.rsample()  # for reparameterization trick (mean + std * N(0,1))
y_t = torch.tanh(x_t)
action = y_t * self.action_scale + self.action_bias
log_prob = normal.log_prob(x_t)
# Enforcing Action Bound
log_prob -= torch.log(self.action_scale * (1 - y_t.pow(2)) + 1e-6)
log_prob = log_prob.sum(1, keepdim=True)
mean = torch.tanh(mean) * self.action_scale + self.action_bias
return action, log_prob, mean

def to(self, device):
self.action_scale = self.action_scale.to(device)
self.action_bias = self.action_bias.to(device)
return super(Actor, self).to(device)

Note that unlike openai/spinningup's implementation which uses LOG_STD_MIN = -20, CleanRL's uses LOG_STD_MIN = -5 instead.

2. sac_continuous_action.py uses different learning rates for the policy and the Soft Q-value networks optimization.

    parser.add_argument("--policy-lr", type=float, default=3e-4,
help="the learning rate of the policy network optimizer")
help="the learning rate of the Q network network optimizer")

while openai/spinningup's uses a single learning rate of lr=1e-3 for both components.

Note that in case it is used, the automatic entropy coefficient $$\alpha$$'s tuning shares the q-lr learning rate:

    # Automatic entropy tuning
if args.autotune:
target_entropy = -torch.prod(torch.Tensor(envs.single_action_space.shape).to(device)).item()
alpha = log_alpha.exp().item()
else:
alpha = args.alpha


3. sac_continuous_action.py uses --batch-size=256 while openai/spinningup's uses --batch-size=100 by default.

### Experiment results

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

The table below compares the results of CleanRL's sac_continuous_action.py with the latest published results by the original authors of the SAC algorithm.

Info

Note that the results table above references the training episodic return for sac_continuous_action.py, the results of Soft Actor-Critic Algorithms and Applications reference evaluation episodic return obtained by running the policy in the deterministic mode.

Environment sac_continuous_action.py SAC: Algorithms and Applications @ 1M steps
HalfCheetah-v2 10310.37 ± 1873.21 ~11,250
Walker2d-v2 4418.15 ± 592.82 ~4,800
Hopper-v2 2685.76 ± 762.16 ~3,250

Learning curves:

Tracked experiments and game play videos:

1. Diederik P Kingma, Max Welling (2016). Auto-Encoding Variational Bayes. ArXiv, abs/1312.6114. https://arxiv.org/abs/1312.6114