## Configuration for Colab¶

In :
import sys

if IN_COLAB:
!apt install python-opengl
!apt install ffmpeg
!apt install xvfb
!pip install pyvirtualdisplay
from pyvirtualdisplay import Display

# Start virtual display
dis = Display(visible=0, size=(600, 400))
dis.start()


# 05. Soft Actor Critic (SAC)¶

The main purpose of SAC is to maximize the actor's entropy while maximizing expected reward. We can expect both sample efficient learning and stability because maximizing entropy provides a substantial improvement in exploration and robustness.

As an extension of standard RL's objective function $\sum_t \mathbb{E}_{(s_t, a_t) \sim \rho_\pi} [r(s_t, a_t)]$, let's consider a more general maximum entropy objective which favors stochastic policies by augmenting the objective with the expected entropy of the policy over $\rho_\pi (s_t)$:

$$J(\pi) = \sum_{t=0}^T \mathbb{E}_{(s_t, a_t) \sim \rho_\pi} [r(s_t, a_t) + \alpha H(\pi(\cdot | s_t))].$$

The temperature parameter $\alpha$ determines the relative importance of the entropy term against the reward, and thus controls the stochasticity of the optimal policy. By this objective, the policy can explore more widely and capture multiple modes of near-optimal behavior. In conclusion, it considerably improves learning speed over other methods that optimize the conventional RL objective function.

In the paper, the authors show that Soft Policy Iteration guarantees convergence based on a tabular setting (4.1), and they extend it to a practical approximation for large continuous domains (4.2). Firstly, the soft value function is trained to minimize the squared residual error:

$$J_V (\psi) = \mathbb{E}_{s_t \sim D} \big[ \frac{1}{2}(v_\psi (s_t) - \mathbb{E}_{a_t \sim \pi_\phi} [Q_\theta(s_t, a_t) - \log_{\pi_\phi}(a_t | s_t)])^2 \big],$$

where $D$ is the distribution of previously sampled states and actions, or a replay buffer. Second, the soft Q-function parameters can be trained to minimize the soft Bellman residual:

$$J_Q (\theta) = \mathbb{E}_{(s_t, a_t) \sim D} \big[ \frac{1}{2} \big( Q_\theta(s_t, a_t) - \hat{Q}(s_t, a_t) \big)^2 \big],$$

with $\hat{Q}(s_t, a_t) = r(s_t, a_t) + \gamma \mathbb{E}_{s_{t+1} \sim \rho} [V_{\tilde{\psi}} (s_{t+1})].$

Finally, the policy paramameters can be learned by directly minimizing the following expected KL-divergence:

$$J_\pi(\phi) = \mathbb{E}_{s_t \sim D} \big[ D_{KL} \big( \pi_{\phi} (\cdot | s_t) \| \frac{\exp(Q_{\theta}(s_t, \cdot))}{Z_\theta(s_t)} \big) \big].$$

We can rewirte the objective as

$$J_\pi(\phi) = \mathbb{E}_{s_t \sim D, \epsilon_t \sim N} [ \log_{\pi_\phi}(f_\phi(\epsilon_t ; s_t) | s_t) - Q_\theta (s_t, f_\phi (\epsilon_t ; s_t))],$$

where $\pi_\phi$ is defined implicitly in terms of $f_\phi$, and the partition function is independent of $\phi$ and can thus be omitted.

One thing to note is that the authors suggest to use two Q-functions to mitigate positive bias in the policy improvement step that is known to degrade performance of value based methods. In particular, we parameterize two Q-functions, with parameters $\theta_i$, and train them independently to optimize $J_Q(\theta_i)$. We then use the minimum of the Q-functions for the value gradient and policy gradient. Two Q-functions can significantly speed up training, especially on harder tasks.

### Can we do better?¶

In Soft Actor Critic paper, the experiment of reward scale shows that SAC's performance quite varies depending on reward scaling. In the follow-up paper , the authors assume that the temperature parameter $\alpha$ needs to be adjusted depending on the magnitude of the reward, and they define the soft policy optimization as a constrained problem.

$$\max_{\pi_{0:T}} \mathbb{E}_{\rho_\pi} \big[ \sum_{t=0}^T r(s_t, a_t) \big] \text{ s.t. } \mathbb{E}_{(s_t, a_t) \sim \rho_\pi} [-\log(\pi_t(a_t|s_t))] \ge H \text{ for all } t,$$

where $H$ is a desired minimum expected entropy. This constrained maximization becomes the following dual problem.

$$\min_{a_T \ge 0} \max_{\pi_T} \mathbb{E} [r(s_T, a_T) - \alpha_T \log \pi(a_t|s_t)] - \alpha_T H,$$

where $\alpha_T$ is the dual variable. Furthermore, it can be rewrited as a optimization problem with regards to $\alpha$.

$$J(\alpha) = \mathbb{E}_{a_t \sim \pi_t} [-\alpha \log \pi_t (a_t | s_t) - \alpha H].$$

By optimizing this dual problem, we can adjust the dual variable $\alpha$, which plays the role of the temperature.

## Import modules¶

In :
import random
from typing import Dict, List, Tuple

import gym
import matplotlib.pyplot as plt
import numpy as np
import torch
import torch.nn as nn
import torch.nn.functional as F
import torch.optim as optim
from IPython.display import clear_output
from torch.distributions import Normal


## Set random seed¶

In :
if torch.backends.cudnn.enabled:
torch.backends.cudnn.benchmark = False
torch.backends.cudnn.deterministic = True

seed = 777
torch.manual_seed(seed)
np.random.seed(seed)
random.seed(seed)


## Replay buffer¶

Typically, people implement replay buffers with one of the following three data structures:

• collections.deque
• list
• numpy.ndarray

deque is very easy to handle once you initialize its maximum length (e.g. deque(maxlen=buffer_size)). However, the indexing operation of deque gets terribly slow as it grows up because it is internally doubly linked list. On the other hands, list is an array, so it is relatively faster than deque when you sample batches at every step. Its amortized cost of Get item is O(1).

Last but not least, let's see numpy.ndarray. numpy.ndarray is even faster than list due to the fact that it is a homogeneous array of fixed-size items, so you can get the benefits of locality of reference, . Whereas list is an array of pointers to objects, even when all of them are of the same type.

Here, we are going to implement a replay buffer using numpy.ndarray.

Reference:

In :
class ReplayBuffer:
"""A simple numpy replay buffer."""

def __init__(self, obs_dim: int, size: int, batch_size: int = 32):
"""Initialize."""
self.obs_buf = np.zeros([size, obs_dim], dtype=np.float32)
self.next_obs_buf = np.zeros([size, obs_dim], dtype=np.float32)
self.acts_buf = np.zeros([size], dtype=np.float32)
self.rews_buf = np.zeros([size], dtype=np.float32)
self.done_buf = np.zeros([size], dtype=np.float32)
self.max_size, self.batch_size = size, batch_size
self.ptr, self.size, = 0, 0

def store(self,
obs: np.ndarray,
act: np.ndarray,
rew: float,
next_obs: np.ndarray,
done: bool,
):
"""Store the transition in buffer."""
self.obs_buf[self.ptr] = obs
self.next_obs_buf[self.ptr] = next_obs
self.acts_buf[self.ptr] = act
self.rews_buf[self.ptr] = rew
self.done_buf[self.ptr] = done
self.ptr = (self.ptr + 1) % self.max_size
self.size = min(self.size + 1, self.max_size)

def sample_batch(self) -> Dict[str, np.ndarray]:
"""Randomly sample a batch of experiences from memory."""
idxs = np.random.choice(self.size, size=self.batch_size, replace=False)
return dict(obs=self.obs_buf[idxs],
next_obs=self.next_obs_buf[idxs],
acts=self.acts_buf[idxs],
rews=self.rews_buf[idxs],
done=self.done_buf[idxs])

def __len__(self) -> int:
return self.size


## Network¶

We are going to use three different networks for policy, Q-function, and V-function. We use two Q-functions to mitigate positive bias and softly update V-function for stable learning. One interesting thing is that the policy network works as Tanh Normal distribution which enforces action bounds. (The details are descibed in Appendix C of .)

In :
def init_layer_uniform(layer: nn.Linear, init_w: float = 3e-3) -> nn.Linear:
"""Init uniform parameters on the single layer."""
layer.weight.data.uniform_(-init_w, init_w)
layer.bias.data.uniform_(-init_w, init_w)

return layer

class Actor(nn.Module):
def __init__(
self,
in_dim: int,
out_dim: int,
log_std_min: float = -20,
log_std_max: float = 2,
):
"""Initialize."""
super(Actor, self).__init__()

# set the log std range
self.log_std_min = log_std_min
self.log_std_max = log_std_max

# set the hidden layers
self.hidden1 = nn.Linear(in_dim, 128)
self.hidden2 = nn.Linear(128, 128)

# set log_std layer
self.log_std_layer = nn.Linear(128, out_dim)
self.log_std_layer = init_layer_uniform(self.log_std_layer)

# set mean layer
self.mu_layer = nn.Linear(128, out_dim)
self.mu_layer = init_layer_uniform(self.mu_layer)

def forward(self, state: torch.Tensor) -> torch.Tensor:
"""Forward method implementation."""
x = F.relu(self.hidden1(state))
x = F.relu(self.hidden2(x))

# get mean
mu = self.mu_layer(x).tanh()

# get std
log_std = self.log_std_layer(x).tanh()
log_std = self.log_std_min + 0.5 * (
self.log_std_max - self.log_std_min
) * (log_std + 1)
std = torch.exp(log_std)

# sample actions
dist = Normal(mu, std)
z = dist.rsample()

# normalize action and log_prob
# see appendix C of 
action = z.tanh()
log_prob = dist.log_prob(z) - torch.log(1 - action.pow(2) + 1e-7)
log_prob = log_prob.sum(-1, keepdim=True)

return action, log_prob

class CriticQ(nn.Module):
def __init__(self, in_dim: int):
"""Initialize."""
super(CriticQ, self).__init__()

self.hidden1 = nn.Linear(in_dim, 128)
self.hidden2 = nn.Linear(128, 128)
self.out = nn.Linear(128, 1)
self.out = init_layer_uniform(self.out)

def forward(
self, state: torch.Tensor, action: torch.Tensor
) -> torch.Tensor:
"""Forward method implementation."""
x = torch.cat((state, action), dim=-1)
x = F.relu(self.hidden1(x))
x = F.relu(self.hidden2(x))
value = self.out(x)

return value

class CriticV(nn.Module):
def __init__(self, in_dim: int):
"""Initialize."""
super(CriticV, self).__init__()

self.hidden1 = nn.Linear(in_dim, 128)
self.hidden2 = nn.Linear(128, 128)
self.out = nn.Linear(128, 1)
self.out = init_layer_uniform(self.out)

def forward(self, state: torch.Tensor) -> torch.Tensor:
"""Forward method implementation."""
x = F.relu(self.hidden1(state))
x = F.relu(self.hidden2(x))
value = self.out(x)

return value


## SAC Agent¶

Here is a summary of SACAgent class.

Method Note
select_action select an action from the input state.
step take an action and return the response of the env.
update_model update the model by gradient descent.
train train the agent during num_frames.
test test the agent (1 episode).
_target_soft_update soft update from the local model to the target model.
_plot plot the training progresses.
In :
class SACAgent:
"""SAC agent interacting with environment.

Attrtibutes:
actor (nn.Module): actor model to select actions
actor_optimizer (Optimizer): optimizer for training actor
vf (nn.Module): critic model to predict state values
vf_target (nn.Module): target critic model to predict state values
vf_optimizer (Optimizer): optimizer for training vf
qf_1 (nn.Module): critic model to predict state-action values
qf_2 (nn.Module): critic model to predict state-action values
qf_1_optimizer (Optimizer): optimizer for training qf_1
qf_2_optimizer (Optimizer): optimizer for training qf_2
env (gym.Env): openAI Gym environment
memory (ReplayBuffer): replay memory
batch_size (int): batch size for sampling
gamma (float): discount factor
tau (float): parameter for soft target update
initial_random_steps (int): initial random action steps
policy_update_freq (int): policy update frequency
device (torch.device): cpu / gpu
target_entropy (int): desired entropy used for the inequality constraint
log_alpha (torch.Tensor): weight for entropy
alpha_optimizer (Optimizer): optimizer for alpha
transition (list): temporory storage for the recent transition
total_step (int): total step numbers
is_test (bool): flag to show the current mode (train / test)
"""

def __init__(
self,
env: gym.Env,
memory_size: int,
batch_size: int,
gamma: float = 0.99,
tau: float = 5e-3,
initial_random_steps: int = int(1e4),
policy_update_freq: int = 2,
):
"""Initialize."""
obs_dim = env.observation_space.shape
action_dim = env.action_space.shape

self.env = env
self.memory = ReplayBuffer(obs_dim, memory_size, batch_size)
self.batch_size = batch_size
self.gamma = gamma
self.tau = tau
self.initial_random_steps = initial_random_steps
self.policy_update_freq = policy_update_freq

# device: cpu / gpu
self.device = torch.device(
"cuda" if torch.cuda.is_available() else "cpu"
)
print(self.device)

# automatic entropy tuning
self.target_entropy = -np.prod((action_dim,)).item()  # heuristic

# actor
self.actor = Actor(obs_dim, action_dim).to(self.device)

# v function
self.vf = CriticV(obs_dim).to(self.device)
self.vf_target = CriticV(obs_dim).to(self.device)

# q function
self.qf_1 = CriticQ(obs_dim + action_dim).to(self.device)
self.qf_2 = CriticQ(obs_dim + action_dim).to(self.device)

# optimizers

# transition to store in memory
self.transition = list()

# total steps count
self.total_step = 0

# mode: train / test
self.is_test = False

def select_action(self, state: np.ndarray) -> np.ndarray:
"""Select an action from the input state."""
# if initial random action should be conducted
if self.total_step < self.initial_random_steps and not self.is_test:
selected_action = self.env.action_space.sample()
else:
selected_action = self.actor(
torch.FloatTensor(state).to(self.device)
).detach().cpu().numpy()

self.transition = [state, selected_action]

return selected_action

def step(self, action: np.ndarray) -> Tuple[np.ndarray, float, bool]:
"""Take an action and return the response of the env."""
next_state, reward, done, _ = self.env.step(action)

if not self.is_test:
self.transition += [reward, next_state, done]
self.memory.store(*self.transition)

return next_state, reward, done

def update_model(self) -> Tuple[torch.Tensor, ...]:
"""Update the model by gradient descent."""
device = self.device  # for shortening the following lines

samples = self.memory.sample_batch()
state = torch.FloatTensor(samples["obs"]).to(device)
next_state = torch.FloatTensor(samples["next_obs"]).to(device)
action = torch.FloatTensor(samples["acts"].reshape(-1, 1)).to(device)
reward = torch.FloatTensor(samples["rews"].reshape(-1, 1)).to(device)
done = torch.FloatTensor(samples["done"].reshape(-1, 1)).to(device)
new_action, log_prob = self.actor(state)

# train alpha (dual problem)
alpha_loss = (
-self.log_alpha.exp() * (log_prob + self.target_entropy).detach()
).mean()

alpha_loss.backward()
self.alpha_optimizer.step()

alpha = self.log_alpha.exp()  # used for the actor loss calculation

# q function loss
q_1_pred = self.qf_1(state, action)
q_2_pred = self.qf_2(state, action)
v_target = self.vf_target(next_state)
q_target = reward + self.gamma * v_target * mask
qf_1_loss = F.mse_loss(q_1_pred, q_target.detach())
qf_2_loss = F.mse_loss(q_2_pred, q_target.detach())

# v function loss
v_pred = self.vf(state)
q_pred = torch.min(
self.qf_1(state, new_action), self.qf_2(state, new_action)
)
v_target = q_pred - alpha * log_prob
vf_loss = F.mse_loss(v_pred, v_target.detach())

if self.total_step % self.policy_update_freq == 0:
# actor loss
actor_loss = (alpha * log_prob - advantage).mean()

# train actor
actor_loss.backward()
self.actor_optimizer.step()

# target update (vf)
self._target_soft_update()
else:
actor_loss = torch.zeros(1)

# train Q functions
qf_1_loss.backward()
self.qf_1_optimizer.step()

qf_2_loss.backward()
self.qf_2_optimizer.step()

qf_loss = qf_1_loss + qf_2_loss

# train V function
vf_loss.backward()
self.vf_optimizer.step()

return actor_loss.data, qf_loss.data, vf_loss.data, alpha_loss.data

def train(self, num_frames: int, plotting_interval: int = 200):
"""Train the agent."""
self.is_test = False

state = self.env.reset()
actor_losses, qf_losses, vf_losses, alpha_losses = [], [], [], []
scores = []
score = 0

for self.total_step in range(1, num_frames + 1):
action = self.select_action(state)
next_state, reward, done = self.step(action)

state = next_state
score += reward

# if episode ends
if done:
state = env.reset()
scores.append(score)
score = 0

if (
len(self.memory) >= self.batch_size
and self.total_step > self.initial_random_steps
):
losses = self.update_model()
actor_losses.append(losses)
qf_losses.append(losses)
vf_losses.append(losses)
alpha_losses.append(losses)

# plotting
if self.total_step % plotting_interval == 0:
self._plot(
self.total_step,
scores,
actor_losses,
qf_losses,
vf_losses,
alpha_losses
)

self.env.close()

def test(self):
"""Test the agent."""
self.is_test = True

state = self.env.reset()
done = False
score = 0

frames = []
while not done:
frames.append(self.env.render(mode="rgb_array"))
action = self.select_action(state)
next_state, reward, done = self.step(action)

state = next_state
score += reward

print("score: ", score)
self.env.close()

return frames

def _target_soft_update(self):
"""Soft-update: target = tau*local + (1-tau)*target."""
tau = self.tau

for t_param, l_param in zip(
self.vf_target.parameters(), self.vf.parameters()
):
t_param.data.copy_(tau * l_param.data + (1.0 - tau) * t_param.data)

def _plot(
self,
frame_idx: int,
scores: List[float],
actor_losses: List[float],
qf_losses: List[float],
vf_losses: List[float],
alpha_losses: List[float],
):
"""Plot the training progresses."""
def subplot(loc: int, title: str, values: List[float]):
plt.subplot(loc)
plt.title(title)
plt.plot(values)

subplot_params = [
(151, f"frame {frame_idx}. score: {np.mean(scores[-10:])}", scores),
(152, "actor_loss", actor_losses),
(153, "qf_loss", qf_losses),
(154, "vf_loss", vf_losses),
(155, "alpha_loss", alpha_losses),
]

clear_output(True)
plt.figure(figsize=(30, 5))
for loc, title, values in subplot_params:
subplot(loc, title, values)
plt.show()


## Environment¶

ActionNormalizer is an action wrapper class to normalize the action values ranged in (-1. 1). Thanks to this class, we can make the agent simply select action values within the zero centered range (-1, 1).

In :
class ActionNormalizer(gym.ActionWrapper):
"""Rescale and relocate the actions."""

def action(self, action: np.ndarray) -> np.ndarray:
"""Change the range (-1, 1) to (low, high)."""
low = self.action_space.low
high = self.action_space.high

scale_factor = (high - low) / 2
reloc_factor = high - scale_factor

action = action * scale_factor + reloc_factor
action = np.clip(action, low, high)

return action

def reverse_action(self, action: np.ndarray) -> np.ndarray:
"""Change the range (low, high) to (-1, 1)."""
low = self.action_space.low
high = self.action_space.high

scale_factor = (high - low) / 2
reloc_factor = high - scale_factor

action = (action - reloc_factor) / scale_factor
action = np.clip(action, -1.0, 1.0)

return action


You can see the code and configurations of Pendulum-v0 from OpenAI's repository.

In :
# environment
env_id = "Pendulum-v0"
env = gym.make(env_id)
env = ActionNormalizer(env)


## Initialize¶

In :
# parameters
num_frames = 50000
memory_size = 100000
batch_size = 128
initial_random_steps = 10000

agent = SACAgent(
env, memory_size, batch_size, initial_random_steps=initial_random_steps
)

cuda


## Train¶

In :
agent.train(num_frames) ## Test¶

Run the trained agent (1 episode).

In :
# test
if IN_COLAB:
agent.env = gym.wrappers.Monitor(agent.env, "videos", force=True)
frames = agent.test()

score:  -231.72002778923985


## Render¶

In :
if IN_COLAB:  # for colab
import base64
import glob
import io
import os

from IPython.display import HTML, display

def ipython_show_video(path: str) -> None:
"""Show a video at path within IPython Notebook."""
if not os.path.isfile(path):
raise NameError("Cannot access: {}".format(path))

encoded = base64.b64encode(video)

display(HTML(
data="""
<video alt="test" controls>
<source src="data:video/mp4;base64,{0}" type="video/mp4"/>
</video>
""".format(encoded.decode("ascii"))
))

list_of_files = glob.glob("videos/*.mp4")
latest_file = max(list_of_files, key=os.path.getctime)
print(latest_file)
ipython_show_video(latest_file)

else:  # for jupyter
from matplotlib import animation
from JSAnimation.IPython_display import display_animation
from IPython.display import display

def display_frames_as_gif(frames):
"""Displays a list of frames as a gif, with controls."""
patch = plt.imshow(frames)
plt.axis('off')

def animate(i):
patch.set_data(frames[i])

anim = animation.FuncAnimation(
plt.gcf(), animate, frames = len(frames), interval=50
)
display(display_animation(anim, default_mode='loop'))

# display
display_frames_as_gif(frames)


Once Loop Reflect
In [ ]: