# Our Blog

Ongoing observations by End Point people

## Self-driving toy car using the Asynchronous Advantage Actor-Critic algorithm

By Kamil Ciemniewski
August 29, 2018

The field of Reinforcement Learning has seen a lot of great improvement in the past years. Researchers at universities and companies like Deep Mind have been developing new and better ways to train intelligent, artificial agents to solve more and more difficult tasks. The algorithms being developed are requiring less time to train. They also are making the training much more stable.

As the subject is both wide and deep, I’m assuming the reader has the relevant background mastered already. Although reading it might be interesting even without understanding most of the notions in use, having a good grasp of them will help you get the most out of it.

Because we’re looking at the Deep Reinforcement Learning, the obvious requirement is to be acquainted with the neural networks. I’m also using different notions known in the field of Reinforcement Learning overall like $Q(a, s)$ and $V(s)$ functions or the n-step return. The mathematical expressions, in particular, are given assuming that the reader already knows what the symbols stand for. Some notions known from other families of RL algorithms are being touched on as well (e.g. experience replay) — to contrast them with the A3C way of solving the same kind of problems. The article along with the source code uses the OpenAI gym, Python, and PyTorch among other Python-related libraries.

### Theory

The A3C algorithm is a part of the greater class of RL algorithms called Policy Gradients.

In this approach, we’re creating a model that approximates the action-choosing policy itself.

Let’s contrast it with value iteration, the goal of which is to learn the value function and have policy emerge as the function that chooses an action transitioning to the state of the greatest value.

With the policy gradient approach, we’re approximating the policy with a differentiable function. Such stated problem requires only a good approximation of the gradient that over time will maximize the rewards.

The unique approach of A3C adds a very clever twist: we’re also learning an approximation of the value function at the same time. This helps us in getting the variance of the gradient down considerably, making the training much more stable.

These two aspects of the algorithm are being personified within its name: actor-critic. The policy function approximation is being called the actor, while the value function is being called the critic.

As we’ve noticed already, in order to improve our policy function approximation, we need a gradient that points at the direction that maximizes the rewards.

I’m not going to reinvent the wheel here. There are some great resources the reader can access to dig deep into the Mathematics of what’s called the Policy Gradient Theorem:

The following equation presents the basic form of the gradient of the policy function:

$$\nabla_{\theta} J(\theta) = E_{\tau}[\,R_{\tau}\cdot\nabla_\theta\,\sum_{t=0}^{T-1}\,log\,\pi(a_t|s_t;\theta)\,]$$

This states that for each sampled trajectory $\tau$, the correct estimate of the gradient is the expected value of the rewards times the action probabilities moved into the log space. Ascending in this direction makes our rewards greater and greater over time.

We can derive all the needed intermediary gradients ourselves by hand of course. Because we’re using PyTorch though, we only need the right loss function.

Let’s figure out the right loss function formula that will produce the gradient as shown above:

$$L_\theta=-J(\theta)$$

Also:

$$J(\theta)=E_\tau[R_\tau\cdot\sum_{t=0}^{T-1}\,log\,\pi(a_t|s_t;\theta)]$$

Hence:

$$L_\theta=-\frac{1}{n}\sum_{t=0}^{n-1}R_t\,\cdot\,log\pi(a_t|s_t;\theta)$$

#### Formalizing the accumulation of rewards

For now, we’ve been using the $R_\tau$ and $R_t$ terms very abstractly. Let’s make this part more intuitive and concrete now.

Its true meaning really is “the quality of the sampled trajectory”. Consider the following equation:

$$R_t=\sum_{i=t}^{N+t}\gamma^{i-t}r_i\,+\,\gamma^{i-t+1}V(s_{t+N+1})$$

Each $r_i$ is the reward received from the environment after each step. Each trajectory consists of multiple steps. Each time, we’re sampling actions based on our policy function. This gives probabilities of a given action being best given the state.

What if we’re taking 5 actions for which we’re not being given any reward but overall it helped us get rewarded in the 6th step? This is exactly the case we’ll be dealing with in this article later when training a toy car to drive based only on pixel values of the scene. In that environment, we’ll be given $-0.1$ “negative” reward each step and something close to $7$ each new “tile” the car stays on the road.

We need a way to still encourage actions that make us earn rewards in a not too distant future. We also need to be smart and discount future rewards somewhat so that the more immediate the reward is to our action, the more emphasis we put on it.

That’s exactly what the above equation does. Notice that $\gamma$ becomes a hyper-​parameter. It makes sense to give it value from $(0, 1)$. Let’s consider the following list of rewards: $[r_1, r_2, r_3, r_4]$. For $r_1$, the formula for the discounted accumulated reward is:

$$R_1=\gamma\,r_1\,+\,\gamma^2r_2\,+\,\gamma^3r_3\,+\,\gamma^4r_4\,+\,\gamma^5V(s_5)$$

For $r_2$ it’s:

$$R_2=\gamma\,r_2\,+\,\gamma^2r_3\,+\,\gamma^3r_4\,+\,\gamma^4V(s_5)$$

And so on... In case when we hit the terminal state, having no “next” state, we substitute $0$ for $V(s_{t+N+1})$.

We’ve said that in A3C we’re learning the value function at the same time. The $R_t$ as described above becomes the target value when training our $V(s)$. The value function becomes an approximation of the average of the rewards given the state (because $R_t$ depends on us sampling actions in this state).

#### Making the gradients more stable

One of the greatest inhibitors of the policy gradient performance is what’s broadly called “high variance”.

I have to admit, the first time I saw that term in this context, I was disoriented. I knew what “variance” was. It’s the “variance of what” that was not clear to me.

Thankfully I found a brilliant answer to this question. It explains the issue simply yet in detail.

Let me cite it here:

When we talk about high variance in the policy gradient method, we’re specifically talking about the facts that the variance of the gradients are high — namely, that $Var(\nabla_{\theta} J(\theta))$ is big.

To put it in simple terms: because we’re sampling trajectories from the space that is stochastic in nature, we’re bound to have those samples give gradients that disagree a lot on the best direction to take our model’s parameters into.

I encourage the reader to pause now and read the above-mentioned answer as it’s very vital. The gist of the solution described in it is that we can subtract a baseline value from each $R_t$. An example of a good baseline that was given was to make it into an average of the sampled accumulated rewards. The A3C algorithm uses this insight in a very, very clever way.

#### Value function as a baseline

To learn the $V(s)$ we’re typically using the MSE or Huber loss against the accumulated rewards for each step. This means that over time we’re averaging those rewards out based on the state we’re finding ourselves in.

Improving our gradient formula with those ideas we now get:

$$\nabla_{\theta} J(\theta) = E_{\tau}[\,\nabla_\theta\,\sum_{t=0}^{T-1}\,log\,\pi(a_t|s_t;\theta)\cdot(R_t-V(s_t))\,]$$

It’s important to treat the $(R_t-V(s_t))$ term as a constant. This means that when using PyTorch or any other deep learning framework, the computation of it should occur outside the graph that influences the gradients.

The enhanced part of the equation is where we get the word “advantage” in the algorithm’s name. The advantage is simply the difference between the accumulated rewards and what those rewards are on average for the given state:

$$A(a_{t..t+n},s_{t..t+n})=R_t(a_{t..t+n},s_{t..t+n})-V(s_t)$$

If we’ll make $R_t$ into $Q(s,a)$ as it’s commonly written in literature, we’ll arrive at the formula:

$$A(s,a)=Q(s,a) - V(s)$$

What’s the intuition here? Imagine that you’re playing chess with a 5-year-old. You win by a huge margin. Your friend who’s watched lots of master-level games observed this one as well. His take is that even though you scored positively, you still made lots of mistakes. You’ve got your critic here. Your score and what it looks like for the “observing critic” combined is what we call the advantage of the actions you took.

#### Guarding against the model’s overconfidence

Although he was warned, Icarus was too young and too enthusiastic about flying. He got excited by the thrill of flying and carried away by the amazing feeling of freedom and started flying high to salute the sun, diving low to the sea, and then up high again. His father Daedalus was trying in vain to make young Icarus to understand that his behavior was dangerous, and Icarus soon saw his wings melting. Icarus fell into the sea and drowned.

The Myth Of Daedalus And Icarus

The job of an “actor” is to output probability values for each possible action the agent can take. The greater the probability, the greater the model’s confidence that this action will result in the highest reward.

What if at some point, the weights are being steered in a way that the model becomes overconfident of some particular action? If this happens before the model learns much, it becomes a huge problem.

Because we’re using the $\pi(a|s;\theta)$ distribution to sample trajectories with, we’re not sampling totally at random. In other words, for $\pi(a|s;\theta) = [0.1, 0.4, 0.2, 0.3]$ our sampling chooses the second option 40% of the time. With any action overwhelming the others, we’re losing the ability to explore different paths and thus learn valuable lessons.

Empirically, I have found myself seeing the process sometimes not even able to escape the “overconfidence” area for long, long hours.

#### Regularizing with entropy

Let’s introduce the notion of an entropy.

In simple words in our case, it’s the measure of how much “knowledge” does given probability distribution posses. It’s being maximized for the uniform distribution. Here’s the formula:

$$H(X)=E[-log_b(P(X))]$$

This expands to the following:

$$H(X)=-\sum_{i=1}^{n}P(x_i)log_b(P(x_i))$$

Let’s look closer at the values this function produces using the following simple Calca code:

uniform = [0.25, 0.25, 0.25, 0.25]
more confident = [0.5, 0.25, 0.15, 0.10]
over confident = [0.95, 0.01, 0.01, 0.03]
super over confident = [0.99, 0.003, 0.004, 0.003]

y(x) = x*log(x, 10)

entropy(dist) = -sum(map(y, dist))

entropy (uniform) => 0.6021
entropy (more confident) => 0.5246
entropy (over confident) => 0.1068
entropy (super over confident) => 0.0291


We can use the above to “punish” the model whenever it’s too confident of its choices. As we’re going to use gradient descend, we’ll be minimizing terms that appear in our loss function. Minimizing the entropy as shown above would encourage more confidence though. We’ll need to make it into a negative in the loss to work the way we intend:

$$L_\theta=-\frac{1}{n}\sum_{t=0}^{n-1}log\pi(a_t|s_t;\theta)\cdot(R_t-V(s_t))\,-\beta\,H(\pi(a_t|s_t;\theta))$$

Where $\beta$ is a hyperparameter scaling the effects of the penalty that the entropy has on the gradients. Choosing the right value for $\beta$ becomes very vital for the model’s convergence. In this article, I’m using $0.01$ as with $0.001$ I was still observing the process stuck being overconfident.

Let’s include the value loss $L_v$ in the loss function formula making it full and ready to be implemented:

$$L_\theta=-\frac{1}{n}\sum_{t=0}^{n-1}log\pi(a_t|s_t;\theta)\cdot(R_t-V(s_t))\,+\alpha\,L_v\,\,-\beta\,H(\pi(a_t|s_t;\theta))$$

#### The last A in A3C

So far we’ve gone from the vanilla policy gradients to using the notion of an advantage. We’ve also improved it with the baseline that intuitively makes the model consist of two parts: the actor and the critic. At this point, we have what’s sometimes called the A2C — Advantage Actor-Critic.

Let us now focus on the last piece of the puzzle: the last A. This last A comes from the word “asynchronous”. It’s been explained very clearly in the original paper on A3C.

This idea I think is the least complex of all that have their place in the approach. I’ll just comment on what was already written:

These approaches share a common idea: the sequence of observed data encountered by an online RL agent is non-stationary, and online RL updates are strongly correlated. By storing the agent’s data in an experience replay memory, the data can be batched (Riedmiller, 2005; Schulman et al., 2015a) or randomly sampled (Mnih et al., 2013; 2015; Van Hasselt et al., 2015) from different time-steps. Aggregating over memory in this way reduces non-stationarity and decorrelates updates, but at the same time limits the methods to off-policy reinforcement learning algorithms.

The A3C unique approach is that it doesn’t use experience replay for de-correlating the updates to the weights of the model. Instead, we’re sampling many different trajectories at the same time in an asynchronous manner.

This means that we’re creating many clones of the environment and we let our agents experience them at the same time. Separate agents share their weights in one way or another. There are implementations with agents sharing those weights very literally — and performing the updates to the weights on their own whenever they need to. There also are implementations with one main agent holding the main weights and doing the updates based on the gradients reported by the “worker” agents. The worker agents are then being updated with the evolved weights. The environments and agents are not being directly synchronized, working at their own speed. As soon as any of them collects the needed rewards to perform the n-step gradients calculations, the gradients are being applied in one way or another.

In this article, I’m preferring the second approach — having one “main” agent and making workers synchronize their weights with it each n-step period.

### Practice

#### The challenge

To present the above theory in practical terms, we’re going to code the A3C to train a toy self-driving game car. The algorithm will only have the game’s pixels as inputs. We’re also going to collect rewards.

Each step, the player will decide how to move the steering wheel, how much throttle to apply and how much brake.

Points are being assigned for each new “tile” that the car enters staying within the road. There’s a small penalty for each other case of $-0.1$ points.

We’re going to use OpenAI Gym and the environment’s called CarRacing.

You can read a bit more about the setup in the environment’s source code on GitHub.

#### Coding the Agent

Our agent is going to output both $\pi(a|s;\theta)$ as well as $V(s)$. We’re going to use the GRU unit to give the agent the ability to remember its previous actions and environments previous features.

I’ve also decided to use PRelu instead of Relu activations as it appeared to me that this way the agent was learning much quicker (although I don’t have any numbers to back this impression up).

Disclaimer: the code presented below has not been refactored in any way. If this was going to be used in production I’d certainly hugely clean it up.

Here’s the full listing of the agent’s class:

class Agent(nn.Module):
def __init__(self, **kwargs):
super(Agent, self).__init__(**kwargs)

self.init_args = kwargs

self.h = torch.zeros(1, 256)

self.norm1 = nn.BatchNorm2d(4)
self.norm2 = nn.BatchNorm2d(32)

self.conv1 = nn.Conv2d(4, 32, 4, stride=2, padding=1)
self.conv2 = nn.Conv2d(32, 32, 3, stride=2, padding=1)
self.conv3 = nn.Conv2d(32, 32, 3, stride=2, padding=1)
self.conv4 = nn.Conv2d(32, 32, 3, stride=2, padding=1)

self.gru = nn.GRUCell(1152, 256)
self.policy = nn.Linear(256, 4)
self.value = nn.Linear(256, 1)

self.prelu1 = nn.PReLU()
self.prelu2 = nn.PReLU()
self.prelu3 = nn.PReLU()
self.prelu4 = nn.PReLU()

nn.init.xavier_uniform_(self.conv1.weight, gain=nn.init.calculate_gain('leaky_relu'))
nn.init.constant_(self.conv1.bias, 0.01)
nn.init.xavier_uniform_(self.conv2.weight, gain=nn.init.calculate_gain('leaky_relu'))
nn.init.constant_(self.conv2.bias, 0.01)
nn.init.xavier_uniform_(self.conv3.weight, gain=nn.init.calculate_gain('leaky_relu'))
nn.init.constant_(self.conv3.bias, 0.01)
nn.init.xavier_uniform_(self.conv4.weight, gain=nn.init.calculate_gain('leaky_relu'))
nn.init.constant_(self.conv4.bias, 0.01)
nn.init.constant_(self.gru.bias_ih, 0)
nn.init.constant_(self.gru.bias_hh, 0)
nn.init.xavier_uniform_(self.policy.weight, gain=nn.init.calculate_gain('leaky_relu'))
nn.init.constant_(self.policy.bias, 0.01)
nn.init.xavier_uniform_(self.value.weight, gain=nn.init.calculate_gain('leaky_relu'))
nn.init.constant_(self.value.bias, 0.01)

self.train()

def reset(self):
self.h = torch.zeros(1, 256)

def clone(self, num=1):
return [ self.clone_one() for _ in range(num) ]

def clone_one(self):
return Agent(**self.init_args)

def forward(self, state):
state = state.view(1, 4, 96, 96)
state = self.norm1(state)

data = self.prelu1(self.conv1(state))
data = self.prelu2(self.conv2(data))
data = self.prelu3(self.conv3(data))
data = self.prelu4(self.conv4(data))

data = self.norm2(data)
data = data.view(1, -1)

h = self.gru(data, self.h)
self.h = h.detach()

pre_policy = h.view(-1)

policy = F.softmax(self.policy(pre_policy))
value = self.value(pre_policy)

return policy, value


You can immediately notice that actor and critic parts share most of the weights. They only differ in the last layer.

Next, I wanted to abstract out the notion of the “runner”. It encapsulates the idea of a “running agent”. Think of it as the game player — with the joystick and its brain to score game points. I’m discretizing the action space the following way:

Action name value
Turn left [-0.8, 0.0, 0.0]
Turn right [0.8, 0.0, 0]
Full throttle [0.0, 0.1, 0.0]
Brake [0.0, 0.0, 0.6]
class Runner:
def __init__(self, agent, ix, train = True, **kwargs):
self.agent = agent
self.train = train
self.ix = ix
self.reset = False
self.states = []

# each runner has its own environment:
self.env = gym.make('CarRacing-v0')

def get_value(self):
"""
Returns just the current state's value.
This is used when approximating the R.
If the last step was
not terminal, then we're substituting the "r"
with V(s) - hence, we need a way to just
get that V(s) without moving forward yet.
"""
_input = self.preprocess(self.states)
_, _, _, value = self.decide(_input)
return value

def run_episode(self, yield_every = 10, do_render = False):
"""
The episode runner written in the generator style.
This is meant to be used in a "for (...) in run_episode(...):" manner.
Each value generated is a tuple of:
step_ix: the current "step" number
rewards: the list of rewards as received from the environment (without discounting yet)
values: the list of V(s) values, as predicted by the "critic"
policies: the list of policies as received from the "actor"
actions: the list of actions as sampled based on policies
terminal: whether we're in a "terminal" state
"""
self.reset = False
step_ix = 0

rewards, values, policies, actions = [[], [], [], []]

self.env.reset()

# we're going to feed the last 4 frames to the neural network that acts as the "actor-critic" duo. We'll use the "deque" to efficiently drop too old frames always keeping its length at 4:
states = deque([ ])

# we're pre-populating the states deque by taking first 4 steps as "full throttle forward":
while len(states) < 4:
_, r, _, _ = self.env.step([0.0, 1.0, 0.0])
state = self.env.render(mode='rgb_array')
states.append(state)
logger.info('Init reward ' + str(r) )

# we need to repeat the following as long as the game is not over yet:
while True:
# the frames need to be preprocessed (I'm explaining the reasons later in the article)
_input = self.preprocess(states)

# asking the neural network for the policy and value predictions:
action, action_ix, policy, value = self.decide(_input, step_ix)

# taking the step and receiving the reward along with info if the game is over:
_, reward, terminal, _ = self.env.step(action)

# explicitly rendering the scene (again, this will be explained later)
state = self.env.render(mode='rgb_array')

# update the last 4 states deque:
states.append(state)
while len(states) > 4:
states.popleft()

# if we've been asked to render into the window (e. g. to capture the video):
if do_render:
self.env.render()

self.states = states
step_ix += 1

rewards.append(reward)
values.append(value)
policies.append(policy)
actions.append(action_ix)

# periodically save the state's screenshot along with the numerical values in an easy to read way:
if self.ix == 2 and step_ix % 200 == 0:
fname = './screens/car-racing/screen-' + str(step_ix) + '-' + str(int(time.time())) + '.jpg'
im = Image.fromarray(state)
im.save(fname)
state.tofile(fname + '.txt', sep=" ")
_input.numpy().tofile(fname + '.input.txt', sep=" ")

# if it's game over or we hit the "yield every" value, yield the values from this generator:
if terminal or step_ix % yield_every == 0:
yield step_ix, rewards, values, policies, actions, terminal
rewards, values, policies, actions = [[], [], [], []]

# following is a very tacky way to allow external using code to mark that it wants us to reset the environment, finishing the episode prematurely. (this would be hugely refactored in the production code but for the sake of playing with the algorithm itself, it's good enough):
if self.reset:
self.reset = False
self.agent.reset()
states = deque([ ])
self.states = deque([ ])
return

if terminal:
self.agent.reset()
states = deque([ ])
return

self.reset = True

def preprocess(self, states):

def preprocess_one(self, image):
"""
Scales the rendered image and makes it grayscale
"""
return rescale(rgb2gray(image), (0.24, 0.16), anti_aliasing=False, mode='edge', multichannel=False)

def choose_action(self, policy, step_ix):
"""
Chooses an action to take based on the policy and whether we're in the training mode or not. During training, it samples based on the probability values in the policy. During the evaluation, it takes the most probable action in a greedy way.
"""
policies = [[-0.8, 0.0, 0.0], [0.8, 0.0, 0], [0.0, 0.1, 0.0], [0.0, 0.0, 0.6]]

if self.train:
action_ix = np.random.choice(4, 1, p=torch.tensor(policy).detach().numpy())[0]
else:
action_ix = np.argmax(torch.tensor(policy).detach().numpy())

logger.info('Step ' + str(step_ix) + ' Runner ' + str(self.ix) + ' Action ix: ' + str(action_ix) + ' From: ' + str(policy))

return np.array(policies[action_ix], dtype=np.float32), action_ix

def decide(self, state, step_ix = 999):
policy, value = self.agent(state)

action, action_ix = self.choose_action(policy, step_ix)

return action, action_ix, policy, value

"""
As we'll have multiple "worker" runners, they will need to be able to sync their agents' weights with the main agent.
This function loads the weights into this runner's agent.
"""


I’m also encapsulating the training process in a class of its own. You can notice the gradients being clipped before being applied. I’m also clipping the rewards into the range of $<-3, 3>$ to help to keep the variance low.

class Trainer:
def __init__(self, gamma, agent, window = 15, workers = 8, **kwargs):
super().__init__(**kwargs)

self.agent = agent
self.window = window
self.gamma = gamma
self.workers = workers

# even though we're loading the weights into worker agents explicitly, I found that still without sharing the weights as following, the algorithm was not converging:
self.agent.share_memory()

def fit(self, episodes = 1000):
"""
The higher level method for training the agents.
It called into the lower level "train" which orchestrates the process itself.
"""
last_update = 0

for ix in range(1, self.workers + 1):
updates[ ix ] = { 'episode': 0, 'step': 0, 'rewards': deque(), 'losses': deque(), 'points': 0, 'mean_reward': 0, 'mean_loss': 0 }

for update in self.train(episodes):
now = time.time()

# you could do something useful here with the updates dict.
# I've opted out as I'm using logging anyways and got more value in just watching the log file, grepping for the desired values

# save the current model's weights every minute:
if now - last_update > 60:
torch.save(self.agent.state_dict(), './checkpoints/car-racing/' + str(int(now)) + '-.pytorch')
last_update = now

def train(self, episodes = 1000):
"""
Lower level training orchestration method. Written in the generator style. Intended to be used with "for update in train(...):"
"""

# create the requested number of background agents and runners:
worker_agents = self.agent.clone(num = self.workers)
runners = [ Runner(agent=agent, ix = ix + 1, train = True) for ix, agent in enumerate(worker_agents) ]

# we're going to communicate the workers' updates via the thread safe queue:
queue = mp.SimpleQueue()

# if we've not been given a number of episodes: assume the process is going to be interrupted with the keyboard interrupt once the user (us) decides so:
if episodes is None:
print('Starting out an infinite training process')

# create the actual background processes, making their entry be the train_one method:
processes = [ mp.Process(target=self.train_one, args=(runners[ix - 1], queue, episodes, ix)) for ix in range(1, self.workers + 1) ]

# run those processes:
for process in processes:
process.start()

try:
# what follows is a rather naive implementation of listening to workers updates. it works though for our purposes:
while any([ process.is_alive() for process in processes ]):
results = queue.get()
yield results
except Exception as e:
logger.error(str(e))

def train_one(self, runner, queue, episodes = 1000, ix = 1):
"""
Orchestrate the training for a single worker runner and agent. This is intended to run in its own background process.
"""

# possibly naive way of trying to de-correlate the weight updates further (I have no hard evidence to prove if it works, other than my subjective observation):
time.sleep(ix)

try:
# we are going to request the episode be reset whenever our agent scores lower than its max points. the same will happen if the agent scores total of -10 points:
max_points = 0
max_eval_points = 0
min_points = 0
max_episode = 0

for episode_ix in itertools.count(start=0, step=1):

if episodes is not None and episode_ix >= episodes:
return

max_episode_points = 0
points = 0

# every 5 episodes lets evaluate the weights we've learned so far by recording the run of the car using the greedy strategy:
if ix == 1 and episode_ix % 5 == 0:
eval_points = self.record_greedy(episode_ix)

if eval_points > max_eval_points:
torch.save(runner.agent.state_dict(), './checkpoints/car-racing/' + str(eval_points) + '-eval-points.pytorch')
max_eval_points = eval_points

# each n-step window, compute the gradients and apply
# also: decide if we shouldn't restart the episode if we don't want to explore too much of the not-useful state space:
for step, rewards, values, policies, action_ixs, terminal in runner.run_episode(yield_every=self.window):
points += sum(rewards)

if ix == 1 and points > max_points:
torch.save(runner.agent.state_dict(), './checkpoints/car-racing/' + str(points) + '-points.pytorch')
max_points = points

if ix == 1 and episode_ix > max_episode:
torch.save(runner.agent.state_dict(), './checkpoints/car-racing/' + str(episode_ix) + '-episode.pytorch')
max_episode = episode_ix

if points < -10 or (max_episode_points > min_points and points < min_points):
terminal = True
max_episode_points = 0
point = 0

if terminal:
logger.info('TERMINAL for ' + str(ix) + ' at step ' + str(step) + ' with total points ' + str(points) + ' max: ' + str(max_episode_points) )

if runner.train:
loss = self.apply_gradients(policies, action_ixs, rewards, values, terminal, runner)

max_episode_points = max(max_episode_points, points)
min_points = max(min_points, points)

# communicate the gathered values to the main process:
queue.put((ix, episode_ix, step, rewards, loss, points, terminal))

except Exception as e:
string = traceback.format_exc()
logger.error(str(e) + ' → ' + string)
queue.put((ix, -1, -1, [-1], -1, str(e) + '<br />' + string, True))

def record_greedy(self, episode_ix):
"""
Records the video of the "greedy" run based on the current weights.
"""
directory = './videos/car-racing/episode-' + str(episode_ix) + '-' + str(int(time.time()))
player = Player(agent=self.agent, directory=directory, train=False)
points = player.play()
logger.info('Evaluation at episode ' + str(episode_ix) + ': ' + str(points) + ' points (' + directory + ')')
return points

def apply_gradients(self, policies, actions, rewards, values, terminal, runner):
worker_agent = runner.agent
actions_one_hot = torch.tensor([[ int(i == action) for i in range(4) ] for action in actions], dtype=torch.float32)

policies = torch.stack(policies)
values = torch.cat(values)

discounted_rewards = self.discount_rewards(runner, rewards, values_nograd[-1], terminal)

logger.info('Runner ' + str(runner.ix) + 'Rewards: ' + str(rewards))
logger.info('Runner ' + str(runner.ix) + 'Discounted Rewards: ' + str(discounted_rewards.numpy()))

log_policies = torch.log(0.00000001 + policies)

one_log_policies = torch.sum(log_policies * actions_one_hot, dim=1)

entropy = torch.sum(policies * -log_policies)

value_loss = F.mse_loss(values, discounted_rewards)

logger.info('Value Loss: ' + str(float(value_loss_nograd)) + ' Policy Loss: ' + str(float(policy_loss_nograd)))

loss = policy_loss + 0.5 * value_loss - 0.01 * entropy
loss.backward()

# the following step is crucial. at this point, all the info about the gradients reside in the worker agent's memory. We need to "move" those gradients into the main agent's memory:

# update the weights with the computed gradients:
self.optimizer.step()

return float(loss.detach())

for param, shared_param in zip(worker_agent.parameters(), self.agent.parameters()):
return

def clip_reward(self, reward):
"""
Clips the rewards into the <-3, 3> range preventing too big of the gradients variance.
"""
return max(min(reward, 3), -3)

def discount_rewards(self, runner, rewards, last_value, terminal):
discounted_rewards = [0 for _ in rewards]
loop_rewards = [ self.clip_reward(reward) for reward in rewards ]

if terminal:
loop_rewards.append(0)
else:
loop_rewards.append(runner.get_value())

for main_ix in range(len(discounted_rewards) - 1, -1, -1):
for inside_ix in range(len(loop_rewards) - 1, -1, -1):
if inside_ix >= main_ix:
reward = loop_rewards[inside_ix]
discounted_rewards[main_ix] += self.gamma**(inside_ix - main_ix) * reward



For the record_greedy method to work we need the following class:

class Player(Runner):
def __init__(self, directory, **kwargs):
super().__init__(ix=999, **kwargs)

self.env = Monitor(self.env, directory)

def play(self):
points = 0
for step, rewards, values, policies, actions, terminal in self.run_episode(yield_every = 1, do_render = True):
points += sum(rewards)
self.env.close()
return points


All the above code can be used as follows (in the Python script):

if __name__ == "__main__":
agent = Agent()

trainer = Trainer(gamma = 0.99, agent = agent)
trainer.fit(episodes=None)


#### The importance of tuning of the n-step window size

Reading the code, you can notice that we’ve chosen $15$ to be the size of the n-step window. We’ve also chosen $\gamma=0.99$. Getting those values right is a subject for tuning. The same ones that work on one game or a challenge will not necessarily work well for the other.

Here’s a quick explanation of how to think about them: We’re going to be penalized most of the time. It’s important for us to give the algorithm a chance to actually find trajectories that score positively. In the “CarRacing” challenge, I’ve found that it can take 10 steps of moving “full throttle” in the correct direction before we’re being rewarded by entering the new “tile”. I’ve just simply added $5$ of the safety net to that number. No mathematical proof follows this thinking here, but I can tell you though that it made a huge difference in the training time for me. The version of the code I’m presenting above starts to score above 700 points after approximately 10 hours on my Ryzen 7 based computing box.

#### Problems with the state being returned from the environment - overcoming with the explicit render

You might have also noticed that I’m not using the state values returned by the step method of the gym environment. This might seem contradictory to how the gym is typically being used. After days of not seeing my model converge though, I have found that the step method was returning one and the same numpy array on each call. You can imagine that it was the absolutely last thing I’ve checked when trying to find that bug.

I’ve found the render(mode='rgb_array') works as intended each time. I just needed to write my own preprocessing code, to scale it down and make it grayscale.

#### How to know when the algorithm converges

I’ve seen some people thinking that their A3C implementation does not converge. The resulting policy did not seem to be working that well, but the training process was taking a bit longer than “some other implementation”. I fell for this kind of thinking myself as well. My humble bit of advice is to stick to what makes sense mathematically. Someone else’s model might be converging faster simply because of the hardware being used or some slight difference in the code around the training (e.g. explicit render needed in my case). This might not have anything to do with the A3C part at all.

How do we “stick to what makes sense mathematically”? Simply by logging the value loss and observing it as the training continues. Intuitively, for the model that has converged, we should see that it has already learned the value function. Those values — representing the average of the discounted rewards — should not make the loss too big most of the time. Still, for some states, the best action will make the $R_t$ much bigger than $V(s_t)$ which means that we still should see the loss spiking from time to time.

Again, the above bit of advice doesn’t come with any mathematical proofs. It’s what I found working and making sense in my case.

### The Results

Instead of presenting hard-core statistics about the model’s performance — which wouldn’t make much sense because I stopped it as soon as the “evaluation” videos started looking cool enough) — I’ll just post three videos of the car driving on its own through the three randomly generated tracks.

Have fun watching and even more fun coding it yourself too!