##### Copyright 2021 The TF-Agents Authors.

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## Introduction

This example shows how to train a Categorical DQN (C51) agent on the Cartpole environment using the TF-Agents library.

Make sure you take a look through the DQN tutorial as a prerequisite. This tutorial will assume familiarity with the DQN tutorial; it will mainly focus on the differences between DQN and C51.

## Setup

If you haven't installed tf-agents yet, run:

`sudo apt-get install -y xvfb ffmpeg`

`pip install -q 'imageio==2.4.0'`

`pip install -q pyvirtualdisplay`

`pip install -q tf-agents`

```
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import base64
import imageio
import IPython
import matplotlib
import matplotlib.pyplot as plt
import PIL.Image
import pyvirtualdisplay
import tensorflow as tf
from tf_agents.agents.categorical_dqn import categorical_dqn_agent
from tf_agents.drivers import dynamic_step_driver
from tf_agents.environments import suite_gym
from tf_agents.environments import tf_py_environment
from tf_agents.eval import metric_utils
from tf_agents.metrics import tf_metrics
from tf_agents.networks import categorical_q_network
from tf_agents.policies import random_tf_policy
from tf_agents.replay_buffers import tf_uniform_replay_buffer
from tf_agents.trajectories import trajectory
from tf_agents.utils import common
tf.compat.v1.enable_v2_behavior()
# Set up a virtual display for rendering OpenAI gym environments.
display = pyvirtualdisplay.Display(visible=0, size=(1400, 900)).start()
```

## Hyperparameters

```
env_name = "CartPole-v1" # @param {type:"string"}
num_iterations = 15000 # @param {type:"integer"}
initial_collect_steps = 1000 # @param {type:"integer"}
collect_steps_per_iteration = 1 # @param {type:"integer"}
replay_buffer_capacity = 100000 # @param {type:"integer"}
fc_layer_params = (100,)
batch_size = 64 # @param {type:"integer"}
learning_rate = 1e-3 # @param {type:"number"}
gamma = 0.99
log_interval = 200 # @param {type:"integer"}
num_atoms = 51 # @param {type:"integer"}
min_q_value = -20 # @param {type:"integer"}
max_q_value = 20 # @param {type:"integer"}
n_step_update = 2 # @param {type:"integer"}
num_eval_episodes = 10 # @param {type:"integer"}
eval_interval = 1000 # @param {type:"integer"}
```

## Environment

Load the environment as before, with one for training and one for evaluation. Here we use CartPole-v1 (vs. CartPole-v0 in the DQN tutorial), which has a larger max reward of 500 rather than 200.

```
train_py_env = suite_gym.load(env_name)
eval_py_env = suite_gym.load(env_name)
train_env = tf_py_environment.TFPyEnvironment(train_py_env)
eval_env = tf_py_environment.TFPyEnvironment(eval_py_env)
```

## Agent

C51 is a Q-learning algorithm based on DQN. Like DQN, it can be used on any environment with a discrete action space.

The main difference between C51 and DQN is that rather than simply predicting the Q-value for each state-action pair, C51 predicts a histogram model for the probability distribution of the Q-value:

By learning the distribution rather than simply the expected value, the algorithm is able to stay more stable during training, leading to improved final performance. This is particularly true in situations with bimodal or even multimodal value distributions, where a single average does not provide an accurate picture.

In order to train on probability distributions rather than on values, C51 must perform some complex distributional computations in order to calculate its loss function. But don't worry, all of this is taken care of for you in TF-Agents!

To create a C51 Agent, we first need to create a `CategoricalQNetwork`

. The API of the `CategoricalQNetwork`

is the same as that of the `QNetwork`

, except that there is an additional argument `num_atoms`

. This represents the number of support points in our probability distribution estimates. (The above image includes 10 support points, each represented by a vertical blue bar.) As you can tell from the name, the default number of atoms is 51.

```
categorical_q_net = categorical_q_network.CategoricalQNetwork(
train_env.observation_spec(),
train_env.action_spec(),
num_atoms=num_atoms,
fc_layer_params=fc_layer_params)
```

We also need an `optimizer`

to train the network we just created, and a `train_step_counter`

variable to keep track of how many times the network was updated.

Note that one other significant difference from vanilla `DqnAgent`

is that we now need to specify `min_q_value`

and `max_q_value`

as arguments. These specify the most extreme values of the support (in other words, the most extreme of the 51 atoms on either side). Make sure to choose these appropriately for your particular environment. Here we use -20 and 20.

```
optimizer = tf.compat.v1.train.AdamOptimizer(learning_rate=learning_rate)
train_step_counter = tf.compat.v2.Variable(0)
agent = categorical_dqn_agent.CategoricalDqnAgent(
train_env.time_step_spec(),
train_env.action_spec(),
categorical_q_network=categorical_q_net,
optimizer=optimizer,
min_q_value=min_q_value,
max_q_value=max_q_value,
n_step_update=n_step_update,
td_errors_loss_fn=common.element_wise_squared_loss,
gamma=gamma,
train_step_counter=train_step_counter)
agent.initialize()
```

One last thing to note is that we also added an argument to use n-step updates with $n$ = 2. In single-step Q-learning ($n$ = 1), we only compute the error between the Q-values at the current time step and the next time step using the single-step return (based on the Bellman optimality equation). The single-step return is defined as:

$G_t = R_{t + 1} + \gamma V(s_{t + 1})$

where we define $V(s) = \max_a{Q(s, a)}$.

N-step updates involve expanding the standard single-step return function $n$ times:

$G_t^n = R_{t + 1} + \gamma R_{t + 2} + \gamma^2 R_{t + 3} + \dots + \gamma^n V(s_{t + n})$

N-step updates enable the agent to bootstrap from further in the future, and with the right value of $n$, this often leads to faster learning.

Although C51 and n-step updates are often combined with prioritized replay to form the core of the Rainbow agent, we saw no measurable improvement from implementing prioritized replay. Moreover, we find that when combining our C51 agent with n-step updates alone, our agent performs as well as other Rainbow agents on the sample of Atari environments we've tested.

## Metrics and Evaluation

The most common metric used to evaluate a policy is the average return. The return is the sum of rewards obtained while running a policy in an environment for an episode, and we usually average this over a few episodes. We can compute the average return metric as follows.

```
def compute_avg_return(environment, policy, num_episodes=10):
total_return = 0.0
for _ in range(num_episodes):
time_step = environment.reset()
episode_return = 0.0
while not time_step.is_last():
action_step = policy.action(time_step)
time_step = environment.step(action_step.action)
episode_return += time_step.reward
total_return += episode_return
avg_return = total_return / num_episodes
return avg_return.numpy()[0]
random_policy = random_tf_policy.RandomTFPolicy(train_env.time_step_spec(),
train_env.action_spec())
compute_avg_return(eval_env, random_policy, num_eval_episodes)
# Please also see the metrics module for standard implementations of different
# metrics.
```

28.7

## Data Collection

As in the DQN tutorial, set up the replay buffer and the initial data collection with the random policy.

```
replay_buffer = tf_uniform_replay_buffer.TFUniformReplayBuffer(
data_spec=agent.collect_data_spec,
batch_size=train_env.batch_size,
max_length=replay_buffer_capacity)
def collect_step(environment, policy):
time_step = environment.current_time_step()
action_step = policy.action(time_step)
next_time_step = environment.step(action_step.action)
traj = trajectory.from_transition(time_step, action_step, next_time_step)
# Add trajectory to the replay buffer
replay_buffer.add_batch(traj)
for _ in range(initial_collect_steps):
collect_step(train_env, random_policy)
# This loop is so common in RL, that we provide standard implementations of
# these. For more details see the drivers module.
# Dataset generates trajectories with shape [BxTx...] where
# T = n_step_update + 1.
dataset = replay_buffer.as_dataset(
num_parallel_calls=3, sample_batch_size=batch_size,
num_steps=n_step_update + 1).prefetch(3)
iterator = iter(dataset)
```

WARNING:tensorflow:From /tmpfs/src/tf_docs_env/lib/python3.6/site-packages/tensorflow/python/autograph/operators/control_flow.py:1218: ReplayBuffer.get_next (from tf_agents.replay_buffers.replay_buffer) is deprecated and will be removed in a future version. Instructions for updating: Use `as_dataset(..., single_deterministic_pass=False) instead.

## Training the agent

The training loop involves both collecting data from the environment and optimizing the agent's networks. Along the way, we will occasionally evaluate the agent's policy to see how we are doing.

The following will take ~7 minutes to run.

```
try:
%%time
except:
pass
# (Optional) Optimize by wrapping some of the code in a graph using TF function.
agent.train = common.function(agent.train)
# Reset the train step
agent.train_step_counter.assign(0)
# Evaluate the agent's policy once before training.
avg_return = compute_avg_return(eval_env, agent.policy, num_eval_episodes)
returns = [avg_return]
for _ in range(num_iterations):
# Collect a few steps using collect_policy and save to the replay buffer.
for _ in range(collect_steps_per_iteration):
collect_step(train_env, agent.collect_policy)
# Sample a batch of data from the buffer and update the agent's network.
experience, unused_info = next(iterator)
train_loss = agent.train(experience)
step = agent.train_step_counter.numpy()
if step % log_interval == 0:
print('step = {0}: loss = {1}'.format(step, train_loss.loss))
if step % eval_interval == 0:
avg_return = compute_avg_return(eval_env, agent.policy, num_eval_episodes)
print('step = {0}: Average Return = {1:.2f}'.format(step, avg_return))
returns.append(avg_return)
```

WARNING:tensorflow:From /tmpfs/src/tf_docs_env/lib/python3.6/site-packages/tensorflow/python/util/dispatch.py:201: calling foldr_v2 (from tensorflow.python.ops.functional_ops) with back_prop=False is deprecated and will be removed in a future version. Instructions for updating: back_prop=False is deprecated. Consider using tf.stop_gradient instead. Instead of: results = tf.foldr(fn, elems, back_prop=False) Use: results = tf.nest.map_structure(tf.stop_gradient, tf.foldr(fn, elems)) step = 200: loss = 3.321129560470581 step = 400: loss = 2.485752820968628 step = 600: loss = 2.0748603343963623 step = 800: loss = 1.899770736694336 step = 1000: loss = 1.9147026538848877 step = 1000: Average Return = 69.10 step = 1200: loss = 1.470450758934021 step = 1400: loss = 1.524451494216919 step = 1600: loss = 1.3602908849716187 step = 1800: loss = 1.3945512771606445 step = 2000: loss = 1.2128956317901611 step = 2000: Average Return = 201.20 step = 2200: loss = 1.2250053882598877 step = 2400: loss = 1.0739798545837402 step = 2600: loss = 1.0344221591949463 step = 2800: loss = 0.9437637329101562 step = 3000: loss = 1.0215129852294922 step = 3000: Average Return = 142.70 step = 3200: loss = 1.0233310461044312 step = 3400: loss = 0.8907231688499451 step = 3600: loss = 0.7526266574859619 step = 3800: loss = 0.6926383972167969 step = 4000: loss = 0.7934644222259521 step = 4000: Average Return = 476.80 step = 4200: loss = 0.791626513004303 step = 4400: loss = 0.8220507502555847 step = 4600: loss = 0.7975851893424988 step = 4800: loss = 0.4139212369918823 step = 5000: loss = 0.7318903207778931 step = 5000: Average Return = 310.40 step = 5200: loss = 0.7830334305763245 step = 5400: loss = 0.7445043921470642 step = 5600: loss = 0.6130998134613037 step = 5800: loss = 0.5654287338256836 step = 6000: loss = 0.6499170064926147 step = 6000: Average Return = 498.00 step = 6200: loss = 0.6856206655502319 step = 6400: loss = 0.613524317741394 step = 6600: loss = 0.5312545299530029 step = 6800: loss = 0.5998117923736572 step = 7000: loss = 0.35336682200431824 step = 7000: Average Return = 419.60 step = 7200: loss = 0.37572816014289856 step = 7400: loss = 0.3268156051635742 step = 7600: loss = 0.3964875340461731 step = 7800: loss = 0.4353790283203125 step = 8000: loss = 0.47257936000823975 step = 8000: Average Return = 209.20 step = 8200: loss = 0.41818156838417053 step = 8400: loss = 0.295656681060791 step = 8600: loss = 0.30348891019821167 step = 8800: loss = 0.2654055655002594 step = 9000: loss = 0.4846675992012024 step = 9000: Average Return = 431.30 step = 9200: loss = 0.281438410282135 step = 9400: loss = 0.23425081372261047 step = 9600: loss = 0.6559126377105713 step = 9800: loss = 0.4217219948768616 step = 10000: loss = 0.3250614404678345 step = 10000: Average Return = 283.80 step = 10200: loss = 0.2797137498855591 step = 10400: loss = 0.3637545108795166 step = 10600: loss = 0.2684471011161804 step = 10800: loss = 0.45216208696365356 step = 11000: loss = 0.26978206634521484 step = 11000: Average Return = 432.80 step = 11200: loss = 0.41701459884643555 step = 11400: loss = 0.39164310693740845 step = 11600: loss = 0.48381370306015015 step = 11800: loss = 0.3856581449508667 step = 12000: loss = 0.2671810984611511 step = 12000: Average Return = 412.60 step = 12200: loss = 0.37253132462501526 step = 12400: loss = 0.24322597682476044 step = 12600: loss = 0.48967045545578003 step = 12800: loss = 0.3843742907047272 step = 13000: loss = 0.3109121024608612 step = 13000: Average Return = 441.30 step = 13200: loss = 0.32548320293426514 step = 13400: loss = 0.3387058675289154 step = 13600: loss = 0.3758728504180908 step = 13800: loss = 0.2936052680015564 step = 14000: loss = 0.35974568128585815 step = 14000: Average Return = 427.80 step = 14200: loss = 0.3430924713611603 step = 14400: loss = 0.49261224269866943 step = 14600: loss = 0.39563947916030884 step = 14800: loss = 0.3216741681098938 step = 15000: loss = 0.3640541434288025 step = 15000: Average Return = 432.10

## Visualization

### Plots

We can plot return vs global steps to see the performance of our agent. In `Cartpole-v1`

, the environment gives a reward of +1 for every time step the pole stays up, and since the maximum number of steps is 500, the maximum possible return is also 500.

```
steps = range(0, num_iterations + 1, eval_interval)
plt.plot(steps, returns)
plt.ylabel('Average Return')
plt.xlabel('Step')
plt.ylim(top=550)
```

(-15.555000400543214, 550.0)

### Videos

It is helpful to visualize the performance of an agent by rendering the environment at each step. Before we do that, let us first create a function to embed videos in this colab.

```
def embed_mp4(filename):
"""Embeds an mp4 file in the notebook."""
video = open(filename,'rb').read()
b64 = base64.b64encode(video)
tag = '''
<video width="640" height="480" controls>
<source src="data:video/mp4;base64,{0}" type="video/mp4">
Your browser does not support the video tag.
</video>'''.format(b64.decode())
return IPython.display.HTML(tag)
```

The following code visualizes the agent's policy for a few episodes:

```
num_episodes = 3
video_filename = 'imageio.mp4'
with imageio.get_writer(video_filename, fps=60) as video:
for _ in range(num_episodes):
time_step = eval_env.reset()
video.append_data(eval_py_env.render())
while not time_step.is_last():
action_step = agent.policy.action(time_step)
time_step = eval_env.step(action_step.action)
video.append_data(eval_py_env.render())
embed_mp4(video_filename)
```

WARNING:root:IMAGEIO FFMPEG_WRITER WARNING: input image is not divisible by macro_block_size=16, resizing from (400, 600) to (400, 608) to ensure video compatibility with most codecs and players. To prevent resizing, make your input image divisible by the macro_block_size or set the macro_block_size to None (risking incompatibility). You may also see a FFMPEG warning concerning speedloss due to data not being aligned.

C51 tends to do slightly better than DQN on CartPole-v1, but the difference between the two agents becomes more and more significant in increasingly complex environments. For example, on the full Atari 2600 benchmark, C51 demonstrates a mean score improvement of 126% over DQN after normalizing with respect to a random agent. Additional improvements can be gained by including n-step updates.

For a deeper dive into the C51 algorithm, see A Distributional Perspective on Reinforcement Learning (2017).