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Reinforcement Learning [Part 2]: The Q-learning Algorithm

by Pau Labarta BajoMarch 28th, 2022

Too Long; Didn't Read

This is part 2 of my hands-on course on reinforcement learning, which takes you from zero to HERO. Today we will learn about Q-learning, a classic RL algorithm born in the 90s. We will use an environment from OpenAI Gym, called the `Taxi-v3` environment. The taxi drives to a random location, picks up the passenger, drives to the passenger’s destination (another one of the four specified locations), and then drops off the passenger.

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featured image - Reinforcement Learning [Part 2]: The Q-learning Algorithm

Q-learning is an RL algorithm to find the optimal q-value function. It is a fundamental algorithm in RL, that lies behind the impressive achievements in this field in the last 10 years.

Welcome to my reinforcement learning course ❤️

This is part 2 of my hands-on course on reinforcement learning, which takes you from zero to HERO 🦸‍♂️.

If you missed part 1, please read it to get the reinforcement learning jargon and basics in place.

Today we will learn about Q-learning, a classic RL algorithm born in the 90s.

And we will train an agent to drive a taxi 🚕🚕🚕!

Well, a simplified version of a taxi environment, but a taxi at the end of the day.

All the code for this lesson is in this Github repo. Git clone it to follow along with today’s problem.

Part 2

The taxi driving problem 🚕
Environment, actions, states, rewards
Random agent baseline 🤖🍷
Q-learning agent 🤖🧠
Hyper-parameter tuning 🎛️
Recap ✨
Homework 📚
What’s next? ❤️

1. The taxi driving problem 🚕

We will teach an agent to drive a taxi using Reinforcement Learning.

Driving a taxi in the real world is a very complex task to start with. Because of this, we will work in a simplified environment that captures the 3 essential things a good taxi driver does, which are:

pick up passengers and drop them at their desired destination.
drive safely, meaning no crashes.
drive them in the shortest time possible.

We will use an environment from OpenAI Gym, called the Taxi-v3 environment.

There are four designated locations in the grid world indicated by R(ed), G(reen), Y(ellow), and B(lue).

When the episode starts, the taxi starts off at a random square and the passenger is at a random location (R, G, Y or B).

The taxi drives to the passenger’s location, picks up the passenger, drives to the passenger’s destination (another one of the four specified locations), and then drops off the passenger. While doing so, our taxi driver needs to drive carefully to avoid hitting any wall, marked as |. Once the passenger is dropped off, the episode ends.

Before we get there, let’s understand well what are the actions, states, and rewards for this environment.

2. Environment, actions, states, rewards

👉🏽 notebooks/00_environment.ipynb

Let’s first load the environment:

import gym
env = gym.make("Taxi-v3").env

What are the actions the agent can choose from at each step?

0 drive down
1 drive up
2 drive right
3 drive left
4 pick up a passenger
5 drop off a passenger

print("Action Space {}".format(env.action_space))

And the states?

25 possible taxi positions, because the world is a 5x5 grid.
5 possible locations of the passenger, which are R, G, Y, B, plus the case when the passenger is in the taxi.
4 destination locations

Which gives us 25 x 5 x 4 = 500 states

print("State Space {}".format(env.observation_space))

What about rewards?

-1 default per-step reward. Why -1, and not simply 0? Because we want to encourage the agent to spend the shortest time, by penalizing each extra step. This is what you expect from a taxi driver, don’t you?
+20 reward for delivering the passenger to the correct destination.
-10 reward for executing a pickup or dropoff at the wrong location.

You can read the rewards and the environment transitions (state, action ) → next_state from env.P.

# env.P is double dictionary.
# - The 1st key represents the state, from 0 to 499
# - The 2nd key represens the action taken by the agent,
#   from 0 to 5

# example
state = 123
action = 0  # move south

# env.P[state][action][0] is a list with 4 elements
# (probability, next_state, reward, done)
# 
#  - probability
#    It is always 1 in this environment, which means
#    there are no external/random factors that determine the
#    next_state
#    apart from the agent's action a.
#
#  - next_state: 223 in this case
# 
#  - reward: -1 in this case
#
#  - done: boolean (True/False) indicates wheter the
#    episode has ended (i.e. the driver has dropped the
#    passenger at the correct destination)
print('env.P[state][action][0]: ', env.P[state][action][0])

By the way, you can render the environment under each state to double-check this env.P vectors make sense:

From state=123

# Need to call reset() at least once before render() will work
env.reset()

env.s = 123
env.render(mode='human')

the agent moves south action=0 to get to state=223

env.s = 223
env.render(mode='human')

And the reward is -1, as neither the episode ended, nor the driver incorrectly picked or dropped.

3. Random agent baseline 🤖🍷

👉🏽 notebooks/01_random_agent_baseline.ipynb

Before you start implementing any complex algorithm, you should always build a baseline model.

This advice applies not only to Reinforcement Learning problems but Machine Learning problems in general.

It is very tempting to jump straight into the complex/fancy algorithms, but unless you are really experienced, you will fail terribly.

Let’s use a random agent 🤖🍷 as a baseline model.

class RandomAgent:
    """
    This taxi driver selects actions randomly.
    You better not get into this taxi!
    """
    def __init__(self, env):
        self.env = env

    def get_action(self, state) -> int:
        """
        We have `state` as an input to keep
        a consistent API for all our agents, but it
        is not used.
        
        i.e. The agent does not consider the state of
        the environment when deciding what to do next.
        This is why we call it "random".
        """
        return self.env.action_space.sample()

agent = RandomAgent(env)

We can see how this agent performs for a given initial state=123

# set initial state of the environment
env.reset()
state = 123
env.s = state

epochs = 0
penalties = 0  # wrong pick up or dropp off
reward = 0

# store frames to latter plot them
frames = []

done = False

while not done:
    
    action = agent.get_action(state)
    
    state, reward, done, info = env.step(action)

    if reward == -10:
        penalties += 1
    
    frames.append({
        'frame': env.render(mode='ansi'),
        'state': state,
        'action': action,
        'reward': reward
        }
    )

    epochs += 1
    
    
print("Timesteps taken: {}".format(epochs))
print("Penalties incurred: {}".format(penalties))

1,420 steps is a lot! 😵

You will get different numbers when you run this code on your laptop, because of the randomness in this agent. But still, the results will be consistently bad.

To get a more representative measure of performance, we can repeat the same evaluation loop n=100times starting each time at a random state.

from tqdm import tqdm

n_episodes = 100

# For plotting metrics
timesteps_per_episode = []
penalties_per_episode = []

for i in tqdm(range(0, n_episodes)):
    
    # reset environment to a random state
    state = env.reset()

    epochs, penalties, reward, = 0, 0, 0
    done = False
    
    while not done:
        
        action = agent.get_action(state)       
        next_state, reward, done, info = env.step(action) 
               
        if reward == -10:
            penalties += 1

        state = next_state
        epochs += 1
    
    timesteps_per_episode.append(epochs)
    penalties_per_episode.append(penalties)

If you plot timesteps_per_episode and penalties_per_episode you can observe that none of them decreases as the agent completes more episodes. In other words, the agent is NOT LEARNING anything.

If you want summary statistics of performance you can take averages:

print(f'Avg steps to complete ride: {np.array(timesteps_per_episode).mean()}')
print(f'Avg penalties to complete ride: {np.array(penalties_per_episode).mean()}')

Implementing agents that learn is the goal of Reinforcement Learning, and of this course too.

Let’s implement our first “intelligent” agent using Q-learning, one of the earliest and most used RL algorithms that exist.

4. Q-learning agent 🤖🧠

👉🏽 notebooks/02_q_agent.ipynb

Q-learning (by Chris Walkins 🧠 and Peter Dayan 🧠) is an algorithm to find the optimal q-value function.

As we said in part 1, the q-value function Q(s, a) associated with a policy **π **is the total reward the agent expects to get when at state s the agent takes action a and follows policy π thereafter.

The optimal q-value function **Q*(s, a) **is the q-value function associated with the optimal policy π*.

If you know Q*(s, a) you can infer π*: i.e. you pick as the next action the one that maximizes Q*(s, a) for the current state s.

Q-learning is an iterative algorithm to compute better and better approximations to the optimal

q-value function Q*(s, a), starting from an arbitrary initial guess Q⁰(s, a)

In a tabular environment like Taxi-v3 with a finite number of states and actions, a q-function is essentially a matrix. It has as many rows as states and columns as actions, i.e. 500 x 6.

Ok, but how exactly do you compute the next approximation Q¹(s, a) from Q⁰(s, a)?

This is the key formula in Q-learning:

As our q-agent navigates the environment and observes the next state s’ and reward r, you update your q-value matrix with this formula.

What is the learning rate 𝛼 in this formula?

The learning rate (as usual in machine learning) is a small number that controls how large are the updates to the q-function. You need to tune it, as too large of a value will cause unstable training, and too small might not be enough to escape local minima.

And this discount factor 𝛾?

The discount factor is a (hyper) parameter between 0 and 1 that determines how much our agent cares about rewards in the distant future relative to those in the immediate future.

When 𝛾=0, the agent only cares about maximizing immediate reward. As it happens in life, maximizing immediate reward is not the best recipe for optimal long-term outcomes. This happens in RL agents too.
When 𝛾=1, the agent evaluates each of its actions based on the sum total of all of its future rewards. In this case the agent weights equally immediate rewards and future rewards.

The discount factor is typically an intermediate value, e.g. 0.6.

To sum up…

if you

train long enough
with a decent learning rate and discount factor
and the agent explores enough the state space
and you update the q-value matrix with the Q-learning formula

your initial approximation will eventually converge to optimal q-matrix.

Voila!

Let’s implement a Python class for a Q-agent then.

import numpy as np

class QAgent:

    def __init__(self, env, alpha, gamma):
        self.env = env

        # table with q-values: n_states * n_actions
        self.q_table = np.zeros([env.observation_space.n,
                                 env.action_space.n])

        # hyper-parameters
        self.alpha = alpha  # learning rate
        self.gamma = gamma  # discount factor

    def get_action(self, state):
        """"""
        return np.argmax(self.q_table[state])

    def update_parameters(self, state, action, reward, next_state):
        """"""
        # Q-learning formula
        old_value = self.q_table[state, action]
        next_max = np.max(self.q_table[next_state])
        new_value = \
            old_value + \
            self.alpha * (reward + self.gamma * next_max - old_value)

        # update the q_table
        self.q_table[state, action] = new_value

Its API is the same as for the RandomAgent above, but with an extra method update_parameters(). This method takes the transition vector (state, action, reward, next_state) and updates the q-value matrix approximation self.q_table using the Q-learning formula from above.

Now, we need to plug this agent into a training loop and call its update_parameters() method every time the agent collects a new experience.

Also, remember we need to guarantee the agent explores enough the state space. Remember the exploration-exploitation parameter we talked about in part 1? This is when the epsilon parameter enters into the game.

Let’s train the agent for n_episodes = 10,000 and use epsilon = 10%

import random
from tqdm import tqdm

# exploration vs exploitation prob
epsilon = 0.1

n_episodes = 10000

# For plotting metrics
timesteps_per_episode = []
penalties_per_episode = []


for i in tqdm(range(0, n_episodes)):
    
    state = env.reset()

    epochs, penalties, reward, = 0, 0, 0
    done = False
    
    while not done:
        
        if random.uniform(0, 1) < epsilon:
            # Explore action space
            action = env.action_space.sample()
        else:
            # Exploit learned values
            action = agent.get_action(state)
        
        next_state, reward, done, info = env.step(action) 
        
        agent.update_parameters(state, action, reward, next_state)
        
        if reward == -10:
            penalties += 1

        state = next_state
        epochs += 1
        
    timesteps_per_episode.append(epochs)
    penalties_per_episode.append(penalties)

And plot timesteps_per_episode and penalties_per_episode

import pandas as pd
import matplotlib.pyplot as plt

fig, ax = plt.subplots(figsize = (12, 4))
ax.set_title("Timesteps to complete ride")    
pd.Series(timesteps_per_episode).plot(kind='line')
plt.show()

fig, ax = plt.subplots(figsize = (12, 4))
ax.set_title("Penalties per ride")    
pd.Series(penalties_per_episode).plot(kind='line')
plt.show()

Nice! These graphs look much much better than for the RandomAgent. Both metrics decrease with training, which means our agent is learning 🎉🎉🎉.

We can actually see how the agent drives starting from the same state = 123 as we used for the RandomAgent.

# set initial state of the environment
state = 123
env.s = state

epochs = 0
penalties = 0
reward = 0

# store frames to latter plot them
frames = []

done = False

while not done:
    
    action = agent.get_action(state)
    
    next_state, reward, done, info = env.step(action)
    agent.update_parameters(state, action, reward, next_state)

    if reward == -10:
        penalties += 1
    
    frames.append({
        'frame': env.render(mode='ansi'),
        'state': state,
        'action': action,
        'reward': reward
        }
    )

    state = next_state

    epochs += 1
    
print("Timesteps taken: {}".format(epochs))
print("Penalties incurred: {}".format(penalties))

If you want to compare hard numbers you can evaluate the performance of the q-agent on, let’s say, 100 random episodes and compute the average number of timestamps and penalties incurred.

A little bit about epsilon-greedy policies

When you evaluate the agent, it is still good practice to use a positive epsilon value, and not epsilon = 0.

Why so? Isn’t our agent fully trained? Why do we need to keep this source of randomness when we choose the next action?

The reason is to prevent overfitting. Even for such a small state, action space in Taxi-v3(i.e. 500 x 6) it is likely that during training our agent has not visited enough certain states.

Hence, its performance in these states might not be 100% optimal, causing the agent to get “caught” in an almost infinite loop of suboptimal actions.

If epsilon is a small positive number (e.g. 5%) we can help the agent escape these infinite loops of suboptimal actions.

By using a small epsilon at evaluation we are adopting a so-called epsilon-greedy strategy.

Let’s evaluate our trained agent on n_episodes = 100 using epsilon = 0.05. Observe how the loop looks almost exactly as the train loop above, but without the call to update_parameters()

import random
from tqdm import tqdm

# exploration vs exploitation prob
epsilon = 0.05

n_episodes = 100

# For plotting metrics
timesteps_per_episode = []
penalties_per_episode = []


for i in tqdm(range(0, n_episodes)):
    
    state = env.reset()      
    
    epochs, penalties, reward, = 0, 0, 0
    done = False
    
    while not done:
        
        if random.uniform(0, 1) < epsilon:
            # Explore action space
            action = env.action_space.sample()
        else:
            # Exploit learned values
            action = agent.get_action(state)
        
        next_state, reward, done, info = env.step(action)
        
        agent.update_parameters(state, action, reward, next_state)
                      
        if reward == -10:
            penalties += 1

        state = next_state
        epochs += 1
            
    timesteps_per_episode.append(epochs)
    penalties_per_episode.append(penalties)

print(f'Avg steps to complete ride: {np.array(timesteps_per_episode).mean()}')
print(f'Avg penalties to complete ride: {np.array(penalties_per_episode).mean()}')

These numbers look much much better than for the RandomAgent.

We can say our agent has learned to drive the taxi!

Q-learning gives us a method to compute optimal q-values. But, what about the hyper-parameters alpha , gamma and epsilon ?

I chose them for you, rather arbitrarily. But in practice, you will need to tune them for your RL problems.

Let’s explore their impact on learning to get a better intuition of what is going on.

5. Hyper-parameter tuning 🎛️

👉🏽 notebooks/03_q_agent_hyperparameters_analysis.ipynb

Let’s train our q-agent using different values for alpha (learning rate) and gamma (discount factor). As for epsilon we keep it at 10%.

In order to keep the code clean, I encapsulated the q-agent definition inside src/q_agent.py and the training loop inside the train() function in src/loops.py

# No need to copy paste the same QAgent
# definition in every notebook, don't you think?
from src.q_agent import QAgent

# hyper-parameters
# RL problems are full of these hyper-parameters.
# For the moment, trust me when I set these values.
# We will later play with these and see how they impact learning.
alphas = [0.01, 0.1, 1]
gammas = [0.1, 0.6, 0.9]

import pandas as pd

from src.loops import train

# exploration vs exploitation prob
# let's start with a constant probability of 10%.
epsilon = 0.1
n_episodes = 1000

results = pd.DataFrame()
for alpha in alphas:
    for gamma in gammas:
        
        print(f'alpha: {alpha}, gamma: {gamma}')
        agent = QAgent(env, alpha, gamma)
        
        _, timesteps, penalties = train(agent,
                                        env,
                                        n_episodes,
                                        epsilon)
        
        # collect timesteps and penalties for this pair
        # of hyper-parameters (alpha, gamma)
        results_ = pd.DataFrame()
        results_['timesteps'] = timesteps
        results_['penalties'] = penalties
        results_['alpha'] = alpha
        results_['gamma'] = gamma
        results = pd.concat([results, results_])

# index -> episode
results = results.reset_index().rename(
    columns={'index': 'episode'})

# add column with the 2 hyper-parameters
results['hyperparameters'] = [
    f'alpha={a}, gamma={g}'
    for (a, g) in zip(results['alpha'], results['gamma'])
]

Let us plot the timesteps per episode for each combination of hyper-parameters.

import seaborn as sns
import matplotlib.pyplot as plt

fig = plt.gcf()
fig.set_size_inches(12, 8)
sns.lineplot('episode', 'timesteps',
             hue='hyperparameters', data=results)

The graph looks artsy-fartsy, but a bit too noisy 😵.

Something you can observe though is that when alpha = 0.01 the learning is slower. alpha (learning rate) controls how much we update the q-values in each iteration. Too small of a value implies slower learning.

Let’s discard alpha = 0.01 and do 10 runs of training for each combination of hyper-parameters. We average the timestepsfor each episode number, from 1 to 1000, using these 10 runs.

I created the function train_many_runs() in src/loops.py to keep the notebook code cleaner:

from src.loops import train_many_runs

alphas = [0.1, 1]
gammas = [0.1, 0.6, 0.9]

epsilon = 0.1
n_episodes = 1000
n_runs = 10

results = pd.DataFrame()
for alpha in alphas:
    for gamma in gammas:
        
        print(f'alpha: {alpha}, gamma: {gamma}')
        agent = QAgent(env, alpha, gamma)
        
        timesteps, penalties = train_many_runs(agent,
                                               env,
                                               n_episodes,
                                               epsilon,
                                               n_runs)
        
        # collect timesteps and penalties for this pair of
        # hyper-parameters (alpha, gamma)
        results_ = pd.DataFrame()
        results_['timesteps'] = timesteps
        results_['penalties'] = penalties
        results_['alpha'] = alpha
        results_['gamma'] = gamma
        results = pd.concat([results, results_])

# index -> episode
results = results.reset_index().rename(
    columns={'index': 'episode'})

results['hyperparameters'] = [
    f'alpha={a}, gamma={g}'
    for (a, g) in zip(results['alpha'], results['gamma'])]

It looks like alpha = 1.0 is the value that works best, while gamma seems to have less of an impact.

Congratulations! You have tuned your first learning rate in this course 🥳

Tunning hyper-parameters can be time-consuming and tedious. There are excellent libraries to automate the manual process we just followed, like Optuna, but this is something we will play with later in the course. For the time being, enjoy the speed-up in training we have just found.

Wait, what happens with this epsilon = 10% that I told you to trust me on?

Is the current 10% value the best?

Let’s check it ourselves.

We take the best alpha and gammawe found, i.e.

alpha = 1.0
gamma = 0.9 (we could have taken 0.1 or 0.6 too)

And train with differentepsilons = [0.01, 0.1, 0.9]

# best hyper-parameters so far
alpha = 1.0
gamma = 0.9

epsilons = [0.01, 0.10, 0.9]
n_runs = 10
n_episodes = 200

results = pd.DataFrame()
for epsilon in epsilons:
        
    print(f'epsilon: {epsilon}')
    agent = QAgent(env, alpha, gamma)

    timesteps, penalties = train_many_runs(agent,
                                           env,
                                           n_episodes,
                                           epsilon,
                                           n_runs)

    # collect timesteps and penalties for this pair of
    # hyper-parameters (alpha, gamma)
    results_ = pd.DataFrame()
    results_['timesteps'] = timesteps
    results_['penalties'] = penalties
    results_['epsilon'] = epsilon
    results = pd.concat([results, results_])

# index -> episode
results = results.reset_index().rename(columns={'index': 'episode'})

And plot the resulting timesteps and penalties curves:

fig = plt.gcf()
fig.set_size_inches(12, 8)
sns.lineplot('episode', 'timesteps', hue='epsilon', data=results)
plt.show()

fig = plt.gcf()
fig.set_size_inches(12, 8)
sns.lineplot('episode', 'penalties', hue='epsilon', data=results)

As you can see, both epsilon = 0.01 and epsilon = 0.1 seem to work equally well, as they strike the right balance between exploration and exploitation.

On the other side, epsilon = 0.9 is too large of a value, causing “too much” randomness during training, and preventing our q-matrix to converge to the optimal one. Observe how the performance plateaus at around 250 timesteps per episode.

In general, the best strategy to choose the epsilon hyper-parameter is progressive epsilon-decay. That is, at the beginning of training, when the agent is very uncertain about its q-value estimation, it is best to visit as many states as possible, and for that, a large epsilon is great (e.g. 50%)

As training progresses, and the agent refines its q-value estimation, it is no longer optimal to explore that much. Instead, by decreasing epsilon the agent can learn to perfect and fine-tune the q-values, to make them converge faster to the optimal ones. Too large of an epsilon can cause convergence issues as we see for epsilon = 0.9.

We will be tunning epsilons along the course, so I will not insist too much for the moment. Again, enjoy what we have done today. It is pretty remarkable.

6. Recap ✨

Congratulations on (probably) solving your first Reinforcement Learning problem.

These are the key learnings I want you to sleep on:

The difficulty of a Reinforcement Learning problem is directly related to the number of possible actions and states. Taxi-v3 is a tabular environment (i.e. finite number of states and actions), so it is an easy one.
Q-learning is a learning algorithm that works excellent for tabular environments.
No matter what RL algorithm you use, there are hyper-parameters you need to tune to make sure your agent learns the optimal strategy.
Tunning hyper-parameters is a time-consuming process but necessary to ensure our agents learn. We will get better at this as the course progresses.