## Revision 2071c852751535b6df1d28671a9dbb31a4fb600d (click the page title to view the current version)

# Reinformcement Learning Part 1

**To be completed**

# Exercises

## Task 1

Last week we implemented the Q-Function:

\[Q(s,a) = \sum_{s'}P(s'|s,a)[R(s,a,s') + \gamma \max_{a'}Q(s',a')]\]

and the function for the optimal policy based on the results from the Q-Function:

\(\pi^*(s) = \mathop{\text{argmax }}\limits_aQ(s,a)\)

We also had an assignment where we moved based on a list of calculated utilities.

In this assignment we will be implementing table based Q-learning, a model-free, off-policy reinforcement learning algorithm. For now, we want to just try to play using a list of calculated q-values instead of the utilities, you can use the following q-values:

```
def create_filled_q_table() -> np.array:
"""Creates a 'filled' q-table for the default frozen-lake environment.
Returns:
Filled q-table
"""
return np.array([
[0.009, 0.192, 0.007, 0.009],
[0.003, 0.002, 0.003, 0.17],
[0.003, 0.002, 0.001, 0.067],
[0.001, 0.001, 0.002, 0.037],
[0.526, 0.002, 0.001, 0.002],
[0., 0., 0., 0.],
[0.046, 0., 0., 0.],
[0., 0., 0., 0.],
[0.002, 0.002, 0.002, 0.709],
[0.001, 0.597, 0.001, 0.001],
[0.945, 0., 0., 0.],
[0., 0., 0., 0.],
[0., 0., 0., 0.],
[0.02, 0.012, 0.898, 0.016],
[0.061, 0.991, 0.092, 0.068],
[0., 0., 0., 0.]
])
```

### Part A

We will use the latter in this task, so if you did not already, ** make a python implementation of this policy**. You can use the following ‘skeleton’:

```
def optimal_policy(q_sa: np.array, state: int) -> int:
"""RL-policy for optimal play.
Args:
q_sa: q-table
state: current state
Returns:
optimal action for current state and q-table.
"""
...
```

### Part B

Below I have implemented two functions `play`

and `test_performance`

.

```
import numpy as np
import gym
from collections.abc import Callable
def play(env: gym.Env, q_sa: np.array, policy: Callable, m_ep_length: int = 100, render: bool = True) -> None:
"""Plays one episode of environment env using an optimal policy from the q-table q_sa.
Args:
env: Gym environment
q_sa: q-table
policy: the policy function to play with
m_ep_length: max episode length, default; 100, should be set higher for large environments
render: render environment ?
Returns:
None
"""
state, info = env.reset(return_info=True)
if render:
env.render()
j = 0
while j < m_ep_length:
j += 1
action = policy(q_sa, state) # TODO: Implement this function first!
new_state, reward, done, info = env.step(action)
state = new_state
if render:
env.render()
if done:
break
def test_performance(env: gym.Env, q_sa: np.array, policy: Callable, n_episodes: int = 1000, m_ep_length: int = 100) -> float:
"""
Args:
env: Gym environment
q_sa: q-table
policy: the policy function to play with
n_episodes: number of episodes
m_ep_length: max episode length
Returns:
average reward
"""
rewards = 0
for i in range(n_episodes):
state, info = env.reset(return_info=True)
reward = 0
j = 0
while j < m_ep_length:
j += 1
action = policy(q_sa, state) # TODO: Implement
new_state, reward, done, info = env.step(action)
state = new_state
if done:
break
rewards += reward
return rewards / n_episodes
if __name__ == "__main__":
environment = gym.make('FrozenLake-v1', desc=None, map_name="4x4", is_slippery=True)
q_table = create_filled_q_table() # TODO: Implement
play(environment, q_table, optimal_policy) # TODO: Implement policy
```

Using either this implementation, or your own:

*Using your optimal policy from part 1, play at least one episode*

– Is it playing “optimally”?

*Calculate the performance of the policy using the*`test_performance`

function.

– What does the number here mean?

## Task 2

In this task we will create functions to update our own q-table, for now you can turn make the environment deterministic by turning of the ‘slippery’ argument when making the environment:

### Part A

First we need to create an empty q-table, as the gym framework supports frozen-lake environment of different sizes, we need to initialize it with the size “state_space x action_space”.

You can get them from:

**Create a function to initialize a q-table**

You can use the following ‘skeleton’:

```
def initialize_q_table(env: gym.Env) -> np.array:
"""Creates and returns an empty q-table of size state_space x action_space.
Args:
env: Gym environment
Returns:
np.array of q-table of size state_space x action_space
"""
...
```

### Part B

Now we need some way to update our q-table.

Recall the Q Temporal-Difference function to update the q-values:

\(Q(s,a) \leftarrow Q(s,a) + \alpha[R(s,a,s') + \gamma \max\limits_{a'}Q(s',a') - Q(s,a)]\)

**Implement a function to calculate the value to be updated (the part on the right side of the arrow)**

You can use the following ‘skeleton’:

```
def q_temporal_difference(q_sa: np.array, action: int, reward: float, start_state: int, end_state: int, alpha: float = 0.85, gamma: float = 0.98) -> float:
"""Calculates the q-update.
Args:
q_sa: q-table
action: action we are taking
reward: result of R(s,a,s')
start_state: start state
end_state: end state (after taking action a)
alpha: learning-rate
gamma: discount
Returns:
q-td update value
"""
...
```

### Part C

Using the functions we implemented above, we want to update the q-table by simply having the agent play the game a lot.

** Implement a q-learning function**, you can use the following ‘skeleton’ and/or take inspiration from the

`test_performance`

implementation.```
def q_learning(env: gym.Env, policy: Callable, n_episodes: int = 10000, m_ep_length: int = 100) -> np.array:
"""q-learning implementation to update a q-table.
Args:
env: gym environment
policy: policy function
n_episodes: number of episodes to train on
m_ep_length: maximum episode length
Returns:
updated q-table
"""
...
```

Note that your agent might behave odd (or not work at all), if you use your optimal policy on an empty q-table, so you may want to edit it to take a random action if it has issues differentiating between actions.

### Part D

** Try out your algorithm**;

- Use the updated q-table with the

`play`

function and the `test_performance`

function.- Print out the q-table

Are there any potential problems?

What if you train it again with `slippery=True`

?

## Task 3

(From now on, we will play with a stochastic environment, set `slippery=True`

).

We need some way to encourage exploration, to prevent the agent from only trying to repeat the first sequence that got him to the goal.

There are multiple ways to implement this;

- We can set a static epsilon \(\epsilon\) value, and set the action to some random action a if some random number n is below \(\epsilon\).

- We usually want to encourage exploration in earlier training phases, and encourage exploitation in the later ones. We can therefore use a similar approach to 1, but with the addition of decaying \(\epsilon\) over time.

- The third option (non-exhaustive) is to create a policy that picks an action based on a weighted probability-distribution created based on the q-values. The weighting can then change over time to encourage exploration early, and exploitation later. A modified version of this function could also be used when ‘playing’ the game, if you want a policy that not necessarily always picks the option with the highest utility.

### Part A

** Implement _one_ of the functions above**, you can use the following ‘skeleton’:

```
def epsilon_policy(q_sa: np.array, state: int, env: gym.Env, eps: float = 0.2) -> int:
"""RL epsilon-greedy policy.
Policy for exploration/exploitation tradeoff.
Args:
q_sa: q-table
state: current state
env: gym environment
eps: epsilon
Returns:
action with a 1-eps chance of being exploitation, eps chance of being exploration
"""
...
```

### Part B

** Try out your algorithm**;

- Use the updated q-table with the

`play`

function and the `test_performance`

function.- Print out the q-table

## Task 4

### Part A

*Modify your q-learning algorithm to call test_performance every n-episode. Save this in a table and plot the result using matplotlib.*

We now have a way to calculate the performance over time/training episodes.

### Part B

*Experiment with the different hyperparameters (epsilon, learning-rate, gamma, etc) and compare them using the method in part A.*

You can also try with other versions of the frozen-lake environment (e.g. the 8x8 map), they have a function to create random maps.

## Task 5 (Extra)

Please inform me if you get to this point early, as I might change the task, but for now:

*Implement SARSA, and repeat similar experiments from task 4 to compare the two.*