To be completed

# Exercises

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:

### 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’:

### 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?

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’:

### 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’:

### 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.

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

- 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 ?

(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;

1. We can set a static epsilon $$\epsilon$$ value, and set the action to some random action a if some random number n is below $$\epsilon$$.
2. 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.
3. 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’:

### Part B

- Use the updated q-table with the play function and the test_performance function.
- Print out the q-table

### 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.