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# Markov Decision Processes

This session is a precursor to Reinforcement Learning.

**Reading:**Russel and Norvig, Chapter 16.1-3 + cursory 16.4- A video demo
- Briefing on MDP
- Eirik’s lecture notes from 2022

# Exercises

## Task 1

- Recall the requirements to model a problem with the MDP framework:
- Sequential decision problem

- Stochastic Environment
- Fully observable
- Markovian Transition model
- the probability distribution depends only on the current state and not past history

- Additive rewards

- Sequential decision problem
- and recoll the model elements
- A set of states \(s \in S\)
- A set of actions \(a \in A\)
- A probabilistic transition function \(T(s,a,s')\)
- A reward function \(R(s,a,s')\)
- A start state \(S_0\)

- Consider one of the following problems
- The Frozen Lake
- The game of Black Jack
- any problem you have previously studied in the semester, e.g. on CodinGame
- The moon lander (see the animation from Gymnasium)

- Form groups of 2-4 students and choose one problem each from the list above.

### 1A. Individual Part

Analyse the chosen probler, by

- checking if it fulfills the requirements above
- identify all the model elements (above)

If the problem does not satisfy the MDP requirements, can you amend it so that it does? E.g. could the game be changed into a random one? If it cannot be adapte, try another problem.

Using the properties of an MDP:

- Make a graphical representation of the (possibly modified) problem from Task A.

### 1B. Group Work

- Present your drawings and analysis to each other
- is the analysis convincing?

Discuss together,

- Does the problem have a finite or infinite horizon?
- If you were to attempt to solve the MDP, could the current horizon pose a problem, why/why not?

- Does the problem have a discounted reward?
- If you were to attempt to solve the MDP, what discount factor would make sense to use for the utility function?

- Does the problem have a finite or infinite horizon?

## Task 2. Getting started with Gymnasium

We will be using the Gymnasium framework to test concepts and ideas from reinforcement learning. You may want to consult the documentation, but you should try playing with it first.

- Gymnasium formerly known as Gym from OpenAI
- Frozen Lake

Install with these statements. (I am not sure if you need pygame or not.)

**Familiarize yourself with the FrozenLake environment** You can import and start the simulation of the Frozen Lake like so:

```
import gymnasium as gym
env = gym.make('FrozenLake-v1', desc=None, map_name="4x4", is_slippery=True,render_mode="human")
obs, info = env.reset(return_info=True)
```

You should now hopefully see a render of the environment.

**Try out some of these functions and see what they do:**

```
env.action_space.sample()
observation, reward, terminated, truncated, info = env.step(env.action_space.sample())
env.P # The MDP
env.P[s] # Transition matrix of state s
env.P[s][a] # Transitions from state s given action a
```

**Try to make a custom Frozen Lake map**

When creating a frozen-lake environment you can add a custom-map with the `desc`

argument, e.g:

```
fl_map = ["HFFFS", "FHHFF", "FFFFH", "HFFHG"]
env = gym.make('FrozenLake-v1', desc=fl_map, is_slippery=True)
```

For a custom 5x4 map.

## Task 3. Hill Climbing.

Given a FrozenLake map, and a list of pre-calculated expected utilities, (e.g.:

for the default FrozenLake 4x4 map).

- Visualise the utilities by drawing the lake and annotating it with the utilities.
- do the utilities look plausible?
- Try to solve the game manually using a hill climbing approach; do the utilities still look plausible?

- Implement a Hill Climbing solver for the Froen Lake Game.

Your code should look something like this:

```
import gymnasium as gym
env = gym.make('FrozenLake-v1', desc=None, map_name="4x4", is_slippery=True,render_mode="human")
env.reset()
state, info = env.reset()
utilities = [0.41,0.38,0.35,0.34,0.43,0,0.12,0,0.45,0.48,0.43,0,0,0.59,0.71,1]
done = False
while not done:
action = # Make your own definition
# Find the expected utility following each action and choose the best one.
state, reward, terminated, truncated, info = env.step(action)
print() # Add some diagnostic output so that you can see your decision
done = terminated or truncated
```

- Test your program; does it work rationally? Does it win the game?
- Reflect on your solution. Can we do better?

## Task 4. Value iteration

Recall the value/utility-function: \[U(s) = \mathop\max\limits_{a \in A(s)} \sum\limits_{s'}P(s'|s,a)[R(s,a,s') + \gamma U(s')]\]

Value iteration starts with an arbitrary postulated utility function \(U_0\), and for each iteration \(i=1,2,3,\ldots\) it calculates a new estimated \(U_i\), by the rule \[U_i(s) = \mathop\max\limits_{a \in A(s)} \sum\limits_{s'}P(s'|s,a)[R(s,a,s') + \gamma U_{i-1}(s')]\]

Implement value iteration in python to estimate the utilities of the Frozen Lake.

**or if you prefer**

Execute value iteration manually, to find decent estiamtes.

## Task 5. Policy iteration (optional)

Recall Q-Function: \[Q(s,a) = \sum\limits_{s'}P(s'|s,a)[R(s,a,s') + \gamma \mathop\max\limits_{a'}Q(s',a')]\] and the function to extract an optimal policy from the Q-Function: $\(\pi^*(s) = \mathop{\mathrm{argmax}}\limits_aQ(s,a)\) Recall also that a policy is a map from state to action.

Policy iteration starts with both a utility estimate \(U_0\) and a tentative policy \(\pi_0\). Each iteration \(i\)

- estimates the utility \(U_{i+1}\) by assuming the policy \(\pi_i\)
- calculate a new policy \(\pi_{i+1}\) by assuming utilitie \(U_{i+1}\).

Implement policy iteration in python.