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# CSE 415 Assignment 5 Markov Decision Processes and Computing Utility Values solved

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## Description

5/5 - (1 vote)

The search theme continues here, except that now our agents operate in a world in which
actions may have uncertain outcomes. The interactions are modeled probabilistically using the
technique of Markov Decision Processes.

Within this framework, the assignment focuses on
two approaches to having the agent maximize its expected utility: (A) by a form of planning, in
which we assume that the parameters of the MDP (especially T and R) are known, which takes
the form of Value Iteration, and (B) by an important form of reinforcement learning, called Qlearning, in which we assume that the agent does not know the MDP parameters.

Part A is
required and worth 100 points, while Part B is for 10 points of extra credit.
The particular MDPs we are working with in this assignment are variations of a “TOH World”,
meaning Towers-of-Hanoi World. We can think of an agent as trying to solve a TOH puzzle
instance by wandering around in the state space that corresponds to that puzzle instance.

If it
solves the puzzle, it will get a big reward. Otherwise, it might get nothing, or perhaps a negative
reward for wasting time. In Part A, the agent is allowed to know the transition model and
reward function. In Part B (optional), the agent is not given that information, but has to learn
it by exploring and seeing what states it can get to and how good they seem to be based on what
rewards it can get and where states seem to lead to.

## What to Do.

Download the starter code, and try running the file TOH_MDP.py. This will
start up a graphical user interface, assuming that you are running the standard distribution of
Python 3.6 or Python 3.7, which includes the Tkinter user-interface technology. You can see

Some of the menu choices are handled by the starter code, such
as setting some of the parameters of the MDP. However, you will need to implement the code
to actually run the Value Iteration and, if you choose to do the optional Part B, the Q-Learning.

Figure 1. An early description of the problem “La Tour d’Hanoi” from the 1895 book by
Edouard Lucas. Your program will determine an optimal policy for solving this puzzle, as well
as compute utility values for each state of the puzzle.

## PART A: Implement Value Iteration (100 points).

There is a starter file called YourUWNetID_VI.py. Rename the file according to this format
and your own UWNetID. Also, change the name within TOH_MDP.py where this file is
imported.

Complete the implementation of the functions there:
1. one_step_of_VI(S, A, T, R, gamma, Vk)
which can compute 1 iteration of Value Iteration from the given MDP information plus
the current state values, which are provided in a dictionary Vk whose keys are states and
whose values are floats.

2. return_Q_values(S, A)
which should return the Q-state values that were computed during the most recent call to
one_step_of_VI. See the starter code for more details.
3. extract_policy(S, A)
Using the most recently computed Q-state values, determine the implied policy to
maximize expected utility. See the starter code for more details.

4. apply_policy(s)
Return the action that your current best policy implies for state s.

## A Sample Test Case.

Once your implementation is complete, you should be able to duplicate
the following example display by setting the following parameters for your MDP and VI

from the MDP Rewards menu: “One goal, R=100”; again from the MDP Rewards menu: Living
R= +0.1; from the Discount menu: “γ = 0.9”; from the Value Iteration menu: “Show state
values (V) from VI”, and “100 Steps of VI” and finally “Show Policy from VI.”

Figure 2. A sample of computed state values. The policy computed from the associated Q
values (not shown here) is displayed using red arrows.

## Report for Part A.

Create a PDF document that includes the following questions (1a, 1b, etc.) but also their

Name the file using this naming pattern: YourUWNedID_A5_Report.pdf. The following tells
you what to do and what questions to answer
1. Using the menu commands, set up the MDP for TOH with 3 disks, no noise, one goal,
and living reward=0.

The agent will use discount factor 1. From the Value Iteration menu
select “Show state values (V) from VI”, and then select “Reset state values (V) and Q
values for VI to 0”.
Use the menu command “1 step of VI” as many times as needed to answer these
questions:

1a. How many iterations of VI are required to turn 1/3 of the states green? (i.e., get their
expected utility values to 100).
1b. How many iterations of VI are required to get all the states, including the start state,
to 100?

1c. From the Value Iteration menu, select “Show Policy from VI”. (The policy at each state
is indicated by the outgoing red arrowhead. If the suggested action is illegal, there could
still be a legal state transition due to noise, but the action could also result in no change
of state.) Describe this policy. Is it a good policy? Explain.

2. Repeat the above setup except for 20% noise.
2a. How many iterations are required for the start state to receive a nonzero value.
2c. At this point, view the policy from VI as before. Is it a good policy? Explain.
2d. Run additional VI steps to find out how many iterations are required for VI to
converge. How many is it?

2e. After convergence, examine the computed best policy once again. Has it changed? If
so, how? If not, why not? Explain.
3. Repeat the above setup, including 20% noise but with 2 goals and discount = 0.5.
3a. Run Value Iteration until convergence. What does the policy indicate? What value
does the start state have? (start state value should be 0.82)

3b. Reset the values to 0, change the discount to 0.9 and rerun Value Iteration until
convergence. What does the policy indicate now? What value does the start state have?
(start state value should be 36.9)

4. Now try simulating the agent following the computed policy. Using the “VI Agent” menu,
select “Reset state to s0”. Then select “Perform 10 actions”. The software should show the
motion of the agent taking the actions shown in the policy.

Since the current setup has
20% noise, you may see the agent deviate from the implied plan. Run this simulation 10
times, observing the agent closely.
4a. In how many of these simulation runs did the agent ever go off the plan?

4b. In how many of these simulation runs did the agent arrive in the goals state (at the
end of the golden path)?
4c. For each run in which the agent did not make it to the goal in 10 steps, how many
steps away from the goal was it?

4d. Are there parts of the state space that seemed never to be visited by the agent? If so,
where (roughly)?
5. Overall reflections.
5a. Since it is having a good policy that is most important to the agent, is it essential that
the values of the states have converged?

5b. If the agent were to have to learn the values of states by exploring the space, rather
than computing with the Value Iteration algorithm, and if getting accurate values
requires re-visiting states a lot, how important would it be that all states be visited a lot?

## (OPTIONAL) PART B: Implement Q­Learning (10 points).

Start by renaming the skeleton file YourUWNetID_Q_Learn.py to be your own file, and change
the TOH_MDP.py file to import the renamed file (as well as your custom Value Iteration file).
Implement Q-learning to work in two contexts.

In one context, a user can “drive” the agent
through the problem space by selecting actions through the GUI. After each action is executed
by the system, your handle_transition function will be called, and you should program it to
use the information provided in the call to do your Q-value update.

In the second context, your function choose_next_action will be called, and it should perform
two things: (1) use the information provided about the last transition to update a Q-value
(similar to in the first context), and (2) select an action for the next transition. The action may
be an optimal action (based on existing Q-values) or better, an action that makes a controlled
compromise between exploration and exploitation.

In order control the compromise, you should implement epsilon-greedy Q-learning. You
should provide two alternatives: (a) fixed epsilon, with the value specified by the GUI or the
system (including possible autograder), and (b) custom epsilon-greedy learning.

In the latter,
your program can set an epsilon value and change it during the session, in order to gradually
have the agent focus more on exploitation and less on exploration.

You are encouraged to develop another means of controlling the exploitation/exploration
tradeoff. Implementing the use of an exploration function is optional and worth 5 points of
extra credit. The GUI provides a means to control whether you are using it or not. However, it
is up to you to handle it when it is selected.

One unusual feature of some of the MDPs that we use in studying AI is goal states (or “exit
states”) and exit actions. In order to make your code consistent with the way we describe these
in lecture, adhere to the following rules when implementing the code that chooses an action to
return:
(a) When the current state is a goal state, always return the “Exit” action and never
return any of the other six actions.

(b) When the current state is NOT a goal state, always return one of the six directional
actions, and never return the Exit action.
Here is an example of how you can test a state s to find out if it is a goal state:
if is_valid_goal_state(s):
print(“It’s a goal state; return the Exit action.”)
elif s==Terminal_state:

print(“It’s not a goal state. But if it’s the special Terminal state, return None.”)
else:
print(“it’s neither a goal nor the Terminal state, so return some ordinary action.”)
Note that the Terminal_state variable is only available in the starter code version issued as a5-
starter-code-Feb-14b.tar or later.
Here is the starter code for this assignment.

### What to Turn In.

For Part A, Turn in your Value Iteration Python file, named with the scheme
YourUWNetID_VI.py. Also, turn in your slightly modified version of TOH_MDP.py, without
changing its name.

(The modification this time is that it imports your Value Iterations file. You
can leave the import of YourUWNetID_Q_Learn as it is in the starter code for this first
submission.) ). Finally, be sure to turn in your report file, named using the pattern
YourUWNetID_A5_Report.pdf.

For the optional Part B, turn your Q_Learn Python file, named with the scheme
YourUWNetID_Q_Learn, and a version of TOH_MDP.py that imports both of your custom