🕵️ Introduction
Consider a 10 × 10 gridworld as shown in Figure

State space: Gridworld has 100 distinct states. There is a special absorbing state called ”Goal”. There are other special nonabsorbing states called ”subgoals” that are scattered around the gridworld. All other states are referred to as ”normal” states.

Actions: In each nonterminal state, you can take 4 actions A = {North, East, South, West}, which moves you one cell in the corresponding direction.

Transition model: The model may be stochastic, i.e. for an action a ∈ {North, East, South, West}, the agent moves in the corresponding direction of a with probability of less than 1 and greater than 0 for the other directions. For instance, if you choose ”North” the agent will move one place northward with probability 0.8, while in the other directions it may move with probability 0.2/3. However, transitions that take you off the grid will not result in any change. There are no transitions available from the ”Goal” state.

Rewards: You will receive a reward of −1 for all transitions except the transition to the ”sub goal” or ”Goal” state. Any transition to the ”Goal” state gives you a reward of +100. Any transition to one of the ”subgoal” states gives you a reward in the range [+5, +25]. However, this reward can only be received once from each ”subgoal” state after which it resets into a ”normal” state, i.e. transitioning into it again won’t give you any special reward apart from the −1 transition cost. Also, the number of ”subgoal” sates lie in the range [4, 8].
Implement the following:

Find optimal values using ANY value iteration algorithm. Note that a faster converging algorithm would be awarded higher marks**. Initialize the value grid J0(s) = 0 and policy grid π0(s) = North, ∀s. Let the discount factor γ = 0.99. (Mention sources/papers if any.) (4 marks)
**Marking scheme: Let us say your algorithm A calculates a value grid JA and the optimal value grid (viz. precalculated) is J∗. Then a tolerance δ is calculated as
ε = max JA(s) − J∗(s). s∈S
If this ε < 1e−3, then it is said that your algorithm has converged and the time taken for such a convergence tA is stored. There will be two cutoffs, i.e. a ”faster” time tf and a ”slower” time ts such that tf < ts. If tA < tf, you will be awarded the full 4 marks, else if tA > ts, you'll will be awared 0 marks. If tf < tA <ts, the marks will be linearly interpolated.
Note that δ will not be checked on evaluation for each iteration and only on the final values. Hence, an appropriate stopping criteria would be required. Also the time tA for your algorithm will be calculated on AIcrowd’s server on a specific hardware to ensure fair results. Code may be checked for any malpracties and appropriate actions will be taken under the discretion of the course instructor.

Find the optimal policy and values for each state using policy iteration algorithm. Start with J0(s) = 0 and π0(s) = North, ∀s. Let the discount factor γ = 0.99. (1.5 marks)

Find the optimal policies for the default grid (in the figure) by now setting γ = 0.999 and for 3 different integer values of the transition reward, i.e. { 1, 5, 10 }. Show the variations in the optimal path visually, starting from the "Start" state for each of these 3 values. (1.5 marks)
Answer the following questions:

Briefly comment on your custom value iteration algorithm and the necessary improvements made from standard value iteration algorithms. Generate plots if required. (1 mark)

How are the different values of the transition reward affecting the optimal path for the different transition rewards? Explain your findings with the necessary plots. (2 marks)
📁 Files
Under the Resources
section you will find data files that contains parameters for the environment for this problem.
🚀 Submission
 Submissions will be made through a notebook following the instructions in the starter notebook.
 Each Team can make 5 successful submissions and 5 failed submissions in a day. Once the limit of failed submission is reached, the submission will be counted in the successful submission.
 The submission limit will reset at 5:30 AM IST everyday.
 At the end of the challenge, you will have to select 1 submission as the final one. You can select that here.
📱 Contact
 RL TAs
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