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IIT-M RL-ASSIGNMENT-2-TAXI

Solution for submission 132245

A detailed solution for submission 132245 submitted for challenge IIT-M RL-ASSIGNMENT-2-TAXI

jaswanthi_mandalapu_ee19d400

What is the notebook about?

Problem - Taxi Environment Algorithms

This problem deals with a taxi environment and stochastic actions. The tasks you have to do are:

  • Implement Policy Iteration
  • Implement Modified Policy Iteration
  • Implement Value Iteration
  • Implement Gauss Seidel Value Iteration
  • Visualize the results
  • Explain the results

How to use this notebook? 📝

  • This is a shared template and any edits you make here will not be saved.You should make a copy in your own drive. Click the "File" menu (top-left), then "Save a Copy in Drive". You will be working in your copy however you like.

  • Update the config parameters. You can define the common variables here

Variable Description
AICROWD_DATASET_PATH Path to the file containing test data. This should be an absolute path.
AICROWD_RESULTS_DIR Path to write the output to.
AICROWD_ASSETS_DIR In case your notebook needs additional files (like model weights, etc.,), you can add them to a directory and specify the path to the directory here (please specify relative path). The contents of this directory will be sent to AIcrowd for evaluation.
AICROWD_API_KEY In order to submit your code to AIcrowd, you need to provide your account's API key. This key is available at https://www.aicrowd.com/participants/me

Setup AIcrowd Utilities 🛠

We use this to bundle the files for submission and create a submission on AIcrowd. Do not edit this block.

In [2]:
!pip install aicrowd-cli > /dev/null

AIcrowd Runtime Configuration 🧷

Get login API key from https://www.aicrowd.com/participants/me

In [3]:
import os

AICROWD_DATASET_PATH = os.getenv("DATASET_PATH", os.getcwd()+"/13d77bb0-b325-4e95-a03b-833eb6694acd_a2_taxi_inputs.zip")
AICROWD_RESULTS_DIR = os.getenv("OUTPUTS_DIR", "results")
In [4]:

API Key valid
Saved API Key successfully!
13d77bb0-b325-4e95-a03b-833eb6694acd_a2_taxi_inputs.zip: 100% 31.2k/31.2k [00:00<00:00, 346kB/s]
In [5]:
!unzip $AICROWD_DATASET_PATH
Archive:  /content/13d77bb0-b325-4e95-a03b-833eb6694acd_a2_taxi_inputs.zip
   creating: inputs/
  inflating: inputs/inputs_base.npy  
  inflating: inputs/inputs_1.npy     
  inflating: inputs/inputs_0.npy     
  inflating: inputs/inputs_2.npy     
   creating: targets/
  inflating: targets/targets_2.npy   
  inflating: targets/targets_0.npy   
  inflating: targets/targets_1.npy   
  inflating: targets/targets_base.npy  
In [6]:
DATASET_DIR = 'inputs/'

Taxi Environment

Read the environment to understand the functions, but do not edit anything

In [30]:
import numpy as np
import copy
import matplotlib.pyplot as plt


class TaxiEnv_HW2:
    def __init__(self, states, actions, probabilities, rewards, initial_policy):
        self.possible_states = states
        self._possible_actions = {st: ac for st, ac in zip(states, actions)}
        self._ride_probabilities = {st: pr for st, pr in zip(states, probabilities)}
        self._ride_rewards = {st: rw for st, rw in zip(states, rewards)}
        self.initial_policy = initial_policy
        self._verify()

    def _check_state(self, state):
        assert state in self.possible_states, "State %s is not a valid state" % state

    def _verify(self):
        """ 
        Verify that data conditions are met:
        Number of actions matches shape of next state and actions
        Every probability distribution adds up to 1 
        """
        ns = len(self.possible_states)
        for state in self.possible_states:
            ac = self._possible_actions[state]
            na = len(ac)

            rp = self._ride_probabilities[state]
            assert np.all(rp.shape == (na, ns)), "Probabilities shape mismatch"
        
            rr = self._ride_rewards[state]
            assert np.all(rr.shape == (na, ns)), "Rewards shape mismatch"

            assert np.allclose(rp.sum(axis=1), 1), "Probabilities don't add up to 1"

    def possible_actions(self, state):
        """ Return all possible actions from a given state """
        self._check_state(state)
        return self._possible_actions[state]

    def ride_probabilities(self, state, action):
        """ 
        Returns all possible ride probabilities from a state for a given action
        For every action a list with the returned with values in the same order as self.possible_states
        """
        actions = self.possible_actions(state)
        ac_idx = actions.index(action)
        return self._ride_probabilities[state][ac_idx]

    def ride_rewards(self, state, action):
        actions = self.possible_actions(state)
        ac_idx = actions.index(action)
        return self._ride_rewards[state][ac_idx]

Example of Environment usage

In [8]:
def check_taxienv():
    # These are the values as used in the pdf, but they may be changed during submission, so do not hardcode anything

    states = ['A', 'B', 'C']

    actions = [['1','2','3'], ['1','2'], ['1','2','3']]

    probs = [np.array([[1/2,  1/4,  1/4],
                    [1/16, 3/4,  3/16],
                    [1/4,  1/8,  5/8]]),

            np.array([[1/2,   0,     1/2],
                    [1/16,  7/8,  1/16]]),

            np.array([[1/4,  1/4,  1/2],
                    [1/8,  3/4,  1/8],
                    [3/4,  1/16, 3/16]]),]

    rewards = [np.array([[10,  4,  8],
                        [ 8,  2,  4],
                        [ 4,  6,  4]]),

            np.array([[14,  0, 18],
                        [ 8, 16,  8]]),

            np.array([[10,  2,  8],
                        [6,   4,  2],
                        [4,   0,  8]]),]
    initial_policy = {'A': '1', 'B': '1','C': '1'}

    env = TaxiEnv_HW2(states, actions, probs, rewards, initial_policy)
    print("All possible states", env.possible_states)
    print("All possible actions from state B", env.possible_actions('B'))
    print("Ride probabilities from state A with action 2", env.ride_probabilities('A', '2'))
    print("Ride rewards from state C with action 3", env.ride_rewards('C', '3'))

    base_kwargs = {"states": states, "actions": actions, 
                "probabilities": probs, "rewards": rewards,
                "initial_policy": initial_policy}
    return base_kwargs

base_kwargs = check_taxienv()
env = TaxiEnv_HW2(**base_kwargs)
All possible states ['A', 'B', 'C']
All possible actions from state B ['1', '2']
Ride probabilities from state A with action 2 [0.0625 0.75   0.1875]
Ride rewards from state C with action 3 [4 0 8]

Task 1 - Policy Iteration

Run policy iteration on the environment and generate the policy and expected reward

In [9]:
# 1.1 Policy Iteration
def policy_iteration(taxienv, gamma):
    # A list of all the states
    states = taxienv.possible_states
    # Initial values
    values = {s: 0 for s in states}

    # This is a dictionary of states to policies -> e.g {'A': '1', 'B': '2', 'C': '1'}
    policy = taxienv.initial_policy.copy()

    ## Begin code here

    # Hints - 
    # Do not hardcode anything
    # Only the final result is required for the results
    # Put any extra data in "extra_info" dictonary for any plots etc
    # Use the helper functions taxienv.ride_rewards, taxienv.ride_probabilities,  taxienv.possible_actions
    # For terminating condition use the condition exactly mentioned in the pdf
    
    policy_while_loop_assertion = False
    delta_array = np.full(len(states),1e-8)
    while policy_while_loop_assertion == False:    
      reward_while_loop_assertion = False
      while reward_while_loop_assertion == False:
        delta = np.zeros(len(states))
        j = 0
        for s in states:
          temp_j = values[s]
          action = policy[s]
          transition_probabilities = taxienv.ride_probabilities(s, action)
          transition_rewards = taxienv.ride_rewards(s, action)
          i=0
          sum = 0
          for s1 in states:
            sum = sum + transition_probabilities[i] * (transition_rewards[i] + gamma * values[s1])
            i = i+1
          values[s] = sum
          delta[j] = np.max([delta[j], np.abs(temp_j - sum)])
          j = j+1
        less_equal_assertion = delta < delta_array
        if np.all(less_equal_assertion) == True:
          reward_while_loop_assertion == True
          break
      done = 1
      for s in states:
        b = policy[s]
        possible_actions = taxienv.possible_actions(s)
        costs = np.zeros(len(possible_actions))
        action_variable = 0
        for action in possible_actions:
          transition_probabilities = taxienv.ride_probabilities(s, action)
          transition_rewards = taxienv.ride_rewards(s, action)
          state_variable = 0
          for s1 in states:
            costs[action_variable] = costs[action_variable] + transition_probabilities[state_variable] * (transition_rewards[state_variable] + gamma * values[s1])
            state_variable = state_variable + 1
          action_variable = action_variable + 1
        policy[s] = possible_actions[np.argmax(costs)]
        if b != policy[s]:
          done = 0
      if done == 1:
        policy_while_loop_assertion = True
        break
    
    # Put your extra information needed for plots etc in this dictionary
    extra_info = {}

    ## Do not edit below this line

    # Final results
    return {"Expected Reward": values, "Policy": policy}, extra_info

Task 2 - Policy Iteration for multiple values of gamma

Ideally this code should run as is

In [10]:
# 1.2 Policy Iteration with different values of gamma
def run_policy_iteration(env):
    gamma_values = np.arange(5, 100, 5)/100
    results, extra_info = {}, {}
    for gamma in gamma_values:
        results[gamma], extra_info[gamma] = policy_iteration(env, gamma)
    return results, extra_info

results, extra_info = run_policy_iteration(env)
In [ ]:
# # Testing
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# target_output = np.load('/content/targets/targets_0.npy')
# np.load = np_load_old
# print(target_output.item().get('policy_iteration'))
In [ ]:
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# data = np.load('/content/inputs/inputs_0.npy')
# np.load = np_load_old
# env = TaxiEnv_HW2(data.item().get('states'), data.item().get('actions'), data.item().get('probabilities'), data.item().get('rewards'), data.item().get('initial_policy'))
# results, extra_info = run_policy_iteration(env)
# print(results)

Task 3 - Modifed Policy Iteration

Implement modified policy iteration (where Value iteration is done for fixed m number of steps)

In [11]:
# 1.3 Modified Policy Iteration
def modified_policy_iteration(taxienv, gamma, m):
    # A list of all the states
    states = taxienv.possible_states
    # Initial values
    values = {s: 0 for s in states}

    # This is a dictionary of states to policies -> e.g {'A': '1', 'B': '2', 'C': '1'}
    policy = taxienv.initial_policy.copy()

    ## Begin code here

    # Hints - 
    # Do not hardcode anything
    # Only the final result is required for the results
    # Put any extra data in "extra_info" dictonary for any plots etc
    # Use the helper functions taxienv.ride_rewards, taxienv.ride_probabilities,  taxienv.possible_actions
    # For terminating condition use the condition exactly mentioned in the pdf

    loop_assertion = False
    while loop_assertion == False:
      values_1 = copy.deepcopy(values)
      for k in range(m):
        for s in states:
          sum = 0
          action = policy[s]
          transition_probabilities = taxienv.ride_probabilities(s, action)
          transition_rewards = taxienv.ride_rewards(s, action)
          i = 0
          for s1 in states:
            sum = sum + transition_probabilities[i] * (transition_rewards[i] + gamma * values_1[s1])
            i = i+1
          values[s] = sum
      done = 1
      for s in states:
        temp_policy = policy[s]
        possible_actions = taxienv.possible_actions(s)
        costs = np.zeros(len(possible_actions))
        c = 0
        for action in possible_actions:
          transition_probabilities = taxienv.ride_probabilities(s, action)
          transition_rewards = taxienv.ride_rewards(s, action)
          t = 0
          for s1 in states:
            costs[c] = costs[c] + transition_probabilities[t] * (transition_rewards[t] + gamma * values[s1])
            t = t+1
          c = c+1
        policy[s] = possible_actions[np.argmax(costs)]
        if temp_policy != policy[s]:
          done = 0
      if done == 1:
        loop_assertion = True
        break

    
    # Put your extra information needed for plots etc in this dictionary
    extra_info = {}

    ## Do not edit below this line


    # Final results
    return {"Expected Reward": values, "Policy": policy}, extra_info

Task 4 Modified policy iteration for multiple values of m

Ideally this code should run as is

In [12]:
def run_modified_policy_iteration(env):
    m_values = np.arange(1, 15)
    gamma = 0.9
    results, extra_info = {}, {}
    for m in m_values:
        #print('m=',m)
        results[m], extra_info[m] = modified_policy_iteration(env, gamma, m)
    return results, extra_info

results, extra_info = run_modified_policy_iteration(env)
In [ ]:
# # Testing
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# target_output = np.load('/content/targets/targets_0.npy')
# np.load = np_load_old
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# data = np.load('/content/inputs/inputs_1.npy')
# np.load = np_load_old
# env = TaxiEnv_HW2(data.item().get('states'), data.item().get('actions'), data.item().get('probabilities'), data.item().get('rewards'), data.item().get('initial_policy'))
# mod_results, extra_info = modified_policy_iteration(env,0.9,10)
# results, extra_info = policy_iteration(env,0.9)
# print(mod_results)
# print(results)

Task 5 Value Iteration

Implement value iteration and find the policy and expected rewards

In [13]:
# 1.4 Value Iteration
def value_iteration(taxienv, gamma):
    # A list of all the states
    states = taxienv.possible_states
    # Initial values
    values = {s: 0 for s in states}

    # This is a dictionary of states to policies -> e.g {'A': '1', 'B': '2', 'C': '1'}
    policy = taxienv.initial_policy.copy()

    ## Begin code here

    # Hints - 
    # Do not hardcode anything
    # Only the final result is required for the results
    # Put any extra data in "extra_info" dictonary for any plots etc
    # Use the helper functions taxienv.ride_rewards, taxienv.ride_probabilities,  taxienv.possible_actions
    # For terminating condition use the condition exactly mentioned in the pdf
    delta_array = np.full(len(states),1e-8)
    loop_assertion = False
    H = {s:0 for s in states}
    temp_policy = {s:'' for s in states}
    while loop_assertion == False:
      delta = np.zeros(len(states))
      i=0
      for s in states:
        possible_actions = taxienv.possible_actions(s)
        costs = np.zeros(len(possible_actions))
        k = 0
        for action in possible_actions:
          transition_probabilities = taxienv.ride_probabilities(s, action)
          transition_rewards = taxienv.ride_rewards(s, action)
          k1 = 0
          for s1 in states:
            costs[k] = costs[k] + (transition_probabilities[k1] * (transition_rewards[k1] + gamma * values[s1]))
            k1 = k1 + 1
          k = k + 1
        H[s] = np.max(costs)
        temp_policy[s] = possible_actions[np.argmax(costs)]
        delta[i] = np.max([delta[i], np.abs(values[s] - H[s])])
        i = i+1
      for s in states:
        values[s] = H[s]
        policy[s] = temp_policy[s]
        
      less_equal_assertion = delta < delta_array
      if np.all(less_equal_assertion) == True:
        loop_assertion = True
        break

    # Put your extra information needed for plots etc in this dictionary
    extra_info = {}

    ## Do not edit below this line

    # Final results
    return {"Expected Reward": values, "Policy": policy}, extra_info

Task 6 Value Iteration with multiple values of gamma

Ideally this code should run as is

In [14]:
def run_value_iteration(env):
    gamma_values = np.arange(5, 100, 5)/100
    results = {}
    results, extra_info = {}, {}
    for gamma in gamma_values:
        results[gamma], extra_info[gamma] = value_iteration(env, gamma)
    return results, extra_info
  
results, extra_info = run_value_iteration(env)
In [ ]:
# # Testing
# import time
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# target_output = np.load('/content/targets/targets_2.npy')
# np.load = np_load_old
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# data = np.load('/content/inputs/inputs_2.npy')
# np.load = np_load_old
# env = TaxiEnv_HW2(data.item().get('states'), data.item().get('actions'), data.item().get('probabilities'), data.item().get('rewards'), data.item().get('initial_policy'))
# start_time = time.time()
# results, extra_info = run_value_iteration(env)
# end_time = time.time()
# print(results)
# print('time:', end_time - start_time)
# # print(target_output.item().get('value_iteration'))

Task 7 Gauss Seidel Value Iteration

Implement Gauss Seidel Value Iteration

In [15]:
# 1.4 Gauss Seidel Value Iteration
def gauss_seidel_value_iteration(taxienv, gamma):
    # A list of all the states
    # For Gauss Seidel Value Iteration - iterate through the values in the same order
    states = taxienv.possible_states

    # Initial values
    values = {s: 0 for s in states}

    # This is a dictionary of states to policies -> e.g {'A': '1', 'B': '2', 'C': '1'}
    policy = taxienv.initial_policy.copy()

    # Hints - 
    # Do not hardcode anything
    # For Gauss Seidel Value Iteration - iterate through the values in the same order as taxienv.possible_states
    # Only the final result is required for the results
    # Put any extra data in "extra_info" dictonary for any plots etc
    # Use the helper functions taxienv.ride_rewards, taxienv.ride_probabilities,  taxienv.possible_actions
    # For terminating condition use the condition exactly mentioned in the pdf

    ## Begin code here
    delta_array = np.full(len(states), 1e-8)
    temp_policy = {s:'' for s in states}
    loop_assertion = False
    while loop_assertion == False:
      delta = np.zeros(len(states))
      state_variable = 0
      for s in states:
        temp_j = values[s]
        possible_actions = taxienv.possible_actions(s)
        costs = np.zeros(len(states))
        k = 0
        for action in possible_actions:
          transition_probabilities = taxienv.ride_probabilities(s, action)
          transition_rewards = taxienv.ride_rewards(s, action)
          i1 = 0
          for s1 in states:
            costs[k] = costs[k] + transition_probabilities[i1] * (transition_rewards[i1] + gamma * values[s1])
            i1 = i1 + 1
          k = k + 1
        values[s] = np.max(costs)
        temp_policy[s] = possible_actions[np.argmax(costs)]
        delta[state_variable] = np.max([delta[state_variable], np.abs(temp_j - values[s])])
        state_variable = state_variable + 1
      less_assertion = delta < delta_array
      if np.all(less_assertion) == True:
        loop_assertion = True
        policy = temp_policy
        break
              
    # Put your extra information needed for plots etc in this dictionary
    extra_info = {}

    ## Do not edit below this line

    # Final results
    return {"Expected Reward": values, "Policy": policy}, extra_info

Task 8 Gauss Seidel Value Iteration with multiple values of gamma

Ideally this code should run as is

In [16]:
def run_gauss_seidel_value_iteration(env):
    gamma_values = np.arange(5, 100, 5)/100
    results = {}
    results, extra_info = {}, {}
    for gamma in gamma_values:
        results[gamma], extra_info[gamma] = gauss_seidel_value_iteration(env, gamma)
    return results, extra_info

results, extra_info = run_gauss_seidel_value_iteration(env)
In [ ]:
# # Testing
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# target_output = np.load('/content/targets/targets_2.npy')
# np.load = np_load_old
# np_load_old = np.load
# # # modify the default parameters of np.load
# np.load = lambda *a,**k: np_load_old(*a, allow_pickle=True, **k)
# data = np.load('/content/inputs/inputs_2.npy')
# np.load = np_load_old
# env = TaxiEnv_HW2(data.item().get('states'), data.item().get('actions'), data.item().get('probabilities'), data.item().get('rewards'), data.item().get('initial_policy'))
# start_time = time.time()
# results, extra_info = run_gauss_seidel_value_iteration(env)
# end_time = time.time()
# print(results)
# print('time:',end_time - start_time)
# print(target_output.item().get('gauss_seidel_iteration'))
In [ ]:

Generate Results ✅

In [17]:
# Do not edit this cell
def get_results(kwargs):

    taxienv = TaxiEnv_HW2(**kwargs)

    policy_iteration_results = run_policy_iteration(taxienv)[0]
    modified_policy_iteration_results = run_modified_policy_iteration(taxienv)[0]
    value_iteration_results = run_value_iteration(taxienv)[0]
    gs_vi_results = run_gauss_seidel_value_iteration(taxienv)[0]

    final_results = {}
    final_results["policy_iteration"] = policy_iteration_results
    final_results["modifed_policy_iteration"] = modified_policy_iteration_results
    final_results["value_iteration"] = value_iteration_results
    final_results["gauss_seidel_iteration"] = gs_vi_results

    return final_results
In [18]:
# Do not edit this cell, generate results with it as is
if not os.path.exists(AICROWD_RESULTS_DIR):
    os.mkdir(AICROWD_RESULTS_DIR)

for params_file in os.listdir(DATASET_DIR):
  kwargs = np.load(os.path.join(DATASET_DIR, params_file), allow_pickle=True).item()
  results = get_results(kwargs)
  idx = params_file.split('_')[-1][:-4]
  np.save(os.path.join(AICROWD_RESULTS_DIR, 'results_' + idx), results)

Check your local score

This score is not your final score, and it doesn't use the marks weightages. This is only for your reference of how arrays are matched and with what tolerance.

In [19]:
# Check your score on the given test cases (There are more private test cases not provided)
target_folder = 'targets'
result_folder = AICROWD_RESULTS_DIR

def check_algo_match(results, targets):
    param_matches = []
    for k in results:
        param_results = results[k]
        param_targets = targets[k]
        policy_match = param_results['Policy'] == param_targets['Policy']
        rv = [v for k, v in param_results['Expected Reward'].items()]
        tv = [v for k, v in param_targets['Expected Reward'].items()]
        rewards_match = np.allclose(rv, tv, rtol=3)
        equal = rewards_match and policy_match
        param_matches.append(equal)
    return np.mean(param_matches)

def check_score(target_folder, result_folder):
    match = []
    for out_file in os.listdir(result_folder):
        res_file = os.path.join(result_folder, out_file)
        results = np.load(res_file, allow_pickle=True).item()
        idx = out_file.split('_')[-1][:-4]  # Extract the file number
        target_file = os.path.join(target_folder, f"targets_{idx}.npy")
        targets = np.load(target_file, allow_pickle=True).item()
        algo_match = []
        for k in targets:
            algo_results = results[k]
            algo_targets = targets[k]
            algo_match.append(check_algo_match(algo_results, algo_targets))
        match.append(np.mean(algo_match))
    return np.mean(match)

if os.path.exists(target_folder):
    print("Shared data Score (normalized to 1):", check_score(target_folder, result_folder))
Shared data Score (normalized to 1): 1.0

Visualize results of Policy Iteration with multiple values of gamma

Add code to visualize the results

In [33]:
# ## Visualize policy iteration with multiple values of gamma
# base_kwargs = check_taxienv()
# env = TaxiEnv_HW2(**base_kwargs)
# policy_iteration_results = run_policy_iteration(env)[0]
# print(policy_iteration_results)
# gamma_values = []
# state_A_values = []
# state_B_values = []
# state_C_values = []
# for dict_1 in policy_iteration_results:
#   #print(dict_1, policy_iteration_results[dict_1]['Expected Reward'])
#   gamma_values.append(dict_1)
#   for state in env.possible_states:
#     if state == 'A':
#       state_A_values.append(policy_iteration_results[dict_1]['Expected Reward'][state])
#     elif state == 'B':
#       state_B_values.append(policy_iteration_results[dict_1]['Expected Reward'][state])
#     else:
#       state_C_values.append(policy_iteration_results[dict_1]['Expected Reward'][state])
# print(gamma_values)
# print(state_A_values)

# plt.plot(gamma_values, state_A_values)
# plt.plot(gamma_values, state_B_values)
# plt.plot(gamma_values, state_C_values)
# plt.legend(["'A'","'B'","'C'"])
# plt.xlabel('gamma_values')
# plt.ylabel('expected reward')
All possible states ['A', 'B', 'C']
All possible actions from state B ['1', '2']
Ride probabilities from state A with action 2 [0.0625 0.75   0.1875]
Ride rewards from state C with action 3 [4 0 8]
{0.05: {'Expected Reward': {'A': 8.511527294546923, 'B': 16.400259909029575, 'C': 7.4988690667106095}, 'Policy': {'A': '1', 'B': '1', 'C': '1'}}, 0.1: {'Expected Reward': {'A': 9.076506149392834, 'B': 16.85636856362452, 'C': 8.050865123980312}, 'Policy': {'A': '1', 'B': '1', 'C': '1'}}, 0.15: {'Expected Reward': {'A': 9.708121492285777, 'B': 17.464503041460713, 'C': 8.669160453984542}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.2: {'Expected Reward': {'A': 10.437030074788021, 'B': 18.482142856612846, 'C': 9.384398496135692}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.25: {'Expected Reward': {'A': 11.274074073456447, 'B': 19.629629628834554, 'C': 10.207407407209406}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.3: {'Expected Reward': {'A': 12.243837242843748, 'B': 20.934065932820292, 'C': 11.162756162382273}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.35: {'Expected Reward': {'A': 13.378714434389575, 'B': 22.43076922849439, 'C': 12.282824024490601}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.4: {'Expected Reward': {'A': 14.722222218062745, 'B': 24.16666666169111, 'C': 13.611111109108625}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.45: {'Expected Reward': {'A': 16.334131265172196, 'B': 26.205533591380167, 'C': 15.20737070397659}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.5: {'Expected Reward': {'A': 18.29870129504681, 'B': 28.636363632167495, 'C': 17.155844153726775}, 'Policy': {'A': '1', 'B': '2', 'C': '1'}}, 0.55: {'Expected Reward': {'A': 20.789988634661945, 'B': 31.607396695666196, 'C': 19.830725213847188}, 'Policy': {'A': '1', 'B': '2', 'C': '2'}}, 0.6: {'Expected Reward': {'A': 24.025686438308284, 'B': 35.32772363825764, 'C': 23.458813102392277}, 'Policy': {'A': '1', 'B': '2', 'C': '2'}}, 0.65: {'Expected Reward': {'A': 28.276692060109887, 'B': 40.096280573890574, 'C': 28.129978785480258}, 'Policy': {'A': '1', 'B': '2', 'C': '2'}}, 0.7: {'Expected Reward': {'A': 34.06193076627213, 'B': 46.43541615340033, 'C': 34.36604101044906}, 'Policy': {'A': '1', 'B': '2', 'C': '2'}}, 0.75: {'Expected Reward': {'A': 42.31741138120016, 'B': 55.28505390685269, 'C': 43.10631739475281}, 'Policy': {'A': '1', 'B': '2', 'C': '2'}}, 0.8: {'Expected Reward': {'A': 55.079365046134924, 'B': 68.55820102458796, 'C': 56.26984124288862}, 'Policy': {'A': '2', 'B': '2', 'C': '2'}}, 0.85: {'Expected Reward': {'A': 77.24651210119615, 'B': 90.81170062152418, 'C': 78.43345572723322}, 'Policy': {'A': '2', 'B': '2', 'C': '2'}}, 0.9: {'Expected Reward': {'A': 121.65347105207364, 'B': 135.30627545205246, 'C': 122.83690301136426}, 'Policy': {'A': '2', 'B': '2', 'C': '2'}}, 0.95: {'Expected Reward': {'A': 255.02290826558783, 'B': 268.7646183427653, 'C': 256.2028492762011}, 'Policy': {'A': '2', 'B': '2', 'C': '2'}}}
[0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95]
[8.511527294546923, 9.076506149392834, 9.708121492285777, 10.437030074788021, 11.274074073456447, 12.243837242843748, 13.378714434389575, 14.722222218062745, 16.334131265172196, 18.29870129504681, 20.789988634661945, 24.025686438308284, 28.276692060109887, 34.06193076627213, 42.31741138120016, 55.079365046134924, 77.24651210119615, 121.65347105207364, 255.02290826558783]
Out[33]:
Text(0, 0.5, 'expected reward')

Subjective questions

1.a How are values of $\gamma$ affecting results of policy iteration

Answer:

As the value of gamma increases, a dependence on the next stage costs effects and the cost value increases. For lesser values of gamma, the cost depends more on the current single stage cost and decides the best policy that gives the best single stage cost at the taken instant.

1.b For modified policy itetaration, do you find any improvement if you choose m=10.

Answer:

Yes. The modified policy iteration converges much faster compared to the original policy iteration algorithm given in algorithm 3.

1.c Compare and contrast the behavior of Value Iteration and Gauss Seidel Value Iteraton

Answer:

In the value iteration algorithm, the update to the cost values happen only after the computation of all the cost variables of the states. (like the update of cost variables lines 8-9 in algorithm 1 happens only after the computation of all the cost variables (lines 4-7 in algorithm 1)). And, for computation of the cost at current step, we use the cost values from the previous iteration. In contrast, the guass siedal value iteration updates the cost of a state at every iteration of a state irrespective of the previous state value computation. Hence, for any state, the guass siedal value iteration uses the present updated cost values for the updated states and also uses the costs from previous iteration for the non-updated states. (evident from lines 4 - 8 in algorithm 2).

However, the guass-siedal value iteration is proved to converge at the same rate as the value iteration except that the computation complexity, computation time and memroy usage are reduced in guass-siedal value iteration.

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