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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Crossentropy method
This notebook will teach you to solve reinforcement learning problems with crossentropy method. We'll follow-up by scaling everything up and using neural network policy.
Step1:  Create stochastic policy
This time our policy should be a probability distribution.
policy[s,a] = P(take action a | in state s)
Since we still use integer state and action representations, you can use a 2-dimensional array to represent the policy.
Please initialize policy uniformly, that is, probabililities of all actions should be equal.
Step3:  Play the game
Just like before, but we also record all states and actions we took.
Step6:  Crossentropy method steps
Step8:  Training loop
Generate sessions, select N best and fit to those.
Step9:  Reflecting on results
You may have noticed that the taxi problem quickly converges from <-1000 to a near-optimal score and then descends back into -50/-100. This is in part because the environment has some innate randomness. Namely, the starting points of passenger/driver change from episode to episode.
In case CEM failed to learn how to win from one distinct starting point, it will simply discard it because no sessions from that starting point will make it into the "elites".
To mitigate that problem, you can either reduce the threshold for elite sessions (duct tape way) or  change the way you evaluate strategy (theoretically correct way). You can first sample an action for every possible state and then evaluate this choice of actions by running several games and averaging rewards.
Submit to coursera
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Python Code:
# In Google Colab, uncomment this:
# !wget https://bit.ly/2FMJP5K -O setup.py && bash setup.py
# XVFB will be launched if you run on a server
import os
if type(os.environ.get("DISPLAY")) is not str or len(os.environ.get("DISPLAY")) == 0:
    !bash ../xvfb start
    os.environ['DISPLAY'] = ':1'
import gym
import numpy as np
import pandas as pd
env = gym.make("Taxi-v2")
env.reset()
env.render()
n_states = env.observation_space.n
n_actions = env.action_space.n
print("n_states=%i, n_actions=%i" % (n_states, n_actions))
Explanation: Crossentropy method
This notebook will teach you to solve reinforcement learning problems with crossentropy method. We'll follow-up by scaling everything up and using neural network policy.
End of explanation
policy = np.ones(shape=(n_states, n_actions)) * 1 / n_actions
assert type(policy) in (np.ndarray, np.matrix)
assert np.allclose(policy, 1./n_actions)
assert np.allclose(np.sum(policy, axis=1), 1)
Explanation: Create stochastic policy
This time our policy should be a probability distribution.
policy[s,a] = P(take action a | in state s)
Since we still use integer state and action representations, you can use a 2-dimensional array to represent the policy.
Please initialize policy uniformly, that is, probabililities of all actions should be equal.
End of explanation
def generate_session(policy, t_max=10**4):
    
    Play game until end or for t_max ticks.
    :param policy: an array of shape [n_states,n_actions] with action probabilities
    :returns: list of states, list of actions and sum of rewards
    
    states, actions = [], []
    total_reward = 0.
    s = env.reset()
    def sample_action(policy, s):
        action_p = policy[s, :].reshape(-1,)
        #highest_p_actions = np.argwhere(action_p == np.amax(action_p)).reshape(-1,)
        #non_zero_p_actions = np.argwhere(action_p > 0).reshape(-1,)
        #random_choice = np.random.choice(highest_p_actions)
        #random_choice = np.random.choice(non_zero_p_actions)
        random_choice = np.random.choice(np.arange(len(action_p)), p=action_p)
        return random_choice
        
    for t in range(t_max):
        a = sample_action(policy, s) #<sample action from policy(hint: use np.random.choice) >
        new_s, r, done, info = env.step(a)
        # Record state, action and add up reward to states,actions and total_reward accordingly.
        states.append(s)
        actions.append(a)
        total_reward += r
        s = new_s
        if done:
            break
    return states, actions, total_reward
s, a, r = generate_session(policy)
assert type(s) == type(a) == list
assert len(s) == len(a)
assert type(r) in [float, np.float]
# let's see the initial reward distribution
import matplotlib.pyplot as plt
%matplotlib inline
sample_rewards = [generate_session(policy, t_max=1000)[-1] for _ in range(200)]
plt.hist(sample_rewards, bins=20)
plt.vlines([np.percentile(sample_rewards, 50)], [0], [100], label="50'th percentile", color='green')
plt.vlines([np.percentile(sample_rewards, 90)], [0], [100], label="90'th percentile", color='red')
plt.legend()
Explanation: Play the game
Just like before, but we also record all states and actions we took.
End of explanation
def select_elites(states_batch, actions_batch, rewards_batch, percentile=50):
    
    Select states and actions from games that have rewards >= percentile
    :param states_batch: list of lists of states, states_batch[session_i][t]
    :param actions_batch: list of lists of actions, actions_batch[session_i][t]
    :param rewards_batch: list of rewards, rewards_batch[session_i]
    :returns: elite_states,elite_actions, both 1D lists of states and respective actions from elite sessions
    Please return elite states and actions in their original order 
    [i.e. sorted by session number and timestep within session]
    If you are confused, see examples below. Please don't assume that states are integers
    (they will become different later).
    
    #<Compute minimum reward for elite sessions. Hint: use np.percentile >
    reward_threshold = np.percentile(rewards_batch, percentile)
    #elite_states = <your code here >
    #elite_actions = <your code here >
    elite_states = []
    elite_actions = []
    for i, reward in enumerate(rewards_batch):
        if reward >= reward_threshold:
            elite_states = elite_states + states_batch[i]
            elite_actions = elite_actions + actions_batch[i]
    return elite_states, elite_actions
states_batch = [
    [1, 2, 3],     # game1
    [4, 2, 0, 2],  # game2
    [3, 1],        # game3
]
actions_batch = [
    [0, 2, 4],     # game1
    [3, 2, 0, 1],  # game2
    [3, 3],        # game3
]
rewards_batch = [
    3,  # game1
    4,  # game2
    5,  # game3
]
test_result_0 = select_elites(
    states_batch, actions_batch, rewards_batch, percentile=0)
test_result_40 = select_elites(
    states_batch, actions_batch, rewards_batch, percentile=30)
test_result_90 = select_elites(
    states_batch, actions_batch, rewards_batch, percentile=90)
test_result_100 = select_elites(
    states_batch, actions_batch, rewards_batch, percentile=100)
assert np.all(test_result_0[0] == [1, 2, 3, 4, 2, 0, 2, 3, 1])  \
    and np.all(test_result_0[1] == [0, 2, 4, 3, 2, 0, 1, 3, 3]),\
    "For percentile 0 you should return all states and actions in chronological order"
assert np.all(test_result_40[0] == [4, 2, 0, 2, 3, 1]) and \
    np.all(test_result_40[1] == [3, 2, 0, 1, 3, 3]),\
    "For percentile 30 you should only select states/actions from two first"
assert np.all(test_result_90[0] == [3, 1]) and \
    np.all(test_result_90[1] == [3, 3]),\
    "For percentile 90 you should only select states/actions from one game"
assert np.all(test_result_100[0] == [3, 1]) and\
    np.all(test_result_100[1] == [3, 3]),\
    "Please make sure you use >=, not >. Also double-check how you compute percentile."
print("Ok!")
def update_policy(elite_states, elite_actions):
    
    Given old policy and a list of elite states/actions from select_elites,
    return new updated policy where each action probability is proportional to
    policy[s_i,a_i] ~ #[occurences of si and ai in elite states/actions]
    Don't forget to normalize policy to get valid probabilities and handle 0/0 case.
    In case you never visited a state, set probabilities for all actions to 1./n_actions
    :param elite_states: 1D list of states from elite sessions
    :param elite_actions: 1D list of actions from elite sessions
    
    new_policy = np.zeros([n_states, n_actions])
    
    #<Your code here: update probabilities for actions given elite states & actions >
    # Don't forget to set 1/n_actions for all actions in unvisited states.
    for state, action in zip(elite_states, elite_actions):
        new_policy[state, action] = new_policy[state, action] + 1
    for state in range(n_states):
        s = np.sum(new_policy[state, :])
        if s == 0:
            new_policy[state, :] = 1. / n_actions
        else:
            new_policy[state, :] = new_policy[state, :] / s
    
    return new_policy
elite_states = [1, 2, 3, 4, 2, 0, 2, 3, 1]
elite_actions = [0, 2, 4, 3, 2, 0, 1, 3, 3]
new_policy = update_policy(elite_states, elite_actions)
assert np.isfinite(new_policy).all(
), "Your new policy contains NaNs or +-inf. Make sure you don't divide by zero."
assert np.all(
    new_policy >= 0), "Your new policy can't have negative action probabilities"
assert np.allclose(new_policy.sum(
    axis=-1), 1), "Your new policy should be a valid probability distribution over actions"
reference_answer = np.array([
    [1.,  0.,  0.,  0.,  0.],
    [0.5,  0.,  0.,  0.5,  0.],
    [0.,  0.33333333,  0.66666667,  0.,  0.],
    [0.,  0.,  0.,  0.5,  0.5]])
assert np.allclose(new_policy[:4, :5], reference_answer)
print("Ok!")
Explanation: Crossentropy method steps
End of explanation
from IPython.display import clear_output
def show_progress(rewards_batch, log, percentile, reward_range=[-990, +10]):
    
    A convenience function that displays training progress. 
    No cool math here, just charts.
    
    mean_reward = np.mean(rewards_batch)
    threshold = np.percentile(rewards_batch, percentile)
    log.append([mean_reward, threshold])
    clear_output(True)
    print("mean reward = %.3f, threshold=%.3f" % (mean_reward, threshold))
    plt.figure(figsize=[8, 4])
    plt.subplot(1, 2, 1)
    plt.plot(list(zip(*log))[0], label='Mean rewards')
    plt.plot(list(zip(*log))[1], label='Reward thresholds')
    plt.legend()
    plt.grid()
    plt.subplot(1, 2, 2)
    plt.hist(rewards_batch, range=reward_range)
    plt.vlines([np.percentile(rewards_batch, percentile)],
               [0], [100], label="percentile", color='red')
    plt.legend()
    plt.grid()
    plt.show()
# reset policy just in case
policy = np.ones([n_states, n_actions]) / n_actions
n_sessions = 250  # sample this many sessions
percentile = 30  # take this percent of session with highest rewards
learning_rate = 0.5  # add this thing to all counts for stability
log = []
for i in range(100):
    %time sessions = [generate_session(policy) for x in range(n_sessions)] #[ < generate a list of n_sessions new sessions > ]
    
    states_batch, actions_batch, rewards_batch = zip(*sessions)
    elite_states, elite_actions = select_elites(states_batch, actions_batch, rewards_batch, percentile=percentile) #<select elite states/actions >
    new_policy = update_policy(elite_states, elite_actions) #<compute new policy >
    policy = learning_rate * new_policy + (1 - learning_rate) * policy
    # display results on chart
    show_progress(rewards_batch, log, percentile)
Explanation: Training loop
Generate sessions, select N best and fit to those.
End of explanation
from submit import submit_taxi
submit_taxi(generate_session, policy, 'tonatiuh_rangel@hotmail.com', '7uvgN7bBzpJzVw9f')
Explanation: Reflecting on results
You may have noticed that the taxi problem quickly converges from <-1000 to a near-optimal score and then descends back into -50/-100. This is in part because the environment has some innate randomness. Namely, the starting points of passenger/driver change from episode to episode.
In case CEM failed to learn how to win from one distinct starting point, it will simply discard it because no sessions from that starting point will make it into the "elites".
To mitigate that problem, you can either reduce the threshold for elite sessions (duct tape way) or  change the way you evaluate strategy (theoretically correct way). You can first sample an action for every possible state and then evaluate this choice of actions by running several games and averaging rewards.
Submit to coursera
End of explanation
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