!pip install gym !pip install git+https://github.com/mimoralea/gym-walk#egg=gym-walk !pip install git+https://github.com/mimoralea/gym-aima#egg=gym-aima !pip install ipywidgets from ipywidgets import IntProgress

import warnings ; warnings.filterwarnings('ignore') import itertools import gym, gym_walk, gym_aima import numpy as np from tabulate import tabulate from pprint import pprint from tqdm import tqdm_notebook as tqdm from itertools import cycle, count import random import matplotlib import matplotlib.pyplot as plt import matplotlib.pylab as pylab SEEDS = (12, 34, 56, 78, 90) %matplotlib inline

plt.style.use('fivethirtyeight') params = { 'figure.figsize': (15, 8), 'font.size': 24, 'legend.fontsize': 20, 'axes.titlesize': 28, 'axes.labelsize': 24, 'xtick.labelsize': 20, 'ytick.labelsize': 20 } pylab.rcParams.update(params) np.set_printoptions(suppress=True)

def value_iteration(P, gamma=1.0, theta=1e-10): V = np.zeros(len(P), dtype=np.float64) while True: Q = np.zeros((len(P), len(P[0])), dtype=np.float64) for s in range(len(P)): for a in range(len(P[s])): for prob, next_state, reward, done in P[s][a]: Q[s][a] += prob * (reward + gamma * V[next_state] * (not done)) if np.max(np.abs(V - np.max(Q, axis=1))) < theta: break V = np.max(Q, axis=1) pi = lambda s: {s:a for s, a in enumerate(np.argmax(Q, axis=1))}[s] return Q, V, pi

def print_policy(pi, P, action_symbols=('<', 'v', '>', '^'), n_cols=4, title='Policy:'): print(title) arrs = {k:v for k,v in enumerate(action_symbols)} for s in range(len(P)): a = pi(s) print("| ", end="") if np.all([done for action in P[s].values() for _, _, _, done in action]): print("".rjust(9), end=" ") else: print(str(s).zfill(2), arrs[a].rjust(6), end=" ") if (s + 1) % n_cols == 0: print("|")

def print_state_value_function(V, P, n_cols=4, prec=3, title='State-value function:'): print(title) for s in range(len(P)): v = V[s] print("| ", end="") if np.all([done for action in P[s].values() for _, _, _, done in action]): print("".rjust(9), end=" ") else: print(str(s).zfill(2), '{}'.format(np.round(v, prec)).rjust(6), end=" ") if (s + 1) % n_cols == 0: print("|")

def print_action_value_function(Q, optimal_Q=None, action_symbols=('<', '>'), prec=3, title='Action-value function:'): vf_types=('',) if optimal_Q is None else ('', '*', 'err') headers = ['s',] + [' '.join(i) for i in list(itertools.product(vf_types, action_symbols))] print(title) states = np.arange(len(Q))[..., np.newaxis] arr = np.hstack((states, np.round(Q, prec))) if not (optimal_Q is None): arr = np.hstack((arr, np.round(optimal_Q, prec), np.round(optimal_Q-Q, prec))) print(tabulate(arr, headers, tablefmt="fancy_grid"))

def get_policy_metrics(env, gamma, pi, goal_state, optimal_Q, n_episodes=100, max_steps=200): random.seed(123); np.random.seed(123) ; env.seed(123) reached_goal, episode_reward, episode_regret = [], [], [] for _ in range(n_episodes): state, done, steps = env.reset(), False, 0 episode_reward.append(0.0) episode_regret.append(0.0) while not done and steps < max_steps: action = pi(state) regret = np.max(optimal_Q[state]) - optimal_Q[state][action] episode_regret[-1] += regret state, reward, done, _ = env.step(action) episode_reward[-1] += (gamma**steps * reward) steps += 1 reached_goal.append(state == goal_state) results = np.array((np.sum(reached_goal)/len(reached_goal)*100, np.mean(episode_reward), np.mean(episode_regret))) return results

def get_metrics_from_tracks(env, gamma, goal_state, optimal_Q, pi_track, coverage=0.1): total_samples = len(pi_track) n_samples = int(total_samples * coverage) samples_e = np.linspace(0, total_samples, n_samples, endpoint=True, dtype=np.int) metrics = [] for e, pi in enumerate(tqdm(pi_track)): if e in samples_e: metrics.append(get_policy_metrics( env, gamma=gamma, pi=lambda s: pi[s], goal_state=goal_state, optimal_Q=optimal_Q)) else: metrics.append(metrics[-1]) metrics = np.array(metrics) success_rate_ma, mean_return_ma, mean_regret_ma = np.apply_along_axis(moving_average, axis=0, arr=metrics).T return success_rate_ma, mean_return_ma, mean_regret_ma

def rmse(x, y, dp=4): return np.round(np.sqrt(np.mean((x - y)**2)), dp)

def moving_average(a, n=100) : ret = np.cumsum(a, dtype=float) ret[n:] = ret[n:] - ret[:-n] return ret[n - 1:] / n

def plot_value_function(title, V_track, V_true=None, log=False, limit_value=0.05, limit_items=5): np.random.seed(123) per_col = 25 linecycler = cycle(["-","--",":","-."]) legends = [] valid_values = np.argwhere(V_track[-1] > limit_value).squeeze() items_idxs = np.random.choice(valid_values, min(len(valid_values), limit_items), replace=False) # draw the true values first if V_true is not None: for i, state in enumerate(V_track.T): if i not in items_idxs: continue if state[-1] < limit_value: continue label = 'v*({})'.format(i) plt.axhline(y=V_true[i], color='k', linestyle='-', linewidth=1) plt.text(int(len(V_track)*1.02), V_true[i]+.01, label) # then the estimates for i, state in enumerate(V_track.T): if i not in items_idxs: continue if state[-1] < limit_value: continue line_type = next(linecycler) label = 'V({})'.format(i) p, = plt.plot(state, line_type, label=label, linewidth=3) legends.append(p) legends.reverse() ls = [] for loc, idx in enumerate(range(0, len(legends), per_col)): subset = legends[idx:idx+per_col] l = plt.legend(subset, [p.get_label() for p in subset], loc='center right', bbox_to_anchor=(1.25, 0.5)) ls.append(l) [plt.gca().add_artist(l) for l in ls[:-1]] if log: plt.xscale('log') plt.title(title) plt.ylabel('State-value function') plt.xlabel('Episodes (log scale)' if log else 'Episodes') plt.show()

def decay_schedule(init_value, min_value, decay_ratio, max_steps, log_start=-2, log_base=10): decay_steps = int(max_steps * decay_ratio) rem_steps = max_steps - decay_steps values = np.logspace(log_start, 0, decay_steps, base=log_base, endpoint=True)[::-1] values = (values - values.min()) / (values.max() - values.min()) values = (init_value - min_value) * values + min_value values = np.pad(values, (0, rem_steps), 'edge') return values

env = gym.make('SlipperyWalkSeven-v0') init_state = env.reset() goal_state = 8 gamma = 0.99 n_episodes = 3000 P = env.env.P n_cols, svf_prec, err_prec, avf_prec=9, 4, 2, 3 action_symbols=('<', '>') limit_items, limit_value = 5, 0.0 cu_limit_items, cu_limit_value, cu_episodes = 10, 0.0, 100

plt.plot(decay_schedule(0.5, 0.01, 0.5, n_episodes), '-', linewidth=2, label='Alpha schedule') plt.plot(decay_schedule(1.0, 0.1, 0.9, n_episodes), ':', linewidth=2, label='Epsilon schedule') plt.legend(loc=1, ncol=1) plt.title('Alpha and epsilon schedules') plt.xlabel('Episodes') plt.ylabel('Hyperparameter values') plt.xticks(rotation=45) plt.show()

optimal_Q, optimal_V, optimal_pi = value_iteration(P, gamma=gamma) print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print() print_action_value_function(optimal_Q, None, action_symbols=action_symbols, prec=avf_prec, title='Optimal action-value function:') print() print_policy(optimal_pi, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_op, mean_return_op, mean_regret_op = get_policy_metrics( env, gamma=gamma, pi=optimal_pi, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_op, mean_return_op, mean_regret_op))

def generate_trajectory(select_action, Q, epsilon, env, max_steps=200): done, trajectory = False, [] while not done: state = env.reset() for t in count(): action = select_action(state, Q, epsilon) next_state, reward, done, _ = env.step(action) experience = (state, action, reward, next_state, done) trajectory.append(experience) if done: break if t >= max_steps - 1: trajectory = [] break state = next_state return np.array(trajectory, np.object)

def mc_control(env, gamma=1.0, init_alpha=0.5, min_alpha=0.01, alpha_decay_ratio=0.5, init_epsilon=1.0, min_epsilon=0.1, epsilon_decay_ratio=0.9, n_episodes=3000, max_steps=200, first_visit=True): nS, nA = env.observation_space.n, env.action_space.n discounts = np.logspace(0, max_steps, num=max_steps, base=gamma, endpoint=False) alphas = decay_schedule(init_alpha, min_alpha, alpha_decay_ratio, n_episodes) epsilons = decay_schedule(init_epsilon, min_epsilon, epsilon_decay_ratio, n_episodes) pi_track = [] Q = np.zeros((nS, nA), dtype=np.float64) Q_track = np.zeros((n_episodes, nS, nA), dtype=np.float64) select_action = lambda state, Q, epsilon: np.argmax(Q[state]) \ if np.random.random() > epsilon \ else np.random.randint(len(Q[state])) for e in tqdm(range(n_episodes), leave=False): trajectory = generate_trajectory(select_action, Q, epsilons[e], env, max_steps) visited = np.zeros((nS, nA), dtype=np.bool) for t, (state, action, reward, _, _) in enumerate(trajectory): if visited[state][action] and first_visit: continue visited[state][action] = True n_steps = len(trajectory[t:]) G = np.sum(discounts[:n_steps] * trajectory[t:, 2]) Q[state][action] = Q[state][action] + alphas[e] * (G - Q[state][action]) Q_track[e] = Q pi_track.append(np.argmax(Q, axis=1)) V = np.max(Q, axis=1) pi = lambda s: {s:a for s, a in enumerate(np.argmax(Q, axis=1))}[s] return Q, V, pi, Q_track, pi_track

Q_mcs, V_mcs, Q_track_mcs = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_mc, V_mc, pi_mc, Q_track_mc, pi_track_mc = mc_control(env, gamma=gamma, n_episodes=n_episodes) Q_mcs.append(Q_mc) ; V_mcs.append(V_mc) ; Q_track_mcs.append(Q_track_mc) Q_mc, V_mc, Q_track_mc = np.mean(Q_mcs, axis=0), np.mean(V_mcs, axis=0), np.mean(Q_track_mcs, axis=0) del Q_mcs ; del V_mcs ; del Q_track_mcs

print_state_value_function(V_mc, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by FVMC:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_mc - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_mc, optimal_V))) print() print_action_value_function(Q_mc, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='FVMC action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_mc, optimal_Q))) print() print_policy(pi_mc, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_mc, mean_return_mc, mean_regret_mc = get_policy_metrics( env, gamma=gamma, pi=pi_mc, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_mc, mean_return_mc, mean_regret_mc))

def sarsa(env, gamma=1.0, init_alpha=0.5, min_alpha=0.01, alpha_decay_ratio=0.5, init_epsilon=1.0, min_epsilon=0.1, epsilon_decay_ratio=0.9, n_episodes=3000): nS, nA = env.observation_space.n, env.action_space.n pi_track = [] Q = np.zeros((nS, nA), dtype=np.float64) Q_track = np.zeros((n_episodes, nS, nA), dtype=np.float64) select_action = lambda state, Q, epsilon: np.argmax(Q[state]) \ if np.random.random() > epsilon \ else np.random.randint(len(Q[state])) alphas = decay_schedule(init_alpha, min_alpha, alpha_decay_ratio, n_episodes) epsilons = decay_schedule(init_epsilon, min_epsilon, epsilon_decay_ratio, n_episodes) for e in tqdm(range(n_episodes), leave=False): state, done = env.reset(), False action = select_action(state, Q, epsilons[e]) while not done: next_state, reward, done, _ = env.step(action) next_action = select_action(next_state, Q, epsilons[e]) td_target = reward + gamma * Q[next_state][next_action] * (not done) td_error = td_target - Q[state][action] Q[state][action] = Q[state][action] + alphas[e] * td_error state, action = next_state, next_action Q_track[e] = Q pi_track.append(np.argmax(Q, axis=1)) V = np.max(Q, axis=1) pi = lambda s: {s:a for s, a in enumerate(np.argmax(Q, axis=1))}[s] return Q, V, pi, Q_track, pi_track

Q_sarsas, V_sarsas, Q_track_sarsas = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_sarsa, V_sarsa, pi_sarsa, Q_track_sarsa, pi_track_sarsa = sarsa(env, gamma=gamma, n_episodes=n_episodes) Q_sarsas.append(Q_sarsa) ; V_sarsas.append(V_sarsa) ; Q_track_sarsas.append(Q_track_sarsa) Q_sarsa = np.mean(Q_sarsas, axis=0) V_sarsa = np.mean(V_sarsas, axis=0) Q_track_sarsa = np.mean(Q_track_sarsas, axis=0) del Q_sarsas ; del V_sarsas ; del Q_track_sarsas

print_state_value_function(V_sarsa, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by Sarsa:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_sarsa - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_sarsa, optimal_V))) print() print_action_value_function(Q_sarsa, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='Sarsa action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_sarsa, optimal_Q))) print() print_policy(pi_sarsa, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_sarsa, mean_return_sarsa, mean_regret_sarsa = get_policy_metrics( env, gamma=gamma, pi=pi_sarsa, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_sarsa, mean_return_sarsa, mean_regret_sarsa))

def q_learning(env, gamma=1.0, init_alpha=0.5, min_alpha=0.01, alpha_decay_ratio=0.5, init_epsilon=1.0, min_epsilon=0.1, epsilon_decay_ratio=0.9, n_episodes=3000): nS, nA = env.observation_space.n, env.action_space.n pi_track = [] Q = np.zeros((nS, nA), dtype=np.float64) Q_track = np.zeros((n_episodes, nS, nA), dtype=np.float64) select_action = lambda state, Q, epsilon: np.argmax(Q[state]) \ if np.random.random() > epsilon \ else np.random.randint(len(Q[state])) alphas = decay_schedule(init_alpha, min_alpha, alpha_decay_ratio, n_episodes) epsilons = decay_schedule(init_epsilon, min_epsilon, epsilon_decay_ratio, n_episodes) for e in tqdm(range(n_episodes), leave=False): state, done = env.reset(), False while not done: action = select_action(state, Q, epsilons[e]) next_state, reward, done, _ = env.step(action) td_target = reward + gamma * Q[next_state].max() * (not done) td_error = td_target - Q[state][action] Q[state][action] = Q[state][action] + alphas[e] * td_error state = next_state Q_track[e] = Q pi_track.append(np.argmax(Q, axis=1)) V = np.max(Q, axis=1) pi = lambda s: {s:a for s, a in enumerate(np.argmax(Q, axis=1))}[s] return Q, V, pi, Q_track, pi_track

Q_qls, V_qls, Q_track_qls = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_ql, V_ql, pi_ql, Q_track_ql, pi_track_ql = q_learning(env, gamma=gamma, n_episodes=n_episodes) Q_qls.append(Q_ql) ; V_qls.append(V_ql) ; Q_track_qls.append(Q_track_ql) Q_ql = np.mean(Q_qls, axis=0) V_ql = np.mean(V_qls, axis=0) Q_track_ql = np.mean(Q_track_qls, axis=0) del Q_qls ; del V_qls ; del Q_track_qls

print_state_value_function(V_ql, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by Q-learning:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_ql - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_ql, optimal_V))) print() print_action_value_function(Q_ql, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='Q-learning action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_ql, optimal_Q))) print() print_policy(pi_ql, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_ql, mean_return_ql, mean_regret_ql = get_policy_metrics( env, gamma=gamma, pi=pi_ql, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_ql, mean_return_ql, mean_regret_ql))

def double_q_learning(env, gamma=1.0, init_alpha=0.5, min_alpha=0.01, alpha_decay_ratio=0.5, init_epsilon=1.0, min_epsilon=0.1, epsilon_decay_ratio=0.9, n_episodes=3000): nS, nA = env.observation_space.n, env.action_space.n pi_track = [] Q1 = np.zeros((nS, nA), dtype=np.float64) Q2 = np.zeros((nS, nA), dtype=np.float64) Q_track1 = np.zeros((n_episodes, nS, nA), dtype=np.float64) Q_track2 = np.zeros((n_episodes, nS, nA), dtype=np.float64) select_action = lambda state, Q, epsilon: np.argmax(Q[state]) \ if np.random.random() > epsilon \ else np.random.randint(len(Q[state])) alphas = decay_schedule(init_alpha, min_alpha, alpha_decay_ratio, n_episodes) epsilons = decay_schedule(init_epsilon, min_epsilon, epsilon_decay_ratio, n_episodes) for e in tqdm(range(n_episodes), leave=False): state, done = env.reset(), False while not done: action = select_action(state, (Q1 + Q2)/2, epsilons[e]) next_state, reward, done, _ = env.step(action) if np.random.randint(2): argmax_Q1 = np.argmax(Q1[next_state]) td_target = reward + gamma * Q2[next_state][argmax_Q1] * (not done) td_error = td_target - Q1[state][action] Q1[state][action] = Q1[state][action] + alphas[e] * td_error else: argmax_Q2 = np.argmax(Q2[next_state]) td_target = reward + gamma * Q1[next_state][argmax_Q2] * (not done) td_error = td_target - Q2[state][action] Q2[state][action] = Q2[state][action] + alphas[e] * td_error state = next_state Q_track1[e] = Q1 Q_track2[e] = Q2 pi_track.append(np.argmax((Q1 + Q2)/2, axis=1)) Q = (Q1 + Q2)/2. V = np.max(Q, axis=1) pi = lambda s: {s:a for s, a in enumerate(np.argmax(Q, axis=1))}[s] return Q, V, pi, (Q_track1 + Q_track2)/2., pi_track

Q_dqls, V_dqls, Q_track_dqls = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_dql, V_dql, pi_dql, Q_track_dql, pi_track_dql = double_q_learning(env, gamma=gamma, n_episodes=n_episodes) Q_dqls.append(Q_dql) ; V_dqls.append(V_dql) ; Q_track_dqls.append(Q_track_dql) Q_dql, V_dql, Q_track_dql = np.mean(Q_dqls, axis=0), np.mean(V_dqls, axis=0), np.mean(Q_track_dqls, axis=0) del Q_dqls ; del V_dqls ; del Q_track_dqls

print_state_value_function(V_dql, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by Double Q-Learning:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_dql - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_dql, optimal_V))) print() print_action_value_function(Q_dql, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='Double Q-Learning action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_dql, optimal_Q))) print() print_policy(pi_dql, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_dql, mean_return_dql, mean_regret_dql = get_policy_metrics( env, gamma=gamma, pi=pi_dql, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_dql, mean_return_dql, mean_regret_dql))

plot_value_function( 'FVMC estimates through time vs. true values', np.max(Q_track_mc, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'FVMC estimates through time vs. true values (log scale)', np.max(Q_track_mc, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'FVMC estimates through time (close up)', np.max(Q_track_mc, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

plot_value_function( 'Sarsa estimates through time vs. true values', np.max(Q_track_sarsa, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'Sarsa estimates through time vs. true values (log scale)', np.max(Q_track_sarsa, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'Sarsa estimates through time (close up)', np.max(Q_track_sarsa, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

plot_value_function( 'Q-Learning estimates through time vs. true values', np.max(Q_track_ql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'Q-Learning estimates through time vs. true values (log scale)', np.max(Q_track_ql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'Q-Learning estimates through time (close up)', np.max(Q_track_ql, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

plot_value_function( 'Double Q-Learning estimates through time vs. true values', np.max(Q_track_dql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'Double Q-Learning estimates through time vs. true values (log scale)', np.max(Q_track_dql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'Double Q-Learning estimates through time (close up)', np.max(Q_track_dql, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

mc_success_rate_ma, mc_mean_return_ma, mc_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_mc)

sarsa_success_rate_ma, sarsa_mean_return_ma, sarsa_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_sarsa)

ql_success_rate_ma, ql_mean_return_ma, ql_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_ql)

dql_success_rate_ma, dql_mean_return_ma, dql_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_dql)

plt.axhline(y=success_rate_op, color='k', linestyle='-', linewidth=1) plt.text(int(len(mc_success_rate_ma)*1.02), success_rate_op*1.01, 'π*') plt.plot(mc_success_rate_ma, '-', linewidth=2, label='FVMC') plt.plot(sarsa_success_rate_ma, '--', linewidth=2, label='Sarsa') plt.plot(ql_success_rate_ma, ':', linewidth=2, label='Q-Learning') plt.plot(dql_success_rate_ma, '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=4, ncol=1) plt.title('Policy success rate (ma 100)') plt.xlabel('Episodes') plt.ylabel('Success rate %') plt.ylim(-1, 101) plt.xticks(rotation=45) plt.show()

plt.axhline(y=mean_return_op, color='k', linestyle='-', linewidth=1) plt.text(int(len(mc_mean_return_ma)*1.02), mean_return_op*1.01, 'π*') plt.plot(mc_mean_return_ma, '-', linewidth=2, label='FVMC') plt.plot(sarsa_mean_return_ma, '--', linewidth=2, label='Sarsa') plt.plot(ql_mean_return_ma, ':', linewidth=2, label='Q-Learning') plt.plot(dql_mean_return_ma, '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=4, ncol=1) plt.title('Policy episode return (ma 100)') plt.xlabel('Episodes') plt.ylabel('Return (Gt:T)') plt.xticks(rotation=45) plt.show()

plt.plot(mc_mean_regret_ma, '-', linewidth=2, label='FVMC') plt.plot(sarsa_mean_regret_ma, '--', linewidth=2, label='Sarsa') plt.plot(ql_mean_regret_ma, ':', linewidth=2, label='Q-Learning') plt.plot(dql_mean_regret_ma, '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=1, ncol=1) plt.title('Policy episode regret (ma 100)') plt.xlabel('Episodes') plt.ylabel('Regret (q* - Q)') plt.xticks(rotation=45) plt.show()

plt.axhline(y=optimal_V[init_state], color='k', linestyle='-', linewidth=1) plt.text(int(len(Q_track_mc)*1.05), optimal_V[init_state]+.01, 'v*({})'.format(init_state)) plt.plot(moving_average(np.max(Q_track_mc, axis=2).T[init_state]), '-', linewidth=2, label='FVMC') plt.plot(moving_average(np.max(Q_track_sarsa, axis=2).T[init_state]), '--', linewidth=2, label='Sarsa') plt.plot(moving_average(np.max(Q_track_ql, axis=2).T[init_state]), ':', linewidth=2, label='Q-Learning') plt.plot(moving_average(np.max(Q_track_dql, axis=2).T[init_state]), '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=4, ncol=1) plt.title('Estimated expected return (ma 100)') plt.xlabel('Episodes') plt.ylabel('Estimated value of initial state V({})'.format(init_state)) plt.xticks(rotation=45) plt.show()

plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_mc, axis=2) - optimal_V), axis=1)), '-', linewidth=2, label='FVMC') plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_sarsa, axis=2) - optimal_V), axis=1)), '--', linewidth=2, label='Sarsa') plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_ql, axis=2) - optimal_V), axis=1)), ':', linewidth=2, label='Q-Learning') plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_dql, axis=2) - optimal_V), axis=1)), '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=1, ncol=1) plt.title('State-value function estimation error (ma 100)') plt.xlabel('Episodes') plt.ylabel('Mean Absolute Error MAE(V, v*)') plt.xticks(rotation=45) plt.show()

plt.plot(moving_average(np.mean(np.abs(Q_track_mc - optimal_Q), axis=(1,2))), '-', linewidth=2, label='FVMC') plt.plot(moving_average(np.mean(np.abs(Q_track_sarsa - optimal_Q), axis=(1,2))), '--', linewidth=2, label='Sarsa') plt.plot(moving_average(np.mean(np.abs(Q_track_ql - optimal_Q), axis=(1,2))), ':', linewidth=2, label='Q-Learning') plt.plot(moving_average(np.mean(np.abs(Q_track_dql - optimal_Q), axis=(1,2))), '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=1, ncol=1) plt.title('Action-value function estimation error (ma 100)') plt.xlabel('Episodes') plt.ylabel('Mean Absolute Error MAE(Q, q*)') plt.xticks(rotation=45) plt.show()

env = gym.make('RussellNorvigGridworld-v0') init_state = env.reset() goal_state = 3 gamma = 1.0 n_episodes = 4000 P = env.env.P n_cols, svf_prec, err_prec, avf_prec=4, 4, 2, 3 action_symbols=('<', 'v', '>', '^') limit_items, limit_value = 5, 0.01 cu_limit_items, cu_limit_value, cu_episodes = 10, 0.0, 1000

plt.plot(decay_schedule(0.5, 0.01, 0.5, n_episodes), '-', linewidth=2, label='Alpha schedule') plt.plot(decay_schedule(1.0, 0.1, 0.9, n_episodes), ':', linewidth=2, label='Epsilon schedule') plt.legend(loc=1, ncol=1) plt.title('Alpha and epsilon schedules') plt.xlabel('Episodes') plt.ylabel('Hyperparameter values') plt.xticks(rotation=45) plt.show()

optimal_Q, optimal_V, optimal_pi = value_iteration(P, gamma=gamma) print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print() print_action_value_function(optimal_Q, None, action_symbols=action_symbols, prec=avf_prec, title='Optimal action-value function:') print() print_policy(optimal_pi, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_op, mean_return_op, mean_regret_op = get_policy_metrics( env, gamma=gamma, pi=optimal_pi, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_op, mean_return_op, mean_regret_op))

Q_mcs, V_mcs, Q_track_mcs = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_mc, V_mc, pi_mc, Q_track_mc, pi_track_mc = mc_control(env, gamma=gamma, n_episodes=n_episodes) Q_mcs.append(Q_mc) ; V_mcs.append(V_mc) ; Q_track_mcs.append(Q_track_mc) Q_mc, V_mc, Q_track_mc = np.mean(Q_mcs, axis=0), np.mean(V_mcs, axis=0), np.mean(Q_track_mcs, axis=0) del Q_mcs ; del V_mcs ; del Q_track_mcs

print_state_value_function(V_mc, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by FVMC:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_mc - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_mc, optimal_V))) print() print_action_value_function(Q_mc, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='FVMC action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_mc, optimal_Q))) print() print_policy(pi_mc, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_mc, mean_return_mc, mean_regret_mc = get_policy_metrics( env, gamma=gamma, pi=pi_mc, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_mc, mean_return_mc, mean_regret_mc))

Q_sarsas, V_sarsas, Q_track_sarsas = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_sarsa, V_sarsa, pi_sarsa, Q_track_sarsa, pi_track_sarsa = sarsa(env, gamma=gamma, n_episodes=n_episodes) Q_sarsas.append(Q_sarsa) ; V_sarsas.append(V_sarsa) ; Q_track_sarsas.append(Q_track_sarsa) Q_sarsa = np.mean(Q_sarsas, axis=0) V_sarsa = np.mean(V_sarsas, axis=0) Q_track_sarsa = np.mean(Q_track_sarsas, axis=0) del Q_sarsas ; del V_sarsas ; del Q_track_sarsas

print_state_value_function(V_sarsa, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by Sarsa:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_sarsa - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_sarsa, optimal_V))) print() print_action_value_function(Q_sarsa, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='Sarsa action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_sarsa, optimal_Q))) print() print_policy(pi_sarsa, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_sarsa, mean_return_sarsa, mean_regret_sarsa = get_policy_metrics( env, gamma=gamma, pi=pi_sarsa, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_sarsa, mean_return_sarsa, mean_regret_sarsa))

Q_qls, V_qls, Q_track_qls = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_ql, V_ql, pi_ql, Q_track_ql, pi_track_ql = q_learning(env, gamma=gamma, n_episodes=n_episodes) Q_qls.append(Q_ql) ; V_qls.append(V_ql) ; Q_track_qls.append(Q_track_ql) Q_ql = np.mean(Q_qls, axis=0) V_ql = np.mean(V_qls, axis=0) Q_track_ql = np.mean(Q_track_qls, axis=0) del Q_qls ; del V_qls ; del Q_track_qls

print_state_value_function(V_ql, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by Q-learning:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_ql - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_ql, optimal_V))) print() print_action_value_function(Q_ql, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='Q-learning action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_ql, optimal_Q))) print() print_policy(pi_ql, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_ql, mean_return_ql, mean_regret_ql = get_policy_metrics( env, gamma=gamma, pi=pi_ql, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_ql, mean_return_ql, mean_regret_ql))

Q_dqls, V_dqls, Q_track_dqls = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_dql, V_dql, pi_dql, Q_track_dql, pi_track_dql = double_q_learning(env, gamma=gamma, n_episodes=n_episodes) Q_dqls.append(Q_dql) ; V_dqls.append(V_dql) ; Q_track_dqls.append(Q_track_dql) Q_dql, V_dql, Q_track_dql = np.mean(Q_dqls, axis=0), np.mean(V_dqls, axis=0), np.mean(Q_track_dqls, axis=0) del Q_dqls ; del V_dqls ; del Q_track_dqls

print_state_value_function(V_dql, P, n_cols=n_cols, prec=svf_prec, title='State-value function found by Double Q-Learning:') print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print_state_value_function(V_dql - optimal_V, P, n_cols=n_cols, prec=err_prec, title='State-value function errors:') print('State-value function RMSE: {}'.format(rmse(V_dql, optimal_V))) print() print_action_value_function(Q_dql, optimal_Q, action_symbols=action_symbols, prec=avf_prec, title='Double Q-Learning action-value function:') print('Action-value function RMSE: {}'.format(rmse(Q_dql, optimal_Q))) print() print_policy(pi_dql, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_dql, mean_return_dql, mean_regret_dql = get_policy_metrics( env, gamma=gamma, pi=pi_dql, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_dql, mean_return_dql, mean_regret_dql))

plot_value_function( 'FVMC estimates through time vs. true values', np.max(Q_track_mc, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'FVMC estimates through time vs. true values (log scale)', np.max(Q_track_mc, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'FVMC estimates through time (close up)', np.max(Q_track_mc, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

plot_value_function( 'Sarsa estimates through time vs. true values', np.max(Q_track_sarsa, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'Sarsa estimates through time vs. true values (log scale)', np.max(Q_track_sarsa, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'Sarsa estimates through time (close up)', np.max(Q_track_sarsa, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

plot_value_function( 'Q-Learning estimates through time vs. true values', np.max(Q_track_ql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'Q-Learning estimates through time vs. true values (log scale)', np.max(Q_track_ql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'Q-Learning estimates through time (close up)', np.max(Q_track_ql, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

plot_value_function( 'Double Q-Learning estimates through time vs. true values', np.max(Q_track_dql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=False)

plot_value_function( 'Double Q-Learning estimates through time vs. true values (log scale)', np.max(Q_track_dql, axis=2), optimal_V, limit_items=limit_items, limit_value=limit_value, log=True)

plot_value_function( 'Double Q-Learning estimates through time (close up)', np.max(Q_track_dql, axis=2)[:cu_episodes], None, limit_items=cu_limit_items, limit_value=cu_limit_value, log=False)

mc_success_rate_ma, mc_mean_return_ma, mc_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_mc)

sarsa_success_rate_ma, sarsa_mean_return_ma, sarsa_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_sarsa)

ql_success_rate_ma, ql_mean_return_ma, ql_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_ql)

dql_success_rate_ma, dql_mean_return_ma, dql_mean_regret_ma = get_metrics_from_tracks( env, gamma, goal_state, optimal_Q, pi_track_dql)

plt.axhline(y=success_rate_op, color='k', linestyle='-', linewidth=1) plt.text(int(len(mc_success_rate_ma)*1.02), success_rate_op*1.01, 'π*') plt.plot(mc_success_rate_ma, '-', linewidth=2, label='FVMC') plt.plot(sarsa_success_rate_ma, '--', linewidth=2, label='Sarsa') plt.plot(ql_success_rate_ma, ':', linewidth=2, label='Q-Learning') plt.plot(dql_success_rate_ma, '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=4, ncol=1) plt.title('Policy success rate (ma 100)') plt.xlabel('Episodes') plt.ylabel('Success rate %') plt.ylim(-1, 101) plt.xticks(rotation=45) plt.show()

plt.axhline(y=mean_return_op, color='k', linestyle='-', linewidth=1) plt.text(int(len(mc_mean_return_ma)*1.02), mean_return_op*1.01, 'π*') plt.plot(mc_mean_return_ma, '-', linewidth=2, label='FVMC') plt.plot(sarsa_mean_return_ma, '--', linewidth=2, label='Sarsa') plt.plot(ql_mean_return_ma, ':', linewidth=2, label='Q-Learning') plt.plot(dql_mean_return_ma, '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=4, ncol=1) plt.title('Policy episode return (ma 100)') plt.xlabel('Episodes') plt.ylabel('Return (Gt:T)') plt.xticks(rotation=45) plt.show()

plt.plot(mc_mean_regret_ma, '-', linewidth=2, label='FVMC') plt.plot(sarsa_mean_regret_ma, '--', linewidth=2, label='Sarsa') plt.plot(ql_mean_regret_ma, ':', linewidth=2, label='Q-Learning') plt.plot(dql_mean_regret_ma, '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=1, ncol=1) plt.title('Policy episode regret (ma 100)') plt.xlabel('Episodes') plt.ylabel('Regret (q* - Q)') plt.xticks(rotation=45) plt.show()

plt.axhline(y=optimal_V[init_state], color='k', linestyle='-', linewidth=1) plt.text(int(len(Q_track_mc)*1.05), optimal_V[init_state]+.01, 'v*({})'.format(init_state)) plt.plot(moving_average(np.max(Q_track_mc, axis=2).T[init_state]), '-', linewidth=2, label='FVMC') plt.plot(moving_average(np.max(Q_track_sarsa, axis=2).T[init_state]), '--', linewidth=2, label='Sarsa') plt.plot(moving_average(np.max(Q_track_ql, axis=2).T[init_state]), ':', linewidth=2, label='Q-Learning') plt.plot(moving_average(np.max(Q_track_dql, axis=2).T[init_state]), '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=4, ncol=1) plt.title('Estimated expected return (ma 100)') plt.xlabel('Episodes') plt.ylabel('Estimated value of initial state V({})'.format(init_state)) plt.xticks(rotation=45) plt.show()

plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_mc, axis=2) - optimal_V), axis=1)), '-', linewidth=2, label='FVMC') plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_sarsa, axis=2) - optimal_V), axis=1)), '--', linewidth=2, label='Sarsa') plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_ql, axis=2) - optimal_V), axis=1)), ':', linewidth=2, label='Q-Learning') plt.plot(moving_average(np.mean(np.abs(np.max(Q_track_dql, axis=2) - optimal_V), axis=1)), '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=1, ncol=1) plt.title('State-value function estimation error (ma 100)') plt.xlabel('Episodes') plt.ylabel('Mean Absolute Error MAE(V, v*)') plt.xticks(rotation=45) plt.show()

plt.plot(moving_average(np.mean(np.abs(Q_track_mc - optimal_Q), axis=(1,2))), '-', linewidth=2, label='FVMC') plt.plot(moving_average(np.mean(np.abs(Q_track_sarsa - optimal_Q), axis=(1,2))), '--', linewidth=2, label='Sarsa') plt.plot(moving_average(np.mean(np.abs(Q_track_ql - optimal_Q), axis=(1,2))), ':', linewidth=2, label='Q-Learning') plt.plot(moving_average(np.mean(np.abs(Q_track_dql - optimal_Q), axis=(1,2))), '-.', linewidth=2, label='Double Q-Learning') plt.legend(loc=1, ncol=1) plt.title('Action-value function estimation error (ma 100)') plt.xlabel('Episodes') plt.ylabel('Mean Absolute Error MAE(Q, q*)') plt.xticks(rotation=45) plt.show()

env = gym.make('FrozenLake-v1') init_state = env.reset() goal_state = 15 gamma = 0.99 n_episodes = 10000 P = env.env.P n_cols, svf_prec, err_prec, avf_prec=4, 4, 2, 3 action_symbols=('<', 'v', '>', '^') limit_items, limit_value = 5, 0.0 cu_limit_items, cu_limit_value, cu_episodes = 10, 0.01, 2000

plt.plot(decay_schedule(0.5, 0.01, 0.5, n_episodes), '-', linewidth=2, label='Alpha schedule') plt.plot(decay_schedule(1.0, 0.1, 0.9, n_episodes), ':', linewidth=2, label='Epsilon schedule') plt.legend(loc=1, ncol=1) plt.title('Alpha and epsilon schedules') plt.xlabel('Episodes') plt.ylabel('Hyperparameter values') plt.xticks(rotation=45) plt.show()

optimal_Q, optimal_V, optimal_pi = value_iteration(P, gamma=gamma) print_state_value_function(optimal_V, P, n_cols=n_cols, prec=svf_prec, title='Optimal state-value function:') print() print_action_value_function(optimal_Q, None, action_symbols=action_symbols, prec=avf_prec, title='Optimal action-value function:') print() print_policy(optimal_pi, P, action_symbols=action_symbols, n_cols=n_cols) success_rate_op, mean_return_op, mean_regret_op = get_policy_metrics( env, gamma=gamma, pi=optimal_pi, goal_state=goal_state, optimal_Q=optimal_Q) print('Reaches goal {:.2f}%. Obtains an average return of {:.4f}. Regret of {:.4f}'.format( success_rate_op, mean_return_op, mean_regret_op))

Q_mcs, V_mcs, Q_track_mcs = [], [], [] for seed in tqdm(SEEDS, desc='All seeds', leave=True): random.seed(seed); np.random.seed(seed) ; env.seed(seed) Q_mc, V_mc, pi_mc, Q_track_mc, pi_track_mc = mc_control(env, gamma=gamma, n_episodes=n_episodes) Q_mcs.append(Q_mc) ; V_mcs.append(V_mc) ; Q_track_mcs.append(Q_track_mc) Q_mc, V_mc, Q_track_mc = np.mean(Q_mcs, axis=0), np.mean(V_mcs, axis=0), np.mean(Q_track_mcs, axis=0) del Q_mcs ; del V_mcs ; del Q_track_mcs