bandit_simulations.py

import numpy as np
from matplotlib import pylab as plt
#from mpltools import style # uncomment for prettier plots
#style.use(['ggplot'])

# generate all bernoulli rewards ahead of time
def generate_bernoulli_bandit_data(num_samples,K):
    CTRs_that_generated_data = np.tile(np.random.rand(K),(num_samples,1))
    true_rewards = np.random.rand(num_samples,K) < CTRs_that_generated_data
    return true_rewards,CTRs_that_generated_data

def bernoulli_mean_and_variance_of_mean(observed_data):
    totals = observed_data.sum(1)
    successes = observed_data[:,0]
    estimated_means = successes/totals # sample mean
    estimated_variances_of_mean = (estimated_means - estimated_means**2)/totals
    return estimated_means, estimated_variances_of_mean

# totally random
def random(observed_data):
    return np.random.randint(0,len(observed_data))

# the naive algorithm
def naive(observed_data,number_to_explore=100):
    totals = observed_data.sum(1) # totals
    if np.any(totals < number_to_explore): # if have been explored less than specified
        least_explored = np.argmin(totals) # return the one least explored
        return least_explored
    else: # return the best mean forever
        successes = observed_data[:,0] # successes
        estimated_means = successes/totals # the current means
        best_mean = np.argmax(estimated_means) # the best mean
        return best_mean

# the epsilon greedy algorithm
def epsilon_greedy(observed_data,epsilon=0.01):
    estimated_means,_ = bernoulli_mean_and_variance_of_mean(observed_data)
    best_mean = np.argmax(estimated_means) # the best mean
    be_exporatory = np.random.rand() < epsilon # should we explore?
    if be_exporatory: # totally random, excluding the best_mean
        other_choice = np.random.randint(0,len(observed_data))
        while other_choice == best_mean:
            other_choice = np.random.randint(0,len(observed_data))
        return other_choice
    else: # take the best mean
        return best_mean

# the UCB algorithm using
# (1 - 1/t) confidence interval using Chernoff-Hoeffding bound)
# for details of this particular confidence bound, see the UCB1-TUNED section, slide 18, of:
# http://lane.compbio.cmu.edu/courses/slides_ucb.pdf
def UCB(observed_data):
    t = float(observed_data.sum()) # total number of rounds so far over all arms
    totals = observed_data.sum(1)
    estimated_means,estimated_variances_of_means= bernoulli_mean_and_variance_of_mean(observed_data)
    #log of total trials vs total of current arm makes it "curious" about lesser known arms with less exploring at later stages (big t)
    UCB = estimated_means + np.sqrt( np.minimum( estimated_variances_of_means + np.sqrt(2*np.log(t)/totals), 0.25 ) * np.log(t)/totals )
    return np.argmax(UCB)

# the UCB algorithm - using fixed 95% confidence intervals
# see slide 8 for details:
# http://dept.stat.lsa.umich.edu/~kshedden/Courses/Stat485/Notes/binomial_confidence_intervals.pdf
def UCB_normal(observed_data):
    estimated_means,estimated_variances_of_means= bernoulli_mean_and_variance_of_mean(observed_data)
    UCB = estimated_means + 1.96*np.sqrt(estimated_variances_of_means)
    return np.argmax(UCB)

# Thompson Sampling
# basic idea: samples from distribution and compares those values for the arms instead
# http://www.economics.uci.edu/~ivan/asmb.874.pdf
# http://camdp.com/blogs/multi-armed-bandits
def thompson_sampling(observed_data):
    return np.argmax( np.random.beta(observed_data[:,0], observed_data[:,1]) )

#instead of sampling from the real distribution of the mean, approximate it with a normal distribution
#(ok if you take many samples, central limit theorem)
def thompson_sampling_normal(observed_data):
    estimated_means,estimated_variances_of_means= bernoulli_mean_and_variance_of_mean(observed_data)
    estimated_deviation=np.sqrt(estimated_variances_of_means)
    sample_points=np.random.normal(estimated_means,estimated_deviation)
    return np.argmax(sample_points)

# the bandit algorithm
def run_bandit_alg(true_rewards,CTRs_that_generated_data,choice_func):
    num_samples,K = true_rewards.shape
    observed_data = np.zeros((K,2))
    # seed the estimated params
    prior_a = 1. # aka successes
    prior_b = 1. # aka failures
    observed_data[:,0] += prior_a # allocating the initial conditions
    observed_data[:,1] += prior_b
    regret = np.zeros(num_samples)

    for i in range(0,num_samples):
        # pulling a lever & updating observed_data
        this_choice = choice_func(observed_data)

        # update parameters
        if true_rewards[i,this_choice] == 1:
            update_ind = 0
        else:
            update_ind = 1

        observed_data[this_choice,update_ind] += 1

        # updated expected regret
        regret[i] = np.max(CTRs_that_generated_data[i,:]) - CTRs_that_generated_data[i,this_choice]

    cum_regret = np.cumsum(regret)

    return cum_regret

# define number of samples and number of choices
num_samples = 10000
K = 5
number_experiments = 100

regret_accumulator = np.zeros((num_samples,7))
for i in range(number_experiments):
    print "Running experiment:", i+1
    true_rewards,CTRs_that_generated_data = generate_bernoulli_bandit_data(num_samples,K)
    regret_accumulator[:,0] += run_bandit_alg(true_rewards,CTRs_that_generated_data,random)
    regret_accumulator[:,1] += run_bandit_alg(true_rewards,CTRs_that_generated_data,naive)
    regret_accumulator[:,2] += run_bandit_alg(true_rewards,CTRs_that_generated_data,epsilon_greedy)
    regret_accumulator[:,3] += run_bandit_alg(true_rewards,CTRs_that_generated_data,UCB)
    regret_accumulator[:,4] += run_bandit_alg(true_rewards,CTRs_that_generated_data,UCB_normal)
    regret_accumulator[:,5] += run_bandit_alg(true_rewards,CTRs_that_generated_data,thompson_sampling)
    regret_accumulator[:,6] += run_bandit_alg(true_rewards,CTRs_that_generated_data,thompson_sampling_normal)

plt.semilogy(regret_accumulator/number_experiments)
plt.title('Simulated Bandit Performance for K = 5')
plt.ylabel('Cumulative Expected Regret')
plt.xlabel('Round Index')
plt.legend(('Random','Naive','Epsilon-Greedy','(1 - 1/t) UCB','95% UCB',"thompson", "thomson normal"),loc='lower right')
plt.show()