This library corresponds to the Deep Bayesian Bandits Showdown: An Empirical Comparison of Bayesian Deep Networks for Thompson Sampling paper, published in ICLR 2018. We provide a benchmark to test decision-making algorithms for contextual-bandits. In particular, the current library implements a variety of algorithms (many of them based on approximate Bayesian Neural Networks and Thompson sampling), and a number of real and syntethic data problems exhibiting a diverse set of properties.
It is a Python library that uses TensorFlow.
We encourage contributors to add new approximate Bayesian Neural Networks or, more generally, contextual bandits algorithms to the library. Also, we would like to extend the data sources over time, so we warmly encourage contributions in this front too!
Please, use the following when citing the code or the paper:
@article{riquelme2018deep, title={Deep Bayesian Bandits Showdown: An Empirical
Comparison of Bayesian Deep Networks for Thompson Sampling},
author={Riquelme, Carlos and Tucker, George and Snoek, Jasper},
journal={International Conference on Learning Representations, ICLR.}, year={2018}}
Contact. This repository is maintained by Carlos Riquelme (rikel). Feel free to reach out directly at rikel@google.com with any questions or comments.
We first briefly introduce contextual bandits, Thompson sampling, enumerate the implemented algorithms, and the available data sources. Then, we provide a simple complete example illustrating how to use the library.
Contextual bandits are a rich decision-making framework where an algorithm has to choose among a set of k actions at every time step t, after observing a context (or side-information) denoted by Xt. The general pseudocode for the process if we use algorithm A is as follows:
At time t = 1, ..., T:
1. Observe new context: X_t
2. Choose action: a_t = A.action(X_t)
3. Observe reward: r_t
4. Update internal state of the algorithm: A.update((X_t, a_t, r_t))
The goal is to maximize the total sum of rewards: ∑t rt
For example, each Xt could encode the properties of a specific user (and the time or day), and we may have to choose an ad, discount coupon, treatment, hyper-parameters, or version of a website to show or provide to the user. Hopefully, over time, we will learn how to match each type of user to the most beneficial personalized action under some metric (the reward).
Thompson Sampling is a meta-algorithm that chooses an action for the contextual bandit in a statistically efficient manner, simultaneously finding the best arm while attempting to incur low cost. Informally speaking, we assume the expected reward is given by some function E[rt | Xt, at] = f(Xt, at). Unfortunately, function f is unknown, as otherwise we could just choose the action with highest expected value: at* = arg maxi f(Xt, at).
The idea behind Thompson Sampling is based on keeping a posterior distribution πt over functions in some family f ∈ F after observing the first t-1 datapoints. Then, at time t, we sample one potential explanation of the underlying process: ft ∼ πt, and act optimally (i.e., greedily) according to ft. In other words, we choose at = arg maxi ft(Xt, ai). Finally, we update our posterior distribution with the new collected datapoint (Xt, at, rt).
The main issue is that keeping an updated posterior πt (or, even, sampling from it) is often intractable for highly parameterized models like deep neural networks. The algorithms we list in the next section provide tractable approximations that can be used in combination with Thompson Sampling to solve the contextual bandit problem.
The Deep Bayesian Bandits library includes the following algorithms (see the paper for further details):
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Linear Algorithms. As a powerful baseline, we provide linear algorithms. In particular, we focus on the exact Bayesian linear regression implementation, while it is easy to derive the greedy OLS version (possibly, with epsilon-greedy exploration). The algorithm is implemented in linear_full_posterior_sampling.py, and it is instantiated as follows:
linear_full = LinearFullPosteriorSampling('MyLinearTS', my_hparams) -
Neural Linear. We introduce an algorithm we call Neural Linear, which operates by learning a neural network to map contexts to rewards for each action, and ---simultaneously--- it updates a Bayesian linear regression in the last layer (i.e., the one that maps the final representation z to the rewards r). Thompson Sampling samples the linear parameters βi for each action i, but keeps the network that computes the representation. Then, both parts (network and Bayesian linear regression) are updated, possibly at different frequencies. The algorithm is implemented in neural_linear_sampling.py, and we create an algorithm instance like this:
neural_linear = NeuralLinearPosteriorSampling('MyNLinear', my_hparams) -
Neural Greedy. Another standard benchmark is to train a neural network that maps contexts to rewards, and at each time t just acts greedily according to the current model. In particular, this approach does not explicitly use Thompson Sampling. However, due to stochastic gradient descent, there is still some randomness in its output. It is straight-forward to add epsilon-greedy exploration to choose random actions with probability ε ∈ (0, 1). The algorithm is implemented in neural_bandit_model.py, and it is used together with PosteriorBNNSampling (defined in posterior_bnn_sampling.py) by calling:
neural_greedy = PosteriorBNNSampling('MyNGreedy', my_hparams, 'RMSProp') -
Stochastic Variational Inference, Bayes by Backpropagation. We implement a Bayesian neural network by modeling each individual weight posterior as a univariate Gaussian distribution: wij ∼ N(μij, σij2). Thompson sampling then samples a network at each time step by sampling each weight independently. The variational approach consists in maximizing a proxy for maximum likelihood of the observed data, the ELBO or variational lower bound, to fit the values of μij, σij2 for every i, j.
See Weight Uncertainty in Neural Networks.
The BNN algorithm is implemented in variational_neural_bandit_model.py, and it is used together with PosteriorBNNSampling (defined in posterior_bnn_sampling.py) by calling:
bbb = PosteriorBNNSampling('myBBB', my_hparams, 'Variational') -
Expectation-Propagation, Black-box alpha-divergence minimization. The family of expectation-propagation algorithms is based on the message passing framework . They iteratively approximate the posterior by updating a single approximation factor (or site) at a time, which usually corresponds to the likelihood of one data point. We focus on methods that directly optimize the global EP objective via stochastic gradient descent, as, for instance, Power EP. For further details see original paper below.
See Black-box alpha-divergence Minimization.
We create an instance of the algorithm like this:
bb_adiv = PosteriorBNNSampling('MyEP', my_hparams, 'AlphaDiv') -
Dropout. Dropout is a training technique where the output of each neuron is independently zeroed out with probability p at each forward pass. Once the network has been trained, dropout can still be used to obtain a distribution of predictions for a specific input. Following the best action with respect to the random dropout prediction can be interpreted as an implicit form of Thompson sampling. The code for dropout is the same as for Neural Greedy (see above), but we need to set two hyper-parameters: use_dropout=True and keep_prob=p where p takes the desired value in (0, 1). Then:
dropout = PosteriorBNNSampling('MyDropout', my_hparams, 'RMSProp') -
Monte Carlo Methods. To be added soon.
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Bootstrapped Networks. This algorithm trains simultaneously and in parallel q neural networks based on different datasets D1, ..., Dq. The way those datasets are collected is by adding each new collected datapoint (Xt, at, rt) to each dataset Di independently and with probability p ∈ (0, 1]. Therefore, the main hyperparameters of the algorithm are (q, p). In order to choose an action for a new context, one of the q networks is first selected with uniform probability (i.e., 1/q). Then, the best action according to the selected network is played.
See Deep Exploration via Bootstrapped DQN.
The algorithm is implemented in bootstrapped_bnn_sampling.py, and we instantiate it as (where my_hparams contains both q and p):
bootstrap = BootstrappedBNNSampling('MyBoot', my_hparams) -
Parameter-Noise. Another approach to approximate a distribution over neural networks (or more generally, models) that map contexts to rewards, consists in randomly perturbing a point estimate trained by Stochastic Gradient Descent on the data. The Parameter-Noise algorithm uses a heuristic to control the amount of noise σt2 it adds independently to the parameters representing a neural network: θt' = θt + ε where ε ∼ N(0, σt2 Id). After using θt' for decision making, the following SGD training steps start again from θt. The key hyperparameters to set are those controlling the noise heuristic.
See Parameter Space Noise for Exploration.
The algorithm is implemented in parameter_noise_sampling.py, and we create an instance by calling:
parameter_noise = ParameterNoiseSampling('MyParamNoise', my_hparams) -
Gaussian Processes. Another standard benchmark are Gaussian Processes, see Gaussian Processes for Machine Learning by Rasmussen and Williams for an introduction. To model the expected reward of different actions, we fit a multitask GP.
See Multi-task Gaussian Process Prediction.
Our implementation is provided in multitask_gp.py, and it is instantiated as follows:
gp = PosteriorBNNSampling('MyMultitaskGP', my_hparams, 'GP')
In the code snippet at the bottom, we show how to instantiate some of these algorithms, and how to run the contextual bandit simulator, and display the high-level results.
In the paper we use two types of contextual datasets: synthetic and based on real-world data.
We provide functions that sample problems from those datasets. In the case of real-world data, you first need to download the raw datasets, and pass the route to the functions. Links for the datasets are provided below.
Synthetic datasets are contained in the synthetic_data_sampler.py file. In particular, it includes:
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Linear data. Provides a number of linear arms, and Gaussian contexts.
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Sparse linear data. Provides a number of sparse linear arms, and Gaussian contexts.
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Wheel bandit data. Provides sampled data from the wheel bandit data, see Section 5.4 in the paper.
Real-world data generating functions are contained in the data_sampler.py file.
In particular, it includes:
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Mushroom data. Each incoming context represents a different type of mushroom, and the actions are eat or no-eat. Eating an edible mushroom provides positive reward, while eating a poisonous one provides positive reward with probability p, and a large negative reward with probability 1-p. All the rewards, and the value of p are customizable. The dataset is part of the UCI repository, and the bandit problem was proposed in Blundell et al. (2015). Data is available here or alternatively here, use the agaricus-lepiota.data file.
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Stock data. We created the Financial Dataset by pulling the stock prices of d = 21 publicly traded companies in NYSE and Nasdaq, for the last 14 years (n = 3713). For each day, the context was the price difference between the beginning and end of the session for each stock. We synthetically created the arms to be a linear combination of the contexts, representing k = 8 different potential portfolios. Data is available here.
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Jester data. We create a recommendation system bandit problem as follows. The Jester Dataset (Goldberg et al., 2001) provides continuous ratings in [-10, 10] for 100 jokes from a total of 73421 users. We find a complete subset of n = 19181 users rating all 40 jokes. Following Riquelme et al. (2017), we take d = 32 of the ratings as the context of the user, and k = 8 as the arms. The agent recommends one joke, and obtains the reward corresponding to the rating of the user for the selected joke. Data is available here.
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Statlog data. The Shuttle Statlog Dataset (Asuncion & Newman, 2007) provides the value of d = 9 indicators during a space shuttle flight, and the goal is to predict the state of the radiator subsystem of the shuttle. There are k = 7 possible states, and if the agent selects the right state, then reward 1 is generated. Otherwise, the agent obtains no reward (r = 0). The most interesting aspect of the dataset is that one action is the optimal one in 80% of the cases, and some algorithms may commit to this action instead of further exploring. In this case, the number of contexts is n = 43500. Data is available here or alternatively here, use shuttle.trn file.
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Adult data. The Adult Dataset (Kohavi, 1996; Asuncion & Newman, 2007) comprises personal information from the US Census Bureau database, and the standard prediction task is to determine if a person makes over 50K a year or not. However, we consider the k = 14 different occupations as feasible actions, based on d = 94 covariates (many of them binarized). As in previous datasets, the agent obtains a reward of 1 for making the right prediction, and 0 otherwise. The total number of contexts is n = 45222. Data is available here or alternatively here, use adult.data file.
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Census data. The US Census (1990) Dataset (Asuncion & Newman, 2007) contains a number of personal features (age, native language, education...) which we summarize in d = 389 covariates, including binary dummy variables for categorical features. Our goal again is to predict the occupation of the individual among k = 9 classes. The agent obtains reward 1 for making the right prediction, and 0 otherwise. Data is available here or alternatively here, use USCensus1990.data.txt file.
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Covertype data. The Covertype Dataset (Asuncion & Newman, 2007) classifies the cover type of northern Colorado forest areas in k = 7 classes, based on d = 54 features, including elevation, slope, aspect, and soil type. Again, the agent obtains reward 1 if the correct class is selected, and 0 otherwise. Data is available here or alternatively here, use covtype.data file.
In datasets 4-7, each feature of the dataset is normalized first.
This library requires Tensorflow, Numpy, and Pandas.
The file example_main.py provides a complete example on how to use the library. We run the code:
python example_main.py
Do not forget to configure the routes to the data files at the top of example_main.py.
For example, we can run the Mushroom bandit for 2000 contexts on a few algorithms as follows:
# Problem parameters
num_contexts = 2000
# Choose data source among:
# {linear, sparse_linear, mushroom, financial, jester,
# statlog, adult, covertype, census, wheel}
data_type = 'mushroom'
# Create dataset
sampled_vals = sample_data(data_type, num_contexts)
dataset, opt_rewards, opt_actions, num_actions, context_dim = sampled_vals
# Define hyperparameters and algorithms
hparams_linear = tf.contrib.training.HParams(num_actions=num_actions,
context_dim=context_dim,
a0=6,
b0=6,
lambda_prior=0.25,
initial_pulls=2)
hparams_dropout = tf.contrib.training.HParams(num_actions=num_actions,
context_dim=context_dim,
init_scale=0.3,
activation=tf.nn.relu,
layer_sizes=[50],
batch_size=512,
activate_decay=True,
initial_lr=0.1,
max_grad_norm=5.0,
show_training=False,
freq_summary=1000,
buffer_s=-1,
initial_pulls=2,
optimizer='RMS',
reset_lr=True,
lr_decay_rate=0.5,
training_freq=50,
training_epochs=100,
keep_prob=0.80,
use_dropout=True)
### Create hyper-parameter configurations for other algorithms
[...]
algos = [
UniformSampling('Uniform Sampling', hparams),
PosteriorBNNSampling('Dropout', hparams_dropout, 'RMSProp'),
PosteriorBNNSampling('BBB', hparams_bbb, 'Variational'),
NeuralLinearPosteriorSampling('NeuralLinear', hparams_nlinear),
LinearFullPosteriorSampling('LinFullPost', hparams_linear),
BootstrappedBNNSampling('BootRMS', hparams_boot),
ParameterNoiseSampling('ParamNoise', hparams_pnoise),
]
# Run contextual bandit problem
t_init = time.time()
results = run_contextual_bandit(context_dim, num_actions, dataset, algos)
_, h_rewards = results
# Display results
display_results(algos, opt_rewards, opt_actions, h_rewards, t_init, data_type)
The previous code leads to final results that look like:
---------------------------------------------------
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mushroom bandit completed after 69.8401839733 seconds.
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0) LinFullPost | total reward = 4365.0.
1) NeuralLinear | total reward = 4110.0.
2) Dropout | total reward = 3430.0.
3) ParamNoise | total reward = 3270.0.
4) BootRMS | total reward = 3050.0.
5) BBB | total reward = 2505.0.
6) Uniform Sampling | total reward = -4930.0.
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Optimal total reward = 5235.
Frequency of optimal actions (action, frequency):
[[0, 953], [1, 1047]]
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