Variational Inference: Bayesian Neural Networks

  1. 2017 by Thomas Wiecki, updated by Maxim Kochurov

Original blog post: http://twiecki.github.io/blog/2016/06/01/bayesian-deep-learning/

Bayesian Neural Networks in PyMC3

Generating data

First, lets generate some toy data – a simple binary classification problem that’s not linearly separable.

In [1]:
%matplotlib inline
import theano
floatX = theano.config.floatX
import pymc3 as pm
import theano.tensor as T
import sklearn
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from warnings import filterwarnings
filterwarnings('ignore')
sns.set_style('white')
from sklearn import datasets
from sklearn.preprocessing import scale
from sklearn.cross_validation import train_test_split
from sklearn.datasets import make_moons
In [2]:
X, Y = make_moons(noise=0.2, random_state=0, n_samples=1000)
X = scale(X)
X = X.astype(floatX)
Y = Y.astype(floatX)
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=.5)
In [3]:
fig, ax = plt.subplots()
ax.scatter(X[Y==0, 0], X[Y==0, 1], label='Class 0')
ax.scatter(X[Y==1, 0], X[Y==1, 1], color='r', label='Class 1')
sns.despine(); ax.legend()
ax.set(xlabel='X', ylabel='Y', title='Toy binary classification data set');
../_images/notebooks_bayesian_neural_network_advi_6_0.png

Model specification

A neural network is quite simple. The basic unit is a perceptron which is nothing more than logistic regression. We use many of these in parallel and then stack them up to get hidden layers. Here we will use 2 hidden layers with 5 neurons each which is sufficient for such a simple problem.

In [4]:
def construct_nn(ann_input, ann_output):
    n_hidden = 5

    # Initialize random weights between each layer
    init_1 = np.random.randn(X.shape[1], n_hidden).astype(floatX)
    init_2 = np.random.randn(n_hidden, n_hidden).astype(floatX)
    init_out = np.random.randn(n_hidden).astype(floatX)

    with pm.Model() as neural_network:
        # Weights from input to hidden layer
        weights_in_1 = pm.Normal('w_in_1', 0, sd=1,
                                 shape=(X.shape[1], n_hidden),
                                 testval=init_1)

        # Weights from 1st to 2nd layer
        weights_1_2 = pm.Normal('w_1_2', 0, sd=1,
                                shape=(n_hidden, n_hidden),
                                testval=init_2)

        # Weights from hidden layer to output
        weights_2_out = pm.Normal('w_2_out', 0, sd=1,
                                  shape=(n_hidden,),
                                  testval=init_out)

        # Build neural-network using tanh activation function
        act_1 = pm.math.tanh(pm.math.dot(ann_input,
                                         weights_in_1))
        act_2 = pm.math.tanh(pm.math.dot(act_1,
                                         weights_1_2))
        act_out = pm.math.sigmoid(pm.math.dot(act_2,
                                              weights_2_out))

        # Binary classification -> Bernoulli likelihood
        out = pm.Bernoulli('out',
                           act_out,
                           observed=ann_output,
                           total_size=Y_train.shape[0] # IMPORTANT for minibatches
                          )
    return neural_network

# Trick: Turn inputs and outputs into shared variables.
# It's still the same thing, but we can later change the values of the shared variable
# (to switch in the test-data later) and pymc3 will just use the new data.
# Kind-of like a pointer we can redirect.
# For more info, see: http://deeplearning.net/software/theano/library/compile/shared.html
ann_input = theano.shared(X_train)
ann_output = theano.shared(Y_train)
neural_network = construct_nn(ann_input, ann_output)

That’s not so bad. The Normal priors help regularize the weights. Usually we would add a constant b to the inputs but I omitted it here to keep the code cleaner.

Variational Inference: Scaling model complexity

We could now just run a MCMC sampler like NUTS which works pretty well in this case, but as I already mentioned, this will become very slow as we scale our model up to deeper architectures with more layers.

Instead, we will use the brand-new ADVI variational inference algorithm which was recently added to PyMC3, and updated to use the operator variational inference (OPVI) framework. This is much faster and will scale better. Note, that this is a mean-field approximation so we ignore correlations in the posterior.

In [5]:
from pymc3.theanof import set_tt_rng, MRG_RandomStreams
set_tt_rng(MRG_RandomStreams(42))
In [6]:
%%time

with neural_network:
    inference = pm.ADVI()
    approx = pm.fit(n=30000, method=inference)
Average Loss = 137.97: 100%|██████████| 30000/30000 [00:09<00:00, 3139.84it/s]
Finished [100%]: Average Loss = 137.82
CPU times: user 11.5 s, sys: 818 ms, total: 12.3 s
Wall time: 15.6 s

And using old interface. Performance is nearly the same

In [7]:
%%time

with neural_network:
    advifit = pm.advi(n=30000)
Average ELBO = -170.65: 100%|██████████| 30000/30000 [00:09<00:00, 3083.15it/s]
Finished [100%]: Average ELBO = -159.53
CPU times: user 11.1 s, sys: 565 ms, total: 11.7 s
Wall time: 11.6 s

~ 12 seconds on my laptop. That’s pretty good considering that NUTS is having a really hard time. Further below we make this even faster. To make it really fly, we probably want to run the Neural Network on the GPU.

As samples are more convenient to work with, we can very quickly draw samples from the variational approximation using the sample method (this is just sampling from Normal distributions, so not at all the same like MCMC):

In [8]:
trace = approx.sample(draws=5000)

Plotting the objective function (ELBO) we can see that the optimization slowly improves the fit over time.

In [9]:
plt.plot(-inference.hist, label='new ADVI', alpha=.3)
plt.plot(advifit.elbo_vals, label='old ADVI', alpha=.3)
plt.legend()
plt.ylabel('ELBO')
plt.xlabel('iteration');
../_images/notebooks_bayesian_neural_network_advi_18_0.png

Now that we trained our model, lets predict on the hold-out set using a posterior predictive check (PPC).

  1. We can use `sample_ppc() <../api/inference.rst>`__ to generate new data (in this case class predictions) from the posterior (sampled from the variational estimation).
  2. It is better to get the node directly and build theano graph using our approximation (approx.sample_node) , we get a lot of speed up
In [10]:
# We can get predicted probability from model
neural_network.out.distribution.p
Out[10]:
sigmoid.0
In [11]:
# create symbolic input
x = T.matrix('X')
# symbolic number of samples is supported, we build vectorized posterior on the fly
n = T.iscalar('n')
# Do not forget test_values or set theano.config.compute_test_value = 'off'
x.tag.test_value = np.empty_like(X_train[:10])
n.tag.test_value = 100
_sample_proba = approx.sample_node(neural_network.out.distribution.p,
                                   size=n,
                                   more_replacements={ann_input: x})
# It is time to compile the function
# No updates are needed for Approximation random generator
# Efficient vectorized form of sampling is used
sample_proba = theano.function([x, n], _sample_proba)

# Create bechmark functions
def production_step1():
    ann_input.set_value(X_test)
    ann_output.set_value(Y_test)
    with neural_network:
        ppc = pm.sample_ppc(trace, samples=500, progressbar=False)

    # Use probability of > 0.5 to assume prediction of class 1
    pred = ppc['out'].mean(axis=0) > 0.5

def production_step2():
    sample_proba(X_test, 500).mean(0) > 0.5

See the difference

In [12]:
%timeit production_step1()
223 ms ± 1.38 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
In [13]:
%timeit production_step2()
53.1 ms ± 311 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)

Let’s go ahead and generate predictions:

In [14]:
pred = sample_proba(X_test, 500).mean(0) > 0.5
In [15]:
fig, ax = plt.subplots()
ax.scatter(X_test[pred==0, 0], X_test[pred==0, 1])
ax.scatter(X_test[pred==1, 0], X_test[pred==1, 1], color='r')
sns.despine()
ax.set(title='Predicted labels in testing set', xlabel='X', ylabel='Y');
../_images/notebooks_bayesian_neural_network_advi_27_0.png
In [16]:
print('Accuracy = {}%'.format((Y_test == pred).mean() * 100))
Accuracy = 93.2%

Hey, our neural network did all right!

Lets look at what the classifier has learned

For this, we evaluate the class probability predictions on a grid over the whole input space.

In [17]:
grid = pm.floatX(np.mgrid[-3:3:100j,-3:3:100j])
grid_2d = grid.reshape(2, -1).T
dummy_out = np.ones(grid.shape[1], dtype=np.int8)
In [18]:
ppc = sample_proba(grid_2d ,500)

Probability surface

In [19]:
cmap = sns.diverging_palette(250, 12, s=85, l=25, as_cmap=True)
fig, ax = plt.subplots(figsize=(16, 9))
contour = ax.contourf(grid[0], grid[1], ppc.mean(axis=0).reshape(100, 100), cmap=cmap)
ax.scatter(X_test[pred==0, 0], X_test[pred==0, 1])
ax.scatter(X_test[pred==1, 0], X_test[pred==1, 1], color='r')
cbar = plt.colorbar(contour, ax=ax)
_ = ax.set(xlim=(-3, 3), ylim=(-3, 3), xlabel='X', ylabel='Y');
cbar.ax.set_ylabel('Posterior predictive mean probability of class label = 0');
../_images/notebooks_bayesian_neural_network_advi_34_0.png

Uncertainty in predicted value

So far, everything I showed we could have done with a non-Bayesian Neural Network. The mean of the posterior predictive for each class-label should be identical to maximum likelihood predicted values. However, we can also look at the standard deviation of the posterior predictive to get a sense for the uncertainty in our predictions. Here is what that looks like:

In [20]:
cmap = sns.cubehelix_palette(light=1, as_cmap=True)
fig, ax = plt.subplots(figsize=(16, 9))
contour = ax.contourf(grid[0], grid[1], ppc.std(axis=0).reshape(100, 100), cmap=cmap)
ax.scatter(X_test[pred==0, 0], X_test[pred==0, 1])
ax.scatter(X_test[pred==1, 0], X_test[pred==1, 1], color='r')
cbar = plt.colorbar(contour, ax=ax)
_ = ax.set(xlim=(-3, 3), ylim=(-3, 3), xlabel='X', ylabel='Y');
cbar.ax.set_ylabel('Uncertainty (posterior predictive standard deviation)');
../_images/notebooks_bayesian_neural_network_advi_36_0.png

We can see that very close to the decision boundary, our uncertainty as to which label to predict is highest. You can imagine that associating predictions with uncertainty is a critical property for many applications like health care. To further maximize accuracy, we might want to train the model primarily on samples from that high-uncertainty region.

Mini-batch ADVI

So far, we have trained our model on all data at once. Obviously this won’t scale to something like ImageNet. Moreover, training on mini-batches of data (stochastic gradient descent) avoids local minima and can lead to faster convergence.

Fortunately, ADVI can be run on mini-batches as well. It just requires some setting up:

In [21]:
minibatch_x = pm.Minibatch(X_train, batch_size=50)
minibatch_y = pm.Minibatch(Y_train, batch_size=50)
neural_network_minibatch = construct_nn(minibatch_x, minibatch_y)
with neural_network_minibatch:
    approx = pm.fit(40000, method=pm.ADVI())
Average Loss = 129.65: 100%|██████████| 40000/40000 [00:06<00:00, 5965.07it/s]
Finished [100%]: Average Loss = 129.5
In [22]:
plt.plot(inference.hist)
plt.ylabel('ELBO')
plt.xlabel('iteration');
../_images/notebooks_bayesian_neural_network_advi_40_0.png

As you can see, mini-batch ADVI’s running time is much lower. It also seems to converge faster.

For fun, we can also look at the trace. The point is that we also get uncertainty of our Neural Network weights.

In [23]:
pm.traceplot(trace);
../_images/notebooks_bayesian_neural_network_advi_42_0.png

Summary

Hopefully this blog post demonstrated a very powerful new inference algorithm available in PyMC3: ADVI. I also think bridging the gap between Probabilistic Programming and Deep Learning can open up many new avenues for innovation in this space, as discussed above. Specifically, a hierarchical neural network sounds pretty bad-ass. These are really exciting times.

Next steps

`Theano <http://deeplearning.net/software/theano/>`__, which is used by PyMC3 as its computational backend, was mainly developed for estimating neural networks and there are great libraries like `Lasagne <https://github.com/Lasagne/Lasagne>`__ that build on top of Theano to make construction of the most common neural network architectures easy. Ideally, we wouldn’t have to build the models by hand as I did above, but use the convenient syntax of Lasagne to construct the architecture, define our priors, and run ADVI.

You can also run this example on the GPU by setting device = gpu and floatX = float32 in your .theanorc.

You might also argue that the above network isn’t really deep, but note that we could easily extend it to have more layers, including convolutional ones to train on more challenging data sets.

I also presented some of this work at PyData London, view the video below:

Finally, you can download this NB here. Leave a comment below, and follow me on twitter.

Acknowledgements

Taku Yoshioka did a lot of work on ADVI in PyMC3, including the mini-batch implementation as well as the sampling from the variational posterior. I’d also like to the thank the Stan guys (specifically Alp Kucukelbir and Daniel Lee) for deriving ADVI and teaching us about it. Thanks also to Chris Fonnesbeck, Andrew Campbell, Taku Yoshioka, and Peadar Coyle for useful comments on an earlier draft.