For probabilistic models with latent variables, autoencoding variational Bayes (AEVB; Kingma and Welling, 2014) is an algorithm which allows us to perform inference efficiently for large datasets with an encoder. In AEVB, the encoder is used to infer variational parameters of approximate posterior on latent variables from given samples. By using tunable and flexible encoders such as multilayer perceptrons (MLPs), AEVB approximates complex variational posterior based on mean-field approximation, which does not utilize analytic representations of the true posterior. Combining AEVB with ADVI (Kucukelbir et al., 2015), we can perform posterior inference on almost arbitrary probabilistic models involving continuous latent variables.
I have implemented AEVB for ADVI with mini-batch on PyMC3. To demonstrate flexibility of this approach, we will apply this to latent dirichlet allocation (LDA; Blei et al., 2003) for modeling documents. In the LDA model, each document is assumed to be generated from a multinomial distribution, whose parameters are treated as latent variables. By using AEVB with an MLP as an encoder, we will fit the LDA model to the 20-newsgroups dataset.
In this example, extracted topics by AEVB seem to be qualitatively comparable to those with a standard LDA implementation, i.e., online VB implemented on scikit-learn. Unfortunately, the predictive accuracy of unseen words is less than the standard implementation of LDA, it might be due to the mean-field approximation. However, the combination of AEVB and ADVI allows us to quickly apply more complex probabilistic models than LDA to big data with the help of mini-batches. I hope this notebook will attract readers, especially practitioners working on a variety of machine learning tasks, to probabilistic programming and PyMC3.
In [1]:
%matplotlib inline
import sys, os
# unfortunately I was not able to run it on GPU due to overflow problems
%env THEANO_FLAGS=device=cpu,floatX=float64
import theano
from collections import OrderedDict
from copy import deepcopy
import numpy as np
from time import time
from sklearn.feature_extraction.text import TfidfVectorizer, CountVectorizer
from sklearn.datasets import fetch_20newsgroups
import matplotlib.pyplot as plt
import seaborn as sns
from theano import shared
import theano.tensor as tt
from theano.sandbox.rng_mrg import MRG_RandomStreams
import pymc3 as pm
from pymc3 import math as pmmath
from pymc3 import Dirichlet
from pymc3.distributions.transforms import t_stick_breaking
Here, we will use the 20-newsgroups dataset. This dataset can be obtained by using functions of scikit-learn. The below code is partially adopted from an example of scikit-learn (http://scikit-learn.org/stable/auto_examples/applications/topics_extraction_with_nmf_lda.html). We set the number of words in the vocabulary to 1000.
In [2]:
# The number of words in the vocaburary
n_words = 1000
print("Loading dataset...")
t0 = time()
dataset = fetch_20newsgroups(shuffle=True, random_state=1,
remove=('headers', 'footers', 'quotes'))
data_samples = dataset.data
print("done in %0.3fs." % (time() - t0))
# Use tf (raw term count) features for LDA.
print("Extracting tf features for LDA...")
tf_vectorizer = CountVectorizer(max_df=0.95, min_df=2, max_features=n_words,
stop_words='english')
t0 = time()
tf = tf_vectorizer.fit_transform(data_samples)
feature_names = tf_vectorizer.get_feature_names()
print("done in %0.3fs." % (time() - t0))
Each document is represented by 1000-dimensional term-frequency vector. Let's check the data.
In [3]:
plt.plot(tf[:10, :].toarray().T);
We split the whole documents into training and test sets. The number of tokens in the training set is 480K. Sparsity of the term-frequency document matrix is 0.025%, which implies almost all components in the term-frequency matrix is zero.
In [4]:
n_samples_tr = 10000
n_samples_te = tf.shape[0] - n_samples_tr
docs_tr = tf[:n_samples_tr, :]
docs_te = tf[n_samples_tr:, :]
print('Number of docs for training = {}'.format(docs_tr.shape[0]))
print('Number of docs for test = {}'.format(docs_te.shape[0]))
n_tokens = np.sum(docs_tr[docs_tr.nonzero()])
print('Number of tokens in training set = {}'.format(n_tokens))
print('Sparsity = {}'.format(
len(docs_tr.nonzero()[0]) / float(docs_tr.shape[0] * docs_tr.shape[1])))
For a document $d$ consisting of tokens $w$, the log-likelihood of the LDA model with $K$ topics is given as \begin{eqnarray} \log p\left(d|\theta_{d},\beta\right) & = & \sum_{w\in d}\log\left[\sum_{k=1}^{K}\exp\left(\log\theta_{d,k} + \log \beta_{k,w}\right)\right]+const, \end{eqnarray} where $\theta_{d}$ is the topic distribution for document $d$ and $\beta$ is the word distribution for the $K$ topics. We define a function that returns a tensor of the log-likelihood of documents given $\theta_{d}$ and $\beta$.
In [5]:
def logp_lda_doc(beta, theta):
"""Returns the log-likelihood function for given documents.
K : number of topics in the model
V : number of words (size of vocabulary)
D : number of documents (in a mini-batch)
Parameters
----------
beta : tensor (K x V)
Word distributions.
theta : tensor (D x K)
Topic distributions for documents.
"""
def ll_docs_f(docs):
dixs, vixs = docs.nonzero()
vfreqs = docs[dixs, vixs]
ll_docs = vfreqs * pmmath.logsumexp(
tt.log(theta[dixs]) + tt.log(beta.T[vixs]), axis=1).ravel()
# Per-word log-likelihood times num of tokens in the whole dataset
return tt.sum(ll_docs) / (tt.sum(vfreqs)+1e-9) * n_tokens
return ll_docs_f
In the inner function, the log-likelihood is scaled for mini-batches by the number of tokens in the dataset.
With the log-likelihood function, we can construct the probabilistic model for LDA. doc_t works as a placeholder to which documents in a mini-batch are set.
For ADVI, each of random variables $\theta$ and $\beta$, drawn from Dirichlet distributions, is transformed into unconstrained real coordinate space. To do this, by default, PyMC3 uses a centered stick-breaking transformation. Since these random variables are on a simplex, the dimension of the unconstrained coordinate space is the original dimension minus 1. For example, the dimension of $\theta_{d}$ is the number of topics (n_topics) in the LDA model, thus the transformed space has dimension (n_topics - 1). It shuold be noted that, in this example, we use t_stick_breaking, which is a numerically stable version of stick_breaking used by default. This is required to work ADVI for the LDA model.
The variational posterior on these transformed parameters is represented by a spherical Gaussian distributions (meanfield approximation). Thus, the number of variational parameters of $\theta_{d}$, the latent variable for each document, is 2 * (n_topics - 1) for means and standard deviations.
In the last line of the below cell, DensityDist class is used to define the log-likelihood function of the model. The second argument is a Python function which takes observations (a document matrix in this example) and returns the log-likelihood value. This function is given as a return value of logp_lda_doc(beta, theta), which has been defined above.
In [6]:
n_topics = 10
# we have sparse dataset. It's better to have dence batch so that all words accure there
minibatch_size = 128
# defining minibatch
doc_t_minibatch = pm.Minibatch(docs_tr.toarray(), minibatch_size)
doc_t = shared(docs_tr.toarray()[:minibatch_size])
with pm.Model() as model:
theta = Dirichlet('theta', a=pm.floatX((1.0 / n_topics) * np.ones((minibatch_size, n_topics))),
shape=(minibatch_size, n_topics), transform=t_stick_breaking(1e-9),
# do not forget scaling
total_size=n_samples_tr)
beta = Dirichlet('beta', a=pm.floatX((1.0 / n_topics) * np.ones((n_topics, n_words))),
shape=(n_topics, n_words), transform=t_stick_breaking(1e-9))
# Note, that we devined likelihood with scaling, se here we need no additional `total_size` kwarg
doc = pm.DensityDist('doc', logp_lda_doc(beta, theta), observed=doc_t)
Given a document, the encoder calculates variational parameters of the (transformed) latent variables, more specifically, parameters of Gaussian distributions in the unconstrained real coordinate space. The encode() method is required to output variational means and stds as a tuple, as shown in the following code. As explained above, the number of variational parameters is 2 * (n_topics) - 1. Specifically, the shape of zs_mean (or zs_std) in the method is (minibatch_size, n_topics - 1). It should be noted that zs_std is defined as $\rho = log(exp(std) - 1)$ in ADVI and bounded to be positive. The inverse parametrization is $std = log(1+exp(\rho))$ and considered to be numericaly stable.
To enhance generalization ability to unseen words, a bernoulli corruption process is applied to the inputted documents. Unfortunately, I have never see any significant improvement with this.
In [7]:
class LDAEncoder:
"""Encode (term-frequency) document vectors to variational means and (log-transformed) stds.
"""
def __init__(self, n_words, n_hidden, n_topics, p_corruption=0, random_seed=1):
rng = np.random.RandomState(random_seed)
self.n_words = n_words
self.n_hidden = n_hidden
self.n_topics = n_topics
self.w0 = shared(0.01 * rng.randn(n_words, n_hidden).ravel(), name='w0')
self.b0 = shared(0.01 * rng.randn(n_hidden), name='b0')
self.w1 = shared(0.01 * rng.randn(n_hidden, 2 * (n_topics - 1)).ravel(), name='w1')
self.b1 = shared(0.01 * rng.randn(2 * (n_topics - 1)), name='b1')
self.rng = MRG_RandomStreams(seed=random_seed)
self.p_corruption = p_corruption
def encode(self, xs):
if 0 < self.p_corruption:
dixs, vixs = xs.nonzero()
mask = tt.set_subtensor(
tt.zeros_like(xs)[dixs, vixs],
self.rng.binomial(size=dixs.shape, n=1, p=1-self.p_corruption)
)
xs_ = xs * mask
else:
xs_ = xs
w0 = self.w0.reshape((self.n_words, self.n_hidden))
w1 = self.w1.reshape((self.n_hidden, 2 * (self.n_topics - 1)))
hs = tt.tanh(xs_.dot(w0) + self.b0)
zs = hs.dot(w1) + self.b1
zs_mean = zs[:, :(self.n_topics - 1)]
zs_rho = zs[:, (self.n_topics - 1):]
return {'mu': zs_mean, 'rho':zs_rho}
def get_params(self):
return [self.w0, self.b0, self.w1, self.b1]
To feed the output of the encoder to the variational parameters of $\theta$, we set an OrderedDict of tuples as below.
In [8]:
encoder = LDAEncoder(n_words=n_words, n_hidden=100, n_topics=n_topics, p_corruption=0.0)
local_RVs = OrderedDict([(theta, encoder.encode(doc_t))])
local_RVs
Out[8]:
theta is the random variable defined in the model creation and is a key of an entry of the OrderedDict. The value (encoder.encode(doc_t), n_samples_tr / minibatch_size) is a tuple of a theano expression and a scalar. The theano expression encoder.encode(doc_t) is the output of the encoder given inputs (documents). The scalar n_samples_tr / minibatch_size specifies the scaling factor for mini-batches.
ADVI optimizes the parameters of the encoder. They are passed to the function for ADVI.
In [9]:
encoder_params = encoder.get_params()
encoder_params
Out[9]:
In [10]:
η = .1
s = shared(η)
def reduce_rate(a, h, i):
s.set_value(η/((i/minibatch_size)+1)**.7)
with model:
approx = pm.MeanField(local_rv=local_RVs)
approx.scale_cost_to_minibatch = False
inference = pm.KLqp(approx)
inference.fit(10000, callbacks=[reduce_rate], obj_optimizer=pm.sgd(learning_rate=s),
more_obj_params=encoder_params, total_grad_norm_constraint=200,
more_replacements={doc_t:doc_t_minibatch})
Out[10]:
In [11]:
print(approx)
In [12]:
plt.plot(approx.hist[10:]);
By using estimated variational parameters, we can draw samples from the variational posterior. To do this, we use function sample_vp(). Here we use this function to obtain posterior mean of the word-topic distribution $\beta$ and show top-10 words frequently appeared in the 10 topics.
To apply the above function for the LDA model, we redefine the probabilistic model because the number of documents to be tested changes. Since variational parameters have already been obtained, we can reuse them for sampling from the approximate posterior distribution.
In [13]:
def print_top_words(beta, feature_names, n_top_words=10):
for i in range(len(beta)):
print(("Topic #%d: " % i) + " ".join([feature_names[j]
for j in beta[i].argsort()[:-n_top_words - 1:-1]]))
doc_t.set_value(docs_tr.toarray())
samples = pm.sample_approx(approx, draws=100)
beta_pymc3 = samples['beta'].mean(axis=0)
print_top_words(beta_pymc3, feature_names)
We compare these topics to those obtained by a standard LDA implementation on scikit-learn, which is based on an online stochastic variational inference (Hoffman et al., 2013). We can see that estimated words in the topics are qualitatively similar.
In [14]:
from sklearn.decomposition import LatentDirichletAllocation
lda = LatentDirichletAllocation(n_topics=n_topics, max_iter=5,
learning_method='online', learning_offset=50.,
random_state=0)
%time lda.fit(docs_tr)
beta_sklearn = lda.components_ / lda.components_.sum(axis=1)[:, np.newaxis]
print_top_words(beta_sklearn, feature_names)
In some papers (e.g., Hoffman et al. 2013), the predictive distribution of held-out words was proposed as a quantitative measure for goodness of the model fitness. The log-likelihood function for tokens of the held-out word can be calculated with posterior means of $\theta$ and $\beta$. The validity of this is explained in (Hoffman et al. 2013).
In [15]:
def calc_pp(ws, thetas, beta, wix):
"""
Parameters
----------
ws: ndarray (N,)
Number of times the held-out word appeared in N documents.
thetas: ndarray, shape=(N, K)
Topic distributions for N documents.
beta: ndarray, shape=(K, V)
Word distributions for K topics.
wix: int
Index of the held-out word
Return
------
Log probability of held-out words.
"""
return ws * np.log(thetas.dot(beta[:, wix]))
def eval_lda(transform, beta, docs_te, wixs):
"""Evaluate LDA model by log predictive probability.
Parameters
----------
transform: Python function
Transform document vectors to posterior mean of topic proportions.
wixs: iterable of int
Word indices to be held-out.
"""
lpss = []
docs_ = deepcopy(docs_te)
thetass = []
wss = []
total_words = 0
for wix in wixs:
ws = docs_te[:, wix].ravel()
if 0 < ws.sum():
# Hold-out
docs_[:, wix] = 0
# Topic distributions
thetas = transform(docs_)
# Predictive log probability
lpss.append(calc_pp(ws, thetas, beta, wix))
docs_[:, wix] = ws
thetass.append(thetas)
wss.append(ws)
total_words += ws.sum()
else:
thetass.append(None)
wss.append(None)
# Log-probability
lp = np.sum(np.hstack(lpss)) / total_words
return {
'lp': lp,
'thetass': thetass,
'beta': beta,
'wss': wss
}
transform() function is defined with sample_vp() function. This function is an argument to the function for calculating log predictive probabilities.
In [16]:
inp = tt.matrix(dtype='int64')
sample_vi_theta = theano.function(
[inp],
approx.sample_node(approx.model.theta, 100, more_replacements={doc_t: inp}).mean(0)
)
def transform_pymc3(docs):
return sample_vi_theta(docs)
In [17]:
%time result_pymc3 = eval_lda(transform_pymc3, beta_pymc3, docs_te.toarray(), np.arange(100))
print('Predictive log prob (pm3) = {}'.format(result_pymc3['lp']))
We compare the result with the scikit-learn LDA implemented The log predictive probability is comparable (-6.24) with AEVB-ADVI, and it shows good set of words in the estimated topics.
In [18]:
def transform_sklearn(docs):
thetas = lda.transform(docs)
return thetas / thetas.sum(axis=1)[:, np.newaxis]
%time result_sklearn = eval_lda(transform_sklearn, beta_sklearn, docs_te.toarray(), np.arange(100))
print('Predictive log prob (sklearn) = {}'.format(result_sklearn['lp']))
We have seen that PyMC3 allows us to estimate random variables of LDA, a probabilistic model with latent variables, based on automatic variational inference. Variational parameters of the local latent variables in the probabilistic model are encoded from observations. The parameters of the encoding model, MLP in this example, are optimized with variational parameters of the global latent variables. Once the probabilistic and the encoding models are defined, parameter optimization is done just by invoking an inference (ADVI()) without need to derive complex update equations.
This notebook shows that even mean field approximation can perform as well as sklearn implementation, which is based on the conjugate priors and thus not relying on the mean field approximation.