Deep Factorization Machines
===========================
Learning effective feature combinations is critical to the success of
click-through rate prediction task. Factorization machines model feature
interactions in a linear paradigm (e.g., bilinear interactions). This is
often insufficient for real-world data where inherent feature crossing
structures are usually very complex and nonlinear. Whatâ€™s worse,
second-order feature interactions are generally used in factorization
machines in practice. Modeling higher degrees of feature combinations
with factorization machines is possible theoretically but it is usually
not adopted due to numerical instability and high computational
complexity.
One effective solution is using deep neural networks. Deep neural
networks are powerful in feature representation learning and have the
potential to learn sophisticated feature interactions. As such, it is
natural to integrate deep neural networks to factorization machines.
Adding nonlinear transformation layers to factorization machines gives
it the capability to model both low-order feature combinations and
high-order feature combinations. Moreover, non-linear inherent
structures from inputs can also be captured with deep neural networks.
In this section, we will introduce a representative model named deep
factorization machines (DeepFM) :cite:`Guo.Tang.Ye.ea.2017` which
combine FM and deep neural networks.
Model Architectures
-------------------
DeepFM consists of an FM component and a deep component which are
integrated in a parallel structure. The FM component is the same as the
2-way factorization machines which is used to model the low-order
feature interactions. The deep component is a multi-layered perceptron
that is used to capture high-order feature interactions and
nonlinearities. These two components share the same inputs/embeddings
and their outputs are summed up as the final prediction. It is worth
pointing out that the spirit of DeepFM resembles that of the Wide & Deep
architecture which can capture both memorization and generalization. The
advantages of DeepFM over the Wide & Deep model is that it reduces the
effort of hand-crafted feature engineering by identifying feature
combinations automatically.
We omit the description of the FM component for brevity and denote the
output as :math:`\hat{y}^{(FM)}`. Readers are referred to the last
section for more details. Let :math:`\mathbf{e}_i \in \mathbb{R}^{k}`
denote the latent feature vector of the :math:`i^\mathrm{th}` field. The
input of the deep component is the concatenation of the dense embeddings
of all fields that are looked up with the sparse categorical feature
input, denoted as:
.. math::
\mathbf{z}^{(0)} = [\mathbf{e}_1, \mathbf{e}_2, ..., \mathbf{e}_f],
where :math:`f` is the number of fields. It is then fed into the
following neural network:
.. math::
\mathbf{z}^{(l)} = \alpha(\mathbf{W}^{(l)}\mathbf{z}^{(l-1)} + \mathbf{b}^{(l)}),
where :math:`\alpha` is the activation function. :math:`\mathbf{W}_{l}`
and :math:`\mathbf{b}_{l}` are the weight and bias at the
:math:`l^\mathrm{th}` layer. Let :math:`y_{DNN}` denote the output of
the prediction. The ultimate prediction of DeepFM is the summation of
the outputs from both FM and DNN. So we have:
.. math::
\hat{y} = \sigma(\hat{y}^{(FM)} + \hat{y}^{(DNN)}),
where :math:`\sigma` is the sigmoid function. The architecture of DeepFM
is illustrated below. |Illustration of the DeepFM model|
It is worth noting that DeepFM is not the only way to combine deep
neural networks with FM. We can also add nonlinear layers over the
feature interactions :cite:`He.Chua.2017`.
.. |Illustration of the DeepFM model| image:: ../img/rec-deepfm.svg
.. code:: python
import os
from mxnet import gluon, init, np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l
npx.set_np()
Implemenation of DeepFM
-----------------------
The implementation of DeepFM is similar to that of FM. We keep the FM
part unchanged and use an MLP block with ``relu`` as the activation
function. Dropout is also used to regularize the model. The number of
neurons of the MLP can be adjusted with the ``mlp_dims`` hyperparameter.
.. code:: python
class DeepFM(nn.Block):
def __init__(self, field_dims, num_factors, mlp_dims, drop_rate=0.1):
super(DeepFM, self).__init__()
num_inputs = int(sum(field_dims))
self.embedding = nn.Embedding(num_inputs, num_factors)
self.fc = nn.Embedding(num_inputs, 1)
self.linear_layer = nn.Dense(1, use_bias=True)
input_dim = self.embed_output_dim = len(field_dims) * num_factors
self.mlp = nn.Sequential()
for dim in mlp_dims:
self.mlp.add(nn.Dense(dim, 'relu', True, in_units=input_dim))
self.mlp.add(nn.Dropout(rate=drop_rate))
input_dim = dim
self.mlp.add(nn.Dense(in_units=input_dim, units=1))
def forward(self, x):
embed_x = self.embedding(x)
square_of_sum = np.sum(embed_x, axis=1)**2
sum_of_square = np.sum(embed_x**2, axis=1)
inputs = np.reshape(embed_x, (-1, self.embed_output_dim))
x = self.linear_layer(self.fc(x).sum(1)) \
+ 0.5 * (square_of_sum - sum_of_square).sum(1, keepdims=True) \
+ self.mlp(inputs)
x = npx.sigmoid(x)
return x
Training and Evaluating the Model
---------------------------------
The data loading process is the same as that of FM. We set the MLP
component of DeepFM to a three-layered dense network with the a pyramid
structure (30-20-10). All other hyperparameters remain the same as FM.
.. code:: python
batch_size = 2048
data_dir = d2l.download_extract('ctr')
train_data = d2l.CTRDataset(os.path.join(data_dir, 'train.csv'))
test_data = d2l.CTRDataset(os.path.join(data_dir, 'test.csv'),
feat_mapper=train_data.feat_mapper,
defaults=train_data.defaults)
field_dims = train_data.field_dims
train_iter = gluon.data.DataLoader(train_data, shuffle=True,
last_batch='rollover',
batch_size=batch_size,
num_workers=d2l.get_dataloader_workers())
test_iter = gluon.data.DataLoader(test_data, shuffle=False,
last_batch='rollover',
batch_size=batch_size,
num_workers=d2l.get_dataloader_workers())
devices = d2l.try_all_gpus()
net = DeepFM(field_dims, num_factors=10, mlp_dims=[30, 20, 10])
net.initialize(init.Xavier(), ctx=devices)
lr, num_epochs, optimizer = 0.01, 30, 'adam'
trainer = gluon.Trainer(net.collect_params(), optimizer,
{'learning_rate': lr})
loss = gluon.loss.SigmoidBinaryCrossEntropyLoss()
d2l.train_ch13(net, train_iter, test_iter, loss, trainer, num_epochs, devices)
.. parsed-literal::
:class: output
loss 0.510, train acc 0.529, test acc 0.542
110133.7 examples/sec on [gpu(0), gpu(1)]
.. figure:: output_deepfm_61b173_5_1.svg
Compared with FM, DeepFM converges faster and achieves better
performance.
Summary
-------
- Integrating neural networks to FM enables it to model complex and
high-order interactions.
- DeepFM outperforms the original FM on the advertising dataset.
Exercises
---------
- Vary the structure of the MLP to check its impact on model
performance.
- Change the dataset to Criteo and compare it with the original FM
model.
`Discussions `__