Factorization Machines¶

Factorization Machines model all interactions between variables using factorized parameters. The model equation for a factorization machine of degree $d=2$ is defined as: $$ \hat{y}(x) = \omega_0 + \sum_{i=1}^n\omega_ix_i + \sum_{i=1}^n\sum_{j=i+1}^n v_i^Tv_jx_ix_j. $$

The complexity of straight forward computation of the equation is in $\mathcal{O}(kn^2)$. However, we have: \begin{align} \sum_{i=1}^n\sum_{j=i+1}^n v_i^Tv_jx_ix_j &= \frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n v_i^Tv_jx_ix_j - \frac{1}{2}\sum_{i=1}^n v_i^Tv_ix_ix_i\\ &= \frac{1}{2}(\sum_{i=1}^n\sum_{j=1}^n\sum_{f=1}^k v_{if}v_{jf}x_ix_j - \sum_{i=1}^n\sum_{f=1}^k v_{if}v_{if}x_ix_i)\\ &= \frac{1}{2}\sum_{f=1}^k((\sum_{i=1}^n v_{if}x_i)^2 - \sum_{i=1}^n v_{if}^2x_i^2) \end{align} This equation has only linear complexity in both $k$ and $n$, i.e., its computation is in $\mathcal{O}(kn)$.

d-way Factorization Machines¶

The 2-way FM described so far can easily be generalized to a d-way FM: $$ \hat{y}(x) = \omega_0 + \sum_{i=1}^n\omega_ix_i + \sum_{l=2}^d\sum_{i_1=1}^n\cdot\cdot\cdot\sum_{i_l=i_{l-1}+1}^n(\prod_{j=1}^l x_{i_j})(\sum_{f=1}^{k_l}\prod_{j=1}^l v_{i_jf}^{(l)}). $$

Implementation¶

from sklearn import metrics
import tensorflow as tf
import numpy as np

# evaluation
def get_auc(y, y_pre):
    fpr, tpr, thresholds = metrics.roc_curve(y.astype(int), y_pre, pos_label=1)
    return metrics.auc(fpr, tpr)

# hyper-parameters
vector_dim = 8
learning_rate = 1e-4
l2_factor = 1e-2
max_training_step = 400
train_rate = 0.8

# split data
data = np.loadtxt(fname='/kaggle/input/data', delimiter='\t')
thredhold = int(train_rate * len(data))
x_train = data[:thredhold, :-1]
y_train = data[:thredhold, -1]
x_test = data[thredhold:, :-1]
y_test = data[thredhold:, -1]
feature_num = len(data[0])-1

# construct graph
# model parameters
w_0 = tf.Variable(0.0)
w = tf.Variable(tf.zeros(shape=[feature_num]))
v = tf.Variable(tf.truncated_normal(shape=[feature_num, vector_dim], mean=0.0, stddev=0.01))

# construct loss
x = tf.placeholder(shape=[None, feature_num], dtype=tf.float32)
y = tf.placeholder(shape=[None], dtype=tf.float32)

linear_term = w_0 + tf.reduce_sum(tf.expand_dims(w, axis=0) * x, axis=1)
square_of_sum = tf.square(tf.reduce_sum(tf.expand_dims(x, axis=2) * tf.expand_dims(v, axis=0), axis=1))
sum_of_square = tf.reduce_sum(tf.square(tf.expand_dims(v, axis=0) * tf.expand_dims(x, axis=2)), axis=1)
y_pre = tf.sigmoid(linear_term + 0.5 * tf.reduce_sum(square_of_sum - sum_of_square, axis=1))

cross_entropy = - y * tf.log(y_pre) - (1 - y) * tf.log(1 - y_pre)
train_op = tf.train.GradientDescentOptimizer(learning_rate).minimize(cross_entropy + l2_factor * tf.add_n([tf.nn.l2_loss(item) for item in [w_0, w, v]]))

accuracy = tf.reduce_mean(tf.cast(tf.less(tf.abs(y - y_pre), 0.5), dtype=tf.float32))

with tf.Session() as sess:
    sess.run(tf.global_variables_initializer())
    for step in range(max_training_step):
        sess.run(train_op, {x: x_train, y: y_train})
        acc, y_pre_value = sess.run([accuracy, y_pre], {x: x_test, y: y_test})
        print('----step%3d accuracy: %.3f, auc: %.3f' % (step, acc, get_auc(y_test, y_pre_value)))

----step  0 accuracy: 0.700, auc: 0.645
----step  1 accuracy: 0.700, auc: 0.656
----step  2 accuracy: 0.700, auc: 0.663
----step  3 accuracy: 0.700, auc: 0.673
----step  4 accuracy: 0.700, auc: 0.681
----step  5 accuracy: 0.700, auc: 0.688
----step  6 accuracy: 0.700, auc: 0.696
----step  7 accuracy: 0.700, auc: 0.702
----step  8 accuracy: 0.700, auc: 0.708
----step  9 accuracy: 0.700, auc: 0.714
----step 10 accuracy: 0.700, auc: 0.722
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----step 17 accuracy: 0.700, auc: 0.750
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----step 25 accuracy: 0.710, auc: 0.768
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----step172 accuracy: 0.780, auc: 0.819
----step173 accuracy: 0.780, auc: 0.820
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----step181 accuracy: 0.780, auc: 0.820
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----step386 accuracy: 0.790, auc: 0.828
----step387 accuracy: 0.790, auc: 0.828
----step388 accuracy: 0.790, auc: 0.828
----step389 accuracy: 0.790, auc: 0.828
----step390 accuracy: 0.790, auc: 0.828
----step391 accuracy: 0.790, auc: 0.828
----step392 accuracy: 0.790, auc: 0.828
----step393 accuracy: 0.790, auc: 0.828
----step394 accuracy: 0.790, auc: 0.828
----step395 accuracy: 0.790, auc: 0.828
----step396 accuracy: 0.790, auc: 0.828
----step397 accuracy: 0.790, auc: 0.827
----step398 accuracy: 0.790, auc: 0.828
----step399 accuracy: 0.790, auc: 0.828

Some References¶

Rendle, Steffen. "Factorization machines." Data Mining (ICDM), 2010 IEEE 10th International Conference on. IEEE, 2010.
Juan, Yuchin, et al. "Field-aware factorization machines for CTR prediction." Proceedings of the 10th ACM Conference on Recommender Systems. ACM, 2016.
Guo, Huifeng, et al. "Deepfm: a factorization-machine based neural network for ctr prediction." arXiv preprint arXiv:1703.04247 (2017).
Qu, Yanru, et al. "Product-based neural networks for user response prediction." Data Mining (ICDM), 2016 IEEE 16th International Conference on. IEEE, 2016.