个性化排序算法实践(一)——FM算法

因子分解机（Factorization Machine，简称FM）算法用于解决大规模稀疏数据下的特征组合问题。FM可以看做带特征交叉的LR。
理论部分可参考FM系列，通过将FM的二次项化简，其复杂度可优化到 $O(kn)$ 。即：

\hat{y} (x) = w_{0} + \sum_{i = 1}^{n} w_{i} x_{i} + \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} ⟨ v i, v j ⟩ x_{i} x_{j} = w_{0} + \sum_{i = 1}^{n} w_{i} x_{i} + \frac{1}{2} \sum_{f = 1}^{k} ⟮ {(\sum_{i = 1}^{n} v_{i, f} x_{i})}^{2} - \sum_{i = 1}^{n} v_{i, f}^{2} x_{i}^{2} ⟯

$\hat y(x) = w_0+\sum_{i=1}^n w_i x_i +\sum_{i=1}^n \sum_{j=i+1}^n ⟨vi,vj⟩ x_i x_j \\ =w_0+\sum_{i=1}^n w_i x_i + \frac{1}{2} \sum_{f=1}^{k} {\left \lgroup \left(\sum_{i=1}^{n} v_{i,f} x_i \right)^2 - \sum_{i=1}^{n} v_{i,f}^2 x_i^2\right \rgroup} \qquad$

我们用随机梯度下降（Stochastic Gradient Descent）法学习模型参数。那么，模型各个参数的梯度如下：

\frac{\partial}{\partial θ} y (x) = {\begin{cases} 1, & if θ is w_{0} (常数项) \\ x_{i} & if θ is w_{i} (线性项) \\ x_{i} \sum_{j = 1}^{n} v_{j, f} x_{j} - v_{i, f} x_{i}^{2}, & if θ is v_{i, f} (交叉项) \end{cases}

$\frac{\partial}{\partial \theta} y(\mathbf{x}) = \begin{cases} 1, & \text{if}\; \theta\; \text{is}\; w_0 \text{(常数项)} \\ x_i & \text{if}\; \theta\; \text{is}\; w_i \text{(线性项)} \\ x_i \sum_{j=1}^{n} v_{j,f} x_j - v_{i,f} x_i^2, & \text{if}\; \theta\; \text{is}\; v_{i,f} \text{(交叉项)} \end{cases}$

这里，我们使用tensorflow实现整个算法。基本步骤如下：
1、构建数据集。这里，令movielens数据集的样本个数为行，令用户ID与itemID为特征，令rating为label，构建数据集。最终通过稀疏矩阵的形式存储，具体方法参考稀疏矩阵在Python中的表示方法。

这里采用用户ID与itemID为特征，进行onehot后，对每一个特征构建隐向量，隐向量维度为(feat_num, vec_dim)。注意这里的特征维度(feat_num),已经不是两维了，而是onehot后的维度。所以，这里的隐向量也可以看做是对每一维的EMbedding的向量，FM算法最终通过EMbedding向量的内积预测label。

2、通过tensorflow构建图，主要注意pred与loss的构建。另外，通过迭代器实现了batcher()方法。

核心代码如下：

x = tf.placeholder(tf.float32, shape=[None, feat_num], name="input_x")
y = tf.placeholder(tf.float32, shape=[None, 1], name="ground_truth")

w0 = tf.get_variable(name="bias", shape=(1), dtype=tf.float32)
W = tf.get_variable(name="linear_w", shape=(feat_num), dtype=tf.float32)
V = tf.get_variable(name="interaction_w", shape=(feat_num, vec_dim), dtype=tf.float32)

linear_part = w0 + tf.reduce_sum(tf.multiply(x, W), axis=1, keep_dims=True)
interaction_part = 0.5 * tf.reduce_sum(tf.square(tf.matmul(x, V)) - tf.matmul(tf.square(x), tf.square(V)), axis=1, keep_dims=True)
y_hat = linear_part + interaction_part
loss = tf.reduce_mean(tf.square(y - y_hat))
train_op = tf.train.AdamOptimizer(learning_rate).minimize(loss)

可以看到，这里定义了三个变量 $w_0$ , $W$ 与 $V$ 分别代表偏移量，一阶权重与EMbedding向量。loss定义为平方损失函数(MSE)，使用 $Adam$ 优化器进行优化。

全部代码如下所示：

#-*-coding:utf-8-*-
"""
author:jamest
date:20191029
FMfunction
"""
# -*- coding:utf-8 -*-
import pandas as pd
import numpy as np
from scipy.sparse import csr
from itertools import count
from collections import defaultdict
import tensorflow as tf


def vectorize_dic(dic, label2index=None, hold_num=None):
    if label2index == None:
        d = count(0)
        label2index = defaultdict(lambda: next(d))  # 数值映射表

    sample_num = len(list(dic.values())[0])  # 样本数
    feat_num = len(list(dic.keys()))  # 特征数
    total_value_num = sample_num * feat_num

    col_ix = np.empty(total_value_num, dtype=int) # 列索引

    i = 0
    for k, lis in dic.items():
        col_ix[i::feat_num] = [label2index[str(k) + str(el)] for el in lis] # 'user'和'item'的映射
        i += 1

    row_ix = np.repeat(np.arange(sample_num), feat_num)

    data = np.ones(total_value_num)

    if hold_num is None:
        hold_num = len(label2index)

    left_data_index = np.where(col_ix < hold_num)  # 为了剔除不在train set中出现的test set数据

    return csr.csr_matrix(
        (data[left_data_index], (row_ix[left_data_index], col_ix[left_data_index])),
        shape=(sample_num, hold_num)), label2index


def batcher(X_, y_, batch_size=-1):
    assert X_.shape[0] == len(y_)

    n_samples = X_.shape[0]
    if batch_size == -1:
        batch_size = n_samples
    if batch_size < 1:
        raise ValueError('Parameter batch_size={} is unsupported'.format(batch_size))

    for i in range(0, n_samples, batch_size):
        upper_bound = min(i + batch_size, n_samples)
        ret_x = X_[i:upper_bound]
        ret_y = y_[i:upper_bound]
        yield (ret_x, ret_y)


def load_dataset():
    cols = ['user', 'item', 'rating', 'timestamp']

    ratingsPath = '../data/ml-1m/ratings.dat'
    ratingsDF = pd.read_csv(ratingsPath, index_col=None, sep='::', header=None,
                            names=cols)[:10000]

    ratingsDF = ratingsDF.sample(frac=1.0)  # 全部打乱
    cut_idx = int(round(0.7 * ratingsDF.shape[0]))
    train, test = ratingsDF.iloc[:cut_idx], ratingsDF.iloc[cut_idx:]

    x_train, label2index = vectorize_dic({'users': train.user.values, 'items': train.item.values})
    x_test, label2index = vectorize_dic({'users': test.user.values, 'items': test.item.values}, label2index,
                                        x_train.shape[1])

    y_train = train.rating.values
    y_test = test.rating.values

    x_train = x_train.todense()
    x_test = x_test.todense()

    return x_train, x_test, y_train, y_test


if __name__ == '__main__':
    x_train, x_test, y_train, y_test = load_dataset()

    print("x_train shape: ", x_train.shape)
    print("x_test shape: ", x_test.shape)
    print("y_train shape: ", y_train.shape)
    print("y_test shape: ", y_test.shape)

    vec_dim = 10
    batch_size = 64
    epochs = 50
    learning_rate = 0.001
    sample_num, feat_num = x_train.shape

    x = tf.placeholder(tf.float32, shape=[None, feat_num], name="input_x")
    y = tf.placeholder(tf.float32, shape=[None, 1], name="ground_truth")

    w0 = tf.get_variable(name="bias", shape=(1), dtype=tf.float32)
    W = tf.get_variable(name="linear_w", shape=(feat_num), dtype=tf.float32)
    V = tf.get_variable(name="interaction_w", shape=(feat_num, vec_dim), dtype=tf.float32)

    linear_part = w0 + tf.reduce_sum(tf.multiply(x, W), axis=1, keep_dims=True)
    interaction_part = 0.5 * tf.reduce_sum(tf.square(tf.matmul(x, V)) - tf.matmul(tf.square(x), tf.square(V)), axis=1,keep_dims=True)
    y_hat = linear_part + interaction_part
    loss = tf.reduce_mean(tf.square(y - y_hat))
    train_op = tf.train.AdamOptimizer(learning_rate).minimize(loss)

    with tf.Session() as sess:
        sess.run(tf.global_variables_initializer())
        for e in range(epochs):
            step = 0
            print("epoch:{}".format(e))
            for batch_x, batch_y in batcher(x_train, y_train, batch_size):
                sess.run(train_op, feed_dict={x: batch_x, y: batch_y.reshape(-1, 1)})
                step += 1
                if step % 10 == 0:
                    for val_x, val_y in batcher(x_test, y_test):
                        train_loss = sess.run(loss, feed_dict={x: batch_x, y: batch_y.reshape(-1, 1)})
                        val_loss = sess.run(loss, feed_dict={x: val_x, y: val_y.reshape(-1, 1)})
                        print("batch train_mse={}, val_mse={}".format(train_loss, val_loss))

        for val_x, val_y in batcher(x_test, y_test):
            val_loss = sess.run(loss, feed_dict={x: val_x, y: val_y.reshape(-1, 1)})
            print("test set rmse = {}".format(np.sqrt(val_loss)))