机器学习·L3W2-协同过滤

协同过滤

评分矩阵Y，左侧索引是名称，栏目是用户名

协同过滤的基本原理就是利用已有的评分数据，对未有的评分数据进行预测，根据评分大小推荐给用户

本质上用的算法还是线性回归那套

计算公式

J ( x ( 0 ) , . . . , x ( n m − 1 ) , w ( 0 ) , b ( 0 ) , . . . , w ( n u − 1 ) , b ( n u − 1 ) ) = [ 1 2 ∑ ( i , j ) : r ( i , j ) = 1 ( w ( j ) ⋅ x ( i ) + b ( j ) − y ( i , j ) ) 2 ] + [ λ 2 ∑ j = 0 n u − 1 ∑ k = 0 n − 1 ( w k ( j ) ) 2 + λ 2 ∑ i = 0 n m − 1 ∑ k = 0 n − 1 ( x k ( i ) ) 2 ] ⏟ r e g u l a r i z a t i o n J({\mathbf{x}^{(0)},...,\mathbf{x}^{(n_m-1)},\mathbf{w}^{(0)},b^{(0)},...,\mathbf{w}^{(n_u-1)},b^{(n_u-1)}})= \left[ \frac{1}{2}\sum_{(i,j):r(i,j)=1}(\mathbf{w}^{(j)} \cdot \mathbf{x}^{(i)} + b^{(j)} - y^{(i,j)})^2 \right]+ \underbrace{\left[\frac{\lambda}{2}\sum_{j=0}^{n_u-1}\sum_{k=0}^{n-1}(\mathbf{w}^{(j)}k)^2+ \frac{\lambda}{2}\sum{i=0}^{n_m-1}\sum_{k=0}^{n-1}(\mathbf{x}k^{(i)})^2\right]}{regularization} J(x(0),...,x(nm−1),w(0),b(0),...,w(nu−1),b(nu−1))= 21(i,j):r(i,j)=1∑(w(j)⋅x(i)+b(j)−y(i,j))2 +regularization [2λj=0∑nu−1k=0∑n−1(wk(j))2+2λi=0∑nm−1k=0∑n−1(xk(i))2]

The first summation in (1) is "for all i i i, j j j where r ( i , j ) r(i,j) r(i,j) equals 1 1 1" and could be written:

= [ 1 2 ∑ j = 0 n u − 1 ∑ i = 0 n m − 1 r ( i , j ) ∗ ( w ( j ) ⋅ x ( i ) + b ( j ) − y ( i , j ) ) 2 ] + regularization = \left[ \frac{1}{2}\sum_{j=0}^{n_u-1} \sum_{i=0}^{n_m-1}r(i,j)*(\mathbf{w}^{(j)} \cdot \mathbf{x}^{(i)} + b^{(j)} - y^{(i,j)})^2 \right] +\text{regularization} =[21j=0∑nu−1i=0∑nm−1r(i,j)∗(w(j)⋅x(i)+b(j)−y(i,j))2]+regularization

代码

本质上就是一个矩阵的运算，利用np.sum()化简代码

自定义计算函数cofi_cost_func_v

python 复制代码

def cofi_cost_func_v(X, W, b, Y, R, lambda_):
    """
    Returns the cost for the content-based filtering
    Vectorized for speed. Uses tensorflow operations to be compatible with custom training loop.
    Args:
      X (ndarray (num_movies,num_features)): matrix of item features
      W (ndarray (num_users,num_features)) : matrix of user parameters
      b (ndarray (1, num_users)            : vector of user parameters
      Y (ndarray (num_movies,num_users)    : matrix of user ratings of movies
      R (ndarray (num_movies,num_users)    : matrix, where R(i, j) = 1 if the i-th movies was rated by the j-th user
      lambda_ (float): regularization parameter
    Returns:
      J (float) : Cost
    """
    j = (tf.linalg.matmul(X, tf.transpose(W)) + b - Y)*R
    J = 0.5 * tf.reduce_sum(j**2) + (lambda_/2) * (tf.reduce_sum(X**2) + tf.reduce_sum(W**2))
    return J

利用tensorflow求导

python 复制代码

iterations = 200
lambda_ = 1
for iter in range(iterations):
    with tf.GradientTape() as tape:

        cost_value = cofi_cost_func_v(X, W, b, Y, R, lambda_)


    grads = tape.gradient( cost_value, [X,W,b] )

    optimizer.apply_gradients( zip(grads, [X,W,b]) )

    if iter % 5 == 0:
        print(f"Training loss at iteration {iter}: {cost_value:0.1f}")

预测变量

矩阵乘法：y_pred=X@W.T+b
合并原始数据：Y_res=R*Y+(1-R)*y_pred

预测二进制变量

原来的 f ( w , b , x ) → f = s i g m o i d ( z ) f(w,b,x)\to f=sigmoid(z) f(w,b,x)→f=sigmoid(z)函数

损失函数改为交叉熵即可！

技巧：平均值正常化

模型评估

不适合冷启动问题
需要额外的信息，很难解释这些额外的信息的含义
梯度下降速度极慢，参数太多