黑马（9）baseline交替最小二乘优化

损失函数
J ( θ ) = ∑ u , i ∈ R ( r u i − μ − b u − b i ) 2 + λ ∗ ( ∑ u b u 2 + ∑ i b i 2 ) J(theta)=sum_{u,iin R}(r_{ui}-mu-b_u-b_i)^2+lambda*(sum_u b_u{^2}+sum_i b_i{^2} ) J(θ)=u,i∈R∑​(rui​−μ−bu​−bi​)2+λ∗(u∑​bu​2+i∑​bi​2)

对损失函数求偏导
∂ J ( θ ) ∂ b u = − 2 ∑ u , i ∈ R ( r u i − μ − b u − b i ) + 2 λ ∗ b u frac{partial{J(theta)}}{partial{b_u}}=-2sum_{u,iin R}(r_{ui}-mu-b_u-b_i)+2lambda * b_u ∂bu​∂J(θ)​=−2u,i∈R∑​(rui​−μ−bu​−bi​)+2λ∗bu​
另偏导等于0，可得：
∑ u , i ∈ R ( r u i − μ − b u − b i ) = λ ∗ b u ∑ u , i ∈ R ( r u i − μ − b i ) = ∑ u , i ∈ R b u + λ ∗ b u sum_{u,iin R}(r_{ui}-mu-b_u-b_i)=lambda * b_u \ sum_{u,iin R}(r_{ui}-mu-b_i)=sum_{u,iin R} b_u+lambda * b_u u,i∈R∑​(rui​−μ−bu​−bi​)=λ∗bu​u,i∈R∑​(rui​−μ−bi​)=u,i∈R∑​bu​+λ∗bu​
为了简化公式，令 ∑ u , i ∈ R b u ≈ ∣ R ( u ) ∣ ∗ b u sum_{u,iin R} b_u approx |R(u)|*b_u ∑u,i∈R​bu​≈∣R(u)∣∗bu​， ∣ R ( u ) ∣ |R(u)| ∣R(u)∣是 u u u总的评分次数，可得：
b u = ∑ u , i ∈ R ( r u i − μ − b i ) ∣ R ( u ) ∣ + λ 1 b_u=frac{sum_{u,iin R}(r_{ui}-mu-b_i)}{|R(u)|+lambda_1} bu​=∣R(u)∣+λ1​∑u,i∈R​(rui​−μ−bi​)​
同理可得：
b i = ∑ u , i ∈ R ( r u i − μ − b u ) ∣ R ( i ) ∣ + λ 2 b_i=frac{sum_{u,iin R}(r_{ui}-mu-b_u)}{|R(i)|+lambda_2} bi​=∣R(i)∣+λ2​∑u,i∈R​(rui​−μ−bu​)​
其中 ∣ R ( i ) ∣ |R(i)| ∣R(i)∣是 i i i总的被评分次数

通过原理推导，我们得到了 b u b_u bu​和 b i b_i bi​的表达式，他们的表达式中又各自包含着对方，可以用交替最小二乘的方法来计算他们的值。

整体代码放在https://github.com/Kitten-Rec/HeiMa/code

def als(self):
    """
    随机梯度下降优化bu, bi
    :return: bu, bi
    """
    # 初始化参数 bu, bi  全部设置为0
    bu = dict(zip(self.user_ratings.index, np.zeros(len(self.user_ratings.index))))
    bi = dict(zip(self.item_ratings.index, np.zeros(len(self.item_ratings.index))))

    # 交替最小二乘法更新参数
    for epoch in range(self.number_epochs):
        print("Epoch: %d" % epoch)
        
		for iid, uid_list, real_rating_list in self.item_ratings.itertuples(index=True):
		            sum = 0
		            for uid, real_rating in zip(uid_list, real_rating_list):
		                sum += real_rating - self.global_mean - bu[uid]
		            bi[iid] = sum / (self.lambda2 * len(uid_list))
            
        for uid, iid_list, real_rating_list in self.user_ratings.itertuples(index=True):
            sum = 0
            for iid, real_rating in zip(iid_list, real_rating_list):
                sum += real_rating - self.global_mean - bi[iid]
            bu[uid] = sum / (self.lambda1 * len(iid_list))
            
   return bu, bi