黑马（9）baseline交替最小二乘优化

交替最小二乘法优化

原理推导

损失函数
J(θ)=∑u,i∈R(rui−μ−bu−bi)2+λ∗(∑ubu2+∑ibi2)J(\theta)=\sum_{u,i\in 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∑bu2+i∑bi2)

对损失函数求偏导
∂J(θ)∂bu=−2∑u,i∈R(rui−μ−bu−bi)+2λ∗bu\frac{\partial{J(\theta)}}{\partial{b_u}}=-2\sum_{u,i\in R}(r_{ui}-\mu-b_u-b_i)+2\lambda * 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(rui−μ−bu−bi)=λ∗bu∑u,i∈R(rui−μ−bi)=∑u,i∈Rbu+λ∗bu\sum_{u,i\in R}(r_{ui}-\mu-b_u-b_i)=\lambda * b_u \\ \sum_{u,i\in R}(r_{ui}-\mu-b_i)=\sum_{u,i\in R} b_u+\lambda * b_u u,i∈R∑(rui−μ−bu−bi)=λ∗buu,i∈R∑(rui−μ−bi)=u,i∈R∑bu+λ∗bu
为了简化公式，令∑u,i∈Rbu≈∣R(u)∣∗bu\sum_{u,i\in R} b_u \approx |R(u)|*b_u∑u,i∈Rbu≈∣R(u)∣∗bu，∣R(u)∣|R(u)|∣R(u)∣是uuu总的评分次数，可得：
bu=∑u,i∈R(rui−μ−bi)∣R(u)∣+λ1b_u=\frac{\sum_{u,i\in R}(r_{ui}-\mu-b_i)}{|R(u)|+\lambda_1} bu=∣R(u)∣+λ1∑u,i∈R(rui−μ−bi)
同理可得：
bi=∑u,i∈R(rui−μ−bu)∣R(i)∣+λ2b_i=\frac{\sum_{u,i\in 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iii总的被评分次数

通过原理推导，我们得到了bub_ubu和bib_ibi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

关键代码如下

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0bu = 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 = 0for 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 = 0for 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