Learning To Rank

Neural Sort

• Real-valued score : $s$
• Permutation : $z\in Z_n$
• Sort : $R_n\rightarrow Z_n $
• Permutation Matrix : $P_z\in R^{n\times n}$

$$\begin{array}{c}{P}_z[i,j]=\{\begin{array}{rlr}1& & ifj=z_i\\ 0& & otherwise\end{array}\end{array}$$

• $A_s[i,j]=|s_i-s_j|$
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• $\widehat{P}_{sort(s)}[i,:] = softmax[(n+1-2i)s - A_s\mathcal{1}] $
• $\mathcal{L}(s;\theta )=f({\hat{P}}_{sort(s)};\theta )$

极大似然估计损失函数：

• $\mathcal{L}(s,z=[z_1,z_2,...,z_n];\theta )=f({\hat{P}}_{sort(s)},z=[z_1,z_2,...,z_n])=-{\sum }_{i=1}^{n}log({\hat{P}}_{sort(s)}[i,z_i])$
• 其中 $z$ 为监督信号，例如用户的点击序列

Top K 最大化：

• $\mathcal{L}(s,w=[w_1,w_2,...,w_n];\theta )=f({\hat{P}}_{sort(s)},w=[w_1,w_2,...,w_n])=-{\hat{P}}_{sort(s)}[1:K,:]\cdot w^T$

ListNet

$$P_s(\pi )=\prod _{j=1}^n\frac{\varphi (s_{{\pi }_j})}{{\sum }_{i=j}^n\varphi (s_{{\pi }_i})}$$
$$P_s(j)=\sum _{\pi (1)=j,\pi \in {\Omega }_n}{P}_s(\pi )$$
$$P_s(j)=\frac{\varphi (s_j)}{{\sum }_{k=1}^n\varphi (s_k)}$$
$$L(y^{(i)},z^{(i)})=-\sum _{j=1}^n{P}_{y^{(i)}}(j)log(P_{z^{(i)}}(j))$$

PPD

• $p(z|s)=\frac{s_{z_1}}{{\sum }_{i=1}^n{s}_i}\cdot \frac{s_{z_2}}{{\sum }_{i=1}^n{s}_i-s_{z_1}}\cdot ...\cdot \frac{s_{z_{n-1}}}{s_{z_{n-1}}+s_{z_n}}$
• $\mathcal{L}=-log(p(z|s))$

BPR

$${\mathcal{L}}_{BPR}:=lnP(\Theta |{>}_u)$$
$$=lnP({>}_u|\Theta )P(\Theta )$$
$$=ln\prod _{(u,i,j)\in D_s}\sigma ({\hat{x}}_{uij})P(\Theta )$$
$$=\sum _{(u,i,j)\in D_s}ln\sigma ({\hat{x}}_{uij})+lnP(\Theta )$$
$$=\sum _{(u,i,j)\in D_s}ln\sigma ({\hat{x}}_{ui}-{\hat{x}}_{uj})-{\lambda }_{\Theta }||\Theta |{|}^2$$

RankNet

$$P_{ij}:=P(y_i>y_j)=\sigma (f(d_i)-f(d_j)),L=-\sum _{ij}[T_{ij}log(\sigma (P_{ij}))+(1-T_{ij}log(1-\sigma (P_{ij})))]$$

FRANK

$$L=1-(\sqrt{T_{ij}{P}_{ij}}+\sqrt{(1-T_{ij})(1-P_{ij})})$$