Bayesian Personalized Ranking

Definition

Bayesian personalized ranking (BPR) is a metric that help evaluate the performance of recommendation system. It focuses on the relative order of items rather than absolute scores.

Let $U$ be a set of all users, $V$ be a set of all items, $E=\{(u,v)|u\in U,v\in V,u\sim v\}$ be a set of observed user-item interactions, and $E_{\text{neg}}=\{(u,v)|u\in U,v\in V,(u,v)\notin E\}$ be a set of negative edges.

For each user $u^{\ast }\in U$, the rooted positive and negative edges are defined as $\begin{array}{rl}E(u^{\ast })& =\{(u^{\ast },v)|u\in U,v\in V,(u^{\ast },v)\in E\}\\ E_{\text{neg}}(u^{\ast })& =\{(u^{\ast },v)|u\in U,v\in V,(u^{\ast },v)\in E_{\text{neg}}\}\end{array}$ The BPR loss for user $u^{\ast }$ is defined as $\mathcal{L}(u^{\ast })=\frac{1}{|E(u^{\ast })|\cdot |E_{\text{neg}}(u^{\ast })|}\sum _{(u^{\ast },v_{\text{pos}})\in E(u^{\ast })}\sum _{(u^{\ast },v_{\text{neg}})\in E_{\text{neg}}(u^{\ast })}-\ln (\sigma (f_{\theta }({\mathbf{u}}^{\ast },{\mathbf{v}}_{\text{pos}})-f_{\theta }({\mathbf{u}}^{\ast },{\mathbf{v}}_{\text{neg}})))$

The final BPR loss is the average of them. $\frac{1}{|U|}\sum _{u^{\ast }\in U}\mathcal{L}(u^{\ast })$ where:

• $\sigma (x)$ is a Sigmoid Function
• $\mathbf{u}$ and $\mathbf{v}$ are the embedding vectors of $u$ and $v$.
• $f_{\theta }(\mathbf{u},\mathbf{v})$ is the score between $u$ and $v$.