Binary Loss

Definition

Binary loss is a metric that help evaluate the performance of recommendation system. It treats the recommendation problem as a binary classification task.

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. The binary loss function is defined as $\mathcal{L}=-\frac{1}{E}\sum _{(u,v)\in E}\ln (\sigma (f_{\theta }(\mathbf{u},\mathbf{v}))-\frac{1}{E_{\text{neg}}}\sum _{(u,v)\in E_{\text{neg}}}\ln (1-\sigma (f_{\theta }(\mathbf{u},\mathbf{v}))$ 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 $v$.

Since the binary loss is non-personalized, the all positive edges are pushed higher than those of all negative edges.