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In regression problem, we may need a loss function to measure the relative ranking accuracy between targets $y$ and predicted values $y_{pred}$. Abviously, the simple MSE does not consider such ranking relations.

A straight choice is the so-called IC (Information Coefficient)

$$IC\propto corr(\text{rank}(y), \text{rank}(y_{pred}))$$

which use the correlation between two ranks. However, rank function is not differentiable, thus it can't be used in loss function for regression which uses gradient propagation to update the parameters.

Another choice might be a modified pairwise ranking loss designed for regression

$$\text{Loss}=\sum_{i, j} \text{max}(0, -(y_{pred}^{(i)}-y_{pred}^{(j)})\cdot(y^{(i)}-y^{(j)}))$$

which is proposed in this paper [https://www.cc.gatech.edu/~zha/papers/fp086-zheng.pdf]. This function is differentiable but also hard to train.

My questions are: (1) Is there any other differentiable loss function that can be applied in ranking problem with regression model; (2) Can rank function be approximated by a differentiable function.

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One option is to use standard framing as learning to rank the relative positions, thus pairwise loss function.

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