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The goal of quantile regression is to learn a model that predict the quantiles of a random variable $y$, perhaps given another random variable $x$. Suppose we want to learn the median of $y$. One way to accomplish this is via the Mean-Absolute Error (MAE) loss function. This is because the value of \(\hat{y}\) that minimizes MAE is the median of \(y\):

\[\begin{align*} 0 &= \partial_{\hat{y}} MAE(\hat{y})\\ &= \int_{\mathbb{R}} \partial_{\hat{y}} | y - \hat{y}| p(y) dy\\ &= -\int_{-\infty}^{\hat{y}} p(y) dy + \int_{\hat{y}}^{\infty} p(y) dy \end{align*}\]- Quantile Regression

Quantile regression generalizes mean absolute error. For \(\tau \in (0, 1)\), the quantile regression loss function for the \(\tau\)th quantile is

\[QR_{\tau}(\hat{y}) = (\hat{y} - y)(\tau - \mathbb{I}(\hat{y}-y < 0))\]The loss function is convex and piece-wise linear.