Loss Functions: Median and General Quantiles

Let

$$x_1,\ldots,x_n$$

be fixed real numbers.

Consider objectives of the form

$$f(a)=\sum_{i=1}^{n}L(x_i-a).$$

The squared-loss case

$$L(u)=u^2$$

produces the sample mean as the minimizing value of $a$.

Now consider the absolute-value loss

$$L(u)=|u|.$$

Subproblem A

For

$$f(a)=\sum_{i=1}^{n}|x_i-a|,$$

which value or values of $a$ minimize $f(a)$?

Subproblem B

Generalize the loss so that the minimizing $a$ is an arbitrary empirical quantile.

For example, construct a loss that returns the 10th percentile or the 75th percentile.

For your submitted answer, pick the loss $L$ below for which, for every $0<\tau<1$, the minimizers of $\sum_i L(x_i-a)$ are exactly the empirical $\tau$-quantiles.