Expected loss for regression tree with absolute loss function in case of choosing labels randomly

Question

Suppose we have a regression tree (binary). The absolute loss function $L(y,\hat{y})=|y-\hat{y}|$. I know that the optimal prediction that minimizes $\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\hat{y}|$ is the median of labels: $$\hat{y}=\text{median}(y_{1},\dots,y_{n}).$$ I'm curious what happens if instead of using optimal prediction for this leaf we output the label from $y_{1},\dots,y_{n}$ at random what will happen with the expected loss on the training sample?