#laplace-noise

Question 1: Single choice

mediumAI-checkedLinear Models & Regularization

Suppose the noise in $y = \mathbf{w}^\top\mathbf{x} + b + \epsilon$ follows the Laplace density $p(\epsilon) = \frac{1}{2}e^{-|\epsilon|}$ instead of a Gaussian. Which loss does maximum likelihood estimation minimize?

The log-cosh error $\sum_i \log\cosh(y_i - \mathbf{w}^\top\mathbf{x}_i - b)$ 1
The squared error plus an $\ell_1$ penalty $\lambda\lVert\mathbf{w}\rVert_1$ 2
The squared error $\sum_i (y_i - \mathbf{w}^\top\mathbf{x}_i - b)^2$ 3
The absolute error $\sum_i |y_i - \mathbf{w}^\top\mathbf{x}_i - b|$ 4

Timed test