Two Weeks, Two Loss Surfaces

A café owner predicts each day's iced-coffee sales $y$ (in dozens) from the afternoon temperature $x$ (in units of $10\,^\circ\text{C}$ ) with a straight line

\hat{y} = w x + b

and scores the line with the mean squared error over the $n$ days of a week:

L(w, b) = \frac{1}{n}\sum_{i=1}^{n} \left(w x_i + b - y_i\right)^2

She fits the line separately on two weeks of data, four days each:

	temperature $x$	sales $y$
Week 1	$1,\ 3,\ 4,\ 4$	$5,\ 9,\ 10,\ 12$
Week 2 (steady weather)	$3,\ 3,\ 3,\ 3$	$8,\ 11,\ 9,\ 12$

To judge the shape of each loss surface, build its curvature table of second partial derivatives (also called the Hessian):

\begin{bmatrix} L_{ww} & L_{wb} \\ L_{bw} & L_{bb} \end{bmatrix}

With two inputs, the one-variable rule " $f''(x) \ge 0$ everywhere" becomes a test on this table:

Convex if, at every point, $L_{ww} \ge 0$ , $L_{bb} \ge 0$ and $L_{ww}L_{bb} - L_{wb}^2 \ge 0$ .
Strictly convex if, at every point, $L_{ww} > 0$ and $L_{ww}L_{bb} - L_{wb}^2 > 0$ . The surface then bends upward in every direction, and a surface built from squares like this one has exactly one lowest point.

Which statement about the two loss surfaces is correct?

Two Weeks, Two Loss Surfaces

Hints