Prove that there are unique values $w,b$ such that $L_2(w,b) = \sum_{i=1}^{n}(wx_i + b - y_i)^2$ is minimized.

Question

I would like to prove following result:

Suppose we have pair of points $D = \{(x_1,x_1),\cdots,(x_n,y_n)\}$. At least two of these points do not overlap (meaning that there is at least one pair of points $(x_j,y_j), (x_i,y_i)$ such that either $x_j≠x_i$ or $y_j ≠ y_i$) Show that there are unique values for $w,b$ such that $L_2(w,b) = \sum_{i=1}^{n}(wx_i + b - y_i)^2$ is minimized.

I've tried to approach the problem using partial derivatives, namely:

1. Find $\frac{\partial L_2}{dw}$, $\frac{\partial L_2}{db}$

2. Set both to zero

3. Try to show that solution for $w,b$ is unique.

However, the approach has led me nowhere.

So how can I prove the result above?

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I've seen similar question on the Cross Validated, but here I'm more intrested in the proof, instead of the intuitive explanation.

Answer 1

The problem statement allows all $x_i=0$ as long as all $y_i$ are distinct. In such a case, $L_2(w,b)$ does not depend on $w$ and hence the claim is false.

Concrete example: $n=2$, $(x_1,y_1)=(0,\pi)$, $(x_1,y_2)=(0,42)$.