Linear regression objective and gradient

Problem

For $X=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ , $\mathbf{y}=(1,3)$ , and $\mathbf{w}=(0,0)$ , use

$L(\mathbf{w})=\frac{1}{2}\lVert X\mathbf{w}-\mathbf{y}\rVert_2^2.$

Compute $L$ , its gradient, and one gradient-descent update with learning rate $0.1$ .

Reveal answer or reference solution

$L=5$ . The gradient is

$\nabla_{\mathbf{w}}L=X^\top(X\mathbf{w}-\mathbf{y})=(-4,-3),$

so the update is $\mathbf{w}=(0.4,0.3)$ .

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Appears in

Machine Learning Diagnostic
Regression Practice
Gradient Descent Practice