ML Prep

Linear regression objective and gradient

Problem

For X=[1011]X=\begin{bmatrix}1&0\\1&1\end{bmatrix}, y=(1,3)\mathbf{y}=(1,3), and w=(0,0)\mathbf{w}=(0,0), use

L(w)=12∥Xw−y∥22.L(\mathbf{w})=\frac{1}{2}\lVert X\mathbf{w}-\mathbf{y}\rVert_2^2.

Compute LL, its gradient, and one gradient-descent update with learning rate 0.10.1.

Reveal answer or reference solution

L=5L=5. The gradient is

∇wL=X⊤(Xw−y)=(−4,−3),\nabla_{\mathbf{w}}L=X^\top(X\mathbf{w}-\mathbf{y})=(-4,-3),

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

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