Gradient of the mean-squared error

Complete the TypeScript function

mseGradient(w: number[], b: number, X: number[][], y: number[]): { gw: number[]; gb: number }

for the linear model $\hat y_i=w^\top x_i+b$ and the loss

$\mathcal L(w,b)=\frac1m\sum_{i=1}^m\big(\hat y_i-y_i\big)^2 .$

Shapes. w has $d$ entries; X has $m$ rows (X[i] is the feature vector $x_i\in\mathbb R^d$); y has $m$ targets. Return gw with $d$ entries, gw[k] $=\partial\mathcal L/\partial w_k$, and the number gb $=\partial\mathcal L/\partial b$. Do not modify the inputs.

What to compute. With residuals $r_i=\hat y_i-y_i$,

$\nabla_w\mathcal L=\frac2m\sum_{i=1}^m r_i\,x_i=\frac2m X^\top r,\qquad \frac{\partial\mathcal L}{\partial b}=\frac2m\sum_{i=1}^m r_i .$

Tests (all compared with absolute tolerance $10^{-9}$):

• worked-example (visible): $w=(0.5,-0.25)$, $b=0.1$, $X=\begin{pmatrix}1&2\\3&4\end{pmatrix}$, $y=(1,2)$. The predictions are $\hat y=(0.1,0.6)$, the residuals $r=(-0.9,-1.4)$, hence gw$$=(-5.1,-7.4)$and `gb`$ $=-2.3$.
• random-1 to random-5 (hidden): random problems with $5\le m\le50$ samples and $1\le d\le5$ features, entries in $[-2,2]$. Only their arguments are sent to your browser; the expected gradients stay on the server.