Equations

yprediction(x0,x1,...,xm−1)=w0x0+w1x1+...+wm−1xm−1
MSE=12nn−1∑i=0(yactual−(w0x0+w1x1+...+wm−1xm−1))2

Gradient of the MSE:

∇MSE(w0,w1,...,wm−1)=∂MSE∂w0,∂MSE∂w1,...,∂MSE∂wm−1

Let’s take the first partial derivative as an example.

u=(yactual−(w0x0+w1x1+...+wm−1xm−1))
MSE=12nn−1∑i=0u2

Using the chain rule:

∂MSE∂w0=∂MSE∂u∗∂u∂w0

The constants are cancelled out.

∂MSE∂u=1nn−1∑i=0u

Only x0 will remain from u=(yactual−(w0x0+w1x1+...+wm−1xm−1)) since all other variables will be treated as a constant except for w0.

∂u∂w0=x0

Back to the other equation:

∂MSE∂w0=∂MSE∂u∗∂u∂w0
∂MSE∂w0=1nn−1∑i=0(u)∗x0

Substitute u.

∂MSE∂w0=1nn−1∑i=0(yactual−(w0x0+w1x1+...+wm−1xm−1))∗x0

Substitute the yprediction function.

∂MSE∂w0=1nn−1∑i=0(yactual−yprediction)∗x0

Then do this for all the weights:

∂MSE∂w1=1nn−1∑i=0(yactual−yprediction)∗x1
...
∂MSE∂wm−1=1nn−1∑i=0(yactual−yprediction)∗xm−1