# Machine Learning

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Machine Learning

## Loss functions

### (Mean) Squared Error

The squared error is:
$$\displaystyle J(\theta) = \sum|h_{\theta}(x^{(i)}) - y^{(i)}|^2$$
If our model is linear regression $$\displaystyle h(x)=w^tx$$ then this is convex.

Proof

\displaystyle \begin{aligned} \nabla_{w} J(w) &= \nabla_{w} \sum (w^tx^{(i)} - y^{(i)})^2\\ &= 2\sum (w^t x^{(i)} - y^{(i)})x \\ \implies \nabla_{w}^2 J(w) &= \nabla 2\sum (w^T x^{(i)} - y^{(i)})x^{(i)}\\ &= 2 \sum x^{(i)}(x^{(i)})^T \end{aligned}
so the hessian is positive semi-definite