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  • $\begingroup$ This loss function does not have any local minima though... $\endgroup$ Commented Jul 28, 2020 at 18:14
  • $\begingroup$ The critical point is $(\hat{\beta}_0, \hat{beta}_1) = (\frac{8}{7}, \frac{9}{14})$. The eigenvalues of the Hessian matrix at $(\frac{8}{7}, \frac{9}{14})$ are $13\pm\sqrt{113}$, both of which are $>0$, making the critical point a minimum. $\endgroup$ Commented Jul 28, 2020 at 22:31
  • $\begingroup$ Yes, of course it has a minimum - but it's a global minimum. OP asked about local minima other than the global minimum. $\endgroup$ Commented Jul 29, 2020 at 5:55
  • 1
    $\begingroup$ The idea that the gradient is zero only for the perfect fit is incorrect. $\endgroup$ Commented Jul 29, 2020 at 10:26