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From Likelihood to Log-Loss and Its Gradient
- Starting from the Bernoulli likelihood for n independent observations, derive the log-loss objective that logistic regression minimizes.
- Show that the gradient with respect to \boldsymbol\beta takes the form \mathbf X^\top(\mathbf p - \mathbf y), and explain in words why this looks so similar to the OLS normal equations.
- Explain concretely why there is no closed-form solution for \hat{\boldsymbol\beta}, and what property of the loss guarantees that iterative solvers still find the global optimum.
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