Chunk #37 — 4 Regularized Multinomial Regression

Source: Regularization Paths for Generalized Linear Models via Coordinate Descent.
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We fit the model (20) by regularized maximum (multinomial) likelihood. Using a similar notation as before, let pℓ(xi) = Pr(G = ℓ|xi), and let gi ∈ {1, 2, …, K} be the ith response. We maximize the penalized log-likelihood (21)max{β0ℓ,βℓ}1K∈ℝK(p+1)[1N∑i=1Nlogpgi(xi)−λ∑ℓ=1KPα(βℓ)].