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f (β) = \frac{1}{n} \sum_{i = 1}^{n} lo g_{2} (1 + e^{- y_{i} x_{i}^{T} β}) .

For

g (z) = lo g_{2} (1 + e^{- z})

g^{''} (z) = \frac{e ^{z}}{l o g ( 2 ) ( 1 + e ^{z} ) ^{2}} \geq 0,

g

is a convex function. Each signed margin

y_{i} x_{i}^{T} β

β

, and composition with an affine function preserves convexity. Equivalently,

\nabla^{2} f (β) = \frac{1}{n} \sum_{i = 1}^{n} g^{''} (y_{i} x_{i}^{T} β) x_{i} x_{i}^{T}

f

is convex.