Why do we care about convexity?

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In machine learning, the best parameters for a model are chosen so as to minimize the training objective. Strictly convex functions are paticularly interesting because they have a unique global minimum.

Furthermore, for strict and non-sctrict convex functions, every local minimum is a global minimum.

Convex functions

Visually, a convex function “curves up”, without any bends the other way.

What is convexity?

A function is convex if and only if a segment joining two points on its curve always stays above the curve. $\forall 0 \leq t \leq 1$ :

$f(t \vec{a} + (1-t) \vec{b}) \leq t f(\vec{a}) + (1-t) f(\vec{b})$

The function is strictly convex when the inequality is strict.

Convex functions

Caracterization of convex functions

Sum of convex functions are also convex.
A differentiable function of one variable is convex on an interval iff it lies above all of its tangents: $f(x) \geq f(y) + f'(y)(x-y)$ .
A differentiable function of several variables is convex on a compact iff it lies above its linearization: $f(v) \geq f(\linparamv) + \nabla f(\linparamv)^{\top}(v-\linparamv)$
A twice differentiable function of one variable is convex on an interval iff its second derivative is non-negative.