Convex Function

A convex function is a function whose epigraph — the set of points lying on or above its graph — forms a convex set. Equivalently, a function f is convex if for any two points x and y in its domain and any t in [0,1], f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y). Geometrically, this means the line segment between any two points on the graph lies above or on the graph. Convexity is the property that makes Jensen's inequality work, and it is the structural reason why variational bounds are tight and why optimization landscapes in machine learning often have a single global minimum. A function is strictly convex if the inequality is strict for t in (0,1), which guarantees uniqueness of minimizers. The world of convex functions is a world where local information tells you everything about global structure — a world that approximate inference desperately wishes the brain lived in.