Definition

A function ff is β\beta smooth if, for all 𝐱,𝐲\mathbf{x},\mathbf{y}:

||∇f(𝐱)−∇f(𝐲)||2≤β||𝐱−𝐲||2||\nabla f(\mathbf{x})-\nabla f(\mathbf{y})||_2 \leq \beta ||\mathbf{x}-\mathbf{y}||_2

For scalar valued function ff, equivalent to f′′(x)≤βf''(x) \leq \beta.

For scalar functions, a twice-differentiable function ff is α-strongly convex and β\beta-smooth if for all xx,

α≤f′′(x)≤β\alpha \leq f''(x) \leq \beta

References:

  1. https://arxiv.org/pdf/1405.4980.pdf