Strong convexity often allows for optimization algorithms that converge very quickly to an ε-optimum (rf. FISTA and NESTA). This post will cover some fundamentals of strongly convex functions.

Convexity

For a convex function, f: \mathbf{R} \to \mathbf{R} and \forall \gamma \in [0, 1],

 f(tx + (1-t)y) \le tf(x) + (1-t)f(y)

Strict convexity:

For a strictly convex function,

 f(tx + (1-t)y) < tf(x) + (1-t)f(y)

Geometrically, the definition of convexity implies that all points on any straight line connecting any two points in a convex set also lie in the set. Strict convexity excludes linear and affine functions, or functions with linear/affine subsets in their boundaries. (How would this extend to non-Euclidean geometries?)

α-strong convexity

A function is α-strongly convex with respect to a norm ||.|| if, for α > 0

f(tx + (1-t)y) < tf(x) + (1-t)f(y) - \frac{\alpha}{2} ||x - y||^2

Alternatively,

(\nabla{f(x)} - \nabla{f(y)})(x - y) \ge \alpha ||x - y||^2

or,

f(y) \ge f(x)+ \nabla{f(x)}(y - x) + \frac{\alpha}{2} ||x - y||^2

Strongly convexity extends strict convexity. For twice-differentiable functions, this implies that \nabla^2f(x) \ge \alpha. As Bubeck explains [1], strongly convex functions speed up convergence of first-order methods. Larger values of α imply larger gradient, and hence step size, when further away from the optimum.

Strong smoothness is another property of certain convex functions:

β-smoothness

A function is β-smoothly convex with respect to a norm ||.|| if, for β > 0, [4]

f(y) \le f(x)+ \nabla{f(x)}(y - x) + \frac{\beta}{2} ||x - y||^2

This definition gives an lower bound on the improvement in one step of (sub)gradient descent [1]:

f(x - \frac{1}{\beta}\nabla{f(x)})- f(x) \le \frac{-1}{2 \beta} || \nabla{f(x)} ||^2

Alternatively, β-smoothly convex function [lemma 3.3, 1]:

f(x) - f(y) \le \nabla{f(x)}(y - x) - \frac{1}{2 \beta} ||\nabla{f(x)} - \nabla{f(y)}||^2

Strong/smooth duality

Under certain conditions, a-strong convexity and β-smoothness are dual notions. For now, we’ll state the result without discussion.

If f is a closed and convex function, then f is α-strongly convex function with respect to a norm ||.|| if and only if f* is 1/α-strongly smooth with respect to the dual norm ||.||* (corollary 7 in [4]).

References:

[1] S. Bubeck, Theory of Convex Optimization for Machine Learning, section 3.4
[2] Wikipedia, Convex function
[3] S. Boyd and G. Vandenberghe, Convex Optimization, section 9.1.2
[4] S. M. Kakade, S. Shalev-Schwartz, A. Tiwari. On the duality of strong convexity and strong smoothness: Learning applications and matrix regularization. Technical Report, 2009

 

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