How will you check whether the given function is strictly concave or strictly convex?
- f is concave if and only if H(x) is negative semidefinite for all x ∈ S.
- if H(x) is negative definite for all x ∈ S then f is strictly concave.
- f is convex if and only if H(x) is positive semidefinite for all x ∈ S.
- if H(x) is positive definite for all x ∈ S then f is strictly convex.
Is logistic regression always convex?
4.5. The method most commonly used for logistic regression is gradient descent. Gradient descent requires convex cost functions. This is because the logistic function isn’t always convex. The logarithm of the likelihood function is however always convex.
How do you show something is convex?
Theorem 1. A function f : Rn → R is convex if and only if the function g : R → R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y ∈ Rn. (The domain of g here is all t for which x + ty is in the domain of f.) Proof: This is straightforward from the definition.
How to determine if a function is concave or convex?
A twice differentiable function is concave up wherever it’s second derivative is positive, and is concave down wherever it is negative. This is known as the concavety test. An inflection point occurs wherever the concavety of changes from up to down or down to up (the second derivative changes sign.)
Is the function your + ∗ 2 convex or concave?
Your function is clearly not convex nor concave on ( R + ∗) 2 but you could search more restricted sets if needed… Here is a picture (from below) of your function (convex near y = 0 and concave when y becomes larger at least in the x=y direction, in the x=-y direction it looks convex…) : so y 2 1 + x y is not convex as long as y ≠ 0.
How to extend the notions of concavity and convexity?
To extend the notions of concavity and convexity to functions of many variables we first define the notion of a convex set. (1−λ) x + λ x ‘ ∈ S whenever x ∈ S, x ‘ ∈ S, and λ ∈ [0,1]. We call (1 − λ) x + λ x ‘ a convex combination of x and x ‘.
Can a symmetric result hold for a convex function?
As for a function of a single variable, a symmetric result holds for convex functions. Like the result for functions of a single variable, it is used to show that stationary points are global maximizers of concave functions and global minimizers of convex functions.