## Do Loss functions need to be convex?

TL;DR – A convex loss function makes it easier to find a global optimum and to know when one is reached. Popular loss functions are convex because a local minimum of a convex function is a global minimum. Also, a strictly convex function has a single global minimum.

## How do you know if a loss function is convex?

If in the whole range it is positive then it is convex if it is negative then it is concave, if it can be both positive and negative (for some sub-range) then it is neither convex nor concave. Linear functions (with second order derivative zero) are both convex and concave.

**Which loss functions are not convex?**

In this paper, we review some important convex loss functions, including hinge loss, square loss, modified square loss, exponential loss, logistic regression loss, as well as some non-convex loss functions, such as sigmoid loss, φ-loss, ramp loss, normalized sigmoid loss, and the loss function of 2 layer neural network …

**What is convex cost function?**

▼ A convex function: given any two points on the curve there will be no intersection with any other points, for non convex function there will be at least one intersection. In terms of cost function with a convex type you are always guaranteed to have a global minimum, whilst for a non convex only local minima.

### Why logistic regression loss function is convex?

To prove for any log(x), 2nd derivative of log(x) is -1/x^2 which is concave. For f function being concave, -f is convex [basic theorem of convexity]. That means, – log(x) is convex, so is – log(1 – x). That is why we use this as our cost function during logistic regression.

### Is sigmoid loss convex?

The logarithm of sigmoid function is NOT CONVEX.

**Which loss function is convex?**

Fortunately, hinge loss, logistic loss and square loss are all convex functions.

**Is hinge loss function convex?**

Hinge loss is a convex upper bound on 0-1 loss.

## What are convex problems?

A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. Linear functions are convex, so linear programming problems are convex problems.

## Which is not convex loss function?

Simplest form I can think of is: log(x) as a non-convex loss function example. The 2nd derivative of it is −1×2. So say f(x)=log(w1x+w2).