Cost function

Definition

For a single piece of data

The cost function associated with a given machine learning problem is a function that takes as input a guess for the function and the observed output and the predicted function value and then associates to that a number measuring how far the observed output is from the predicted function value.

• For prediction problems associated with continuous variables, both the predicted value and the actual value are continuous variables. The cost function is a function $C(u,v)$ of two variables $u,v$ (the predicted value and actual value) satisfying the following conditions:

• • $C(u,u)=0$ for all $u\in \mathbb {R}$
  • For $u\leq v\leq w$, $C(u,v)\leq C(u,w)$ and $C(v,w)\leq C(u,w)$

The cost function need not satisfy the triangle inequality; in fact, typical cost functions penalize bigger errors superlinearly.

• For prediction problems associated with discrete variables, the predicted value is a probability and the actual value is simply a discrete value (0 or 1). The cost function is a function $C(p,v)$ of two variables (the predicted probability and actual value) satisfying the following conditions:
  • $C(1,1)=0$
  • $C(0,0)=0$
  • For $p\leq q$, $C(q,1)\leq C(p,1)$
  • For $p\leq q$, $C(p,0)\leq C(q,0)$