What Is Function Approximation?

Functions are the heart and soul of applied mathematics. It's impossible to imagine a world without functions—they're everywhere! But sometimes, we need to approximate something we don't have a function, like an MDP. MDPs are complex systems of agents and states and decision-making processes but what if you wanted to get a closer approximation of that system? What if you wanted to know how likely an agent would make a particular decision? What if you wanted to know how much time they spent in each state? This is where function approximation comes in. Function approximation is all about selecting functions in a class that matches target functions so that you can better understand what's going on with your MDPs. It's helpful in applied mathematics and computer science because it gives us insight into complex systems like this one! To understand function approximation well, it's essential to know that in this term, the word "function" doesn't refer to an object-oriented programming function that takes a variable and provides a result. The word "function" refers to the mathematical use of a function, where a function matches one item in a data set to another single thing in another data set. The most common example is when we're trying to estimate a value using other known values. We might have an unknown height and weight for an individual, but we know their weight can be estimated with some accuracy if you know their age and gender. So when we say "estimating," we're not talking about estimation from one variable—we're talking about taking one variable and using it as an input for another variable. So when you're talking about function approximation, keep in mind that it's not just about functions; it's about taking one piece of information (like your age) and using it as an input for another part of the information (like your weight).