| Calculus essentially discusses how things change smoothly and it has a very nice mechanism for talking about smooth changes algebraically. A system which is at an optimum will, at that exact point, be no longer increasing or decreasing: a metal sheet balanced at the peak of a hill rests flat. Many problems in ML are optimization problems: given some set of constraints, what choices of unknown parameters minimizes error? This can be very hard (NP-hard) in general, but if you design your situation to be "smooth" then you can use calculus and its very nice set of algebraic solutions. You also need multivariate calculus because typically while you're only trying to minimize "error", you do so by changing many, many parameters at once. This means that you've got to talk about smooth changes in a high-dimensional space. -- The other side of calculus is integration which talks about "measuring" how big things are. Most of probability is discussing very generalized ratios: of the total, "how big is this piece" is analogous to "what are the odds this will happen". The general discussion of measure is complex and essentially the only tool to tackle it involves gigantic (infinite, really) sums of small, well-behaved pieces to form a complex whole. It just happens to turn out (and this is the big secret of calculus) that this machinery (integration) is dual to the study of smooth changes and you can knock them both out together. -- So ultimately, ML hinges upon being able to measure things (integration) and talk about how they change (derivation). Those two happen to be the same concept in a way and they are essentially what you study in calculus. |