MSE Min

Aathreya Kadambi
February 19, 2025

This is just a short post about one of the homework problems on the first problem set.

Do all even degree polynomials have a global minimum?

The answer is yes, as you might guess if you look at pictures of many even degree polynomials. Let’s take a look at some:

Some reasoning you could have is that all even degree polynomials seem to go to positive infinity, as you go to the left or the right. As such, somewhere in the middle, they have to reach a minimum! If they didn’t, they would have to go to $-\infty$ in the middle, but that wouldn’t work since they have to go to positive infinity as the extremes.

This is a vague sort of reasoning, but it’s essentially the right idea! It misses one slight nuance though, which is that it relies on our polynomials being continuous (this is how we can make our argument that the function can’t both go to $-\infty$ and $+\infty$). With that information, we can actually guarantee a global minimum. As long as our function is bounded below and continuous, it must have a minimum.

Why is this important? Well, at the very least, it tells us that looking for a minimum with any of our $\ell^p$ norms won’t be futile, since all norms are bounded below by zero! In particular, MSE is continuous and bounded below, so it has a global minimum. This will be good to know for the future when use MSE time and time again as a loss function.

built with Astro, taking heavy stylistic inspiration from Kelly Tran's website!