The Math Behind Machine Learning: Why Should We Care?

As a STEM major at New York University, I enjoy learning about a subject that 80% of our world wouldn't even dare touch: mathematics.

Simply put, mathematics is everywhere; it resides in the transactions we do at the bank, in the stats we gloss over for the World Cup, and even in the amount we choose to cook for a simple meal.

Math is also prevalent in many technical fields, forming the basis for a plethora of revolutionary, expandable applications. One such discipline, which represents the foundation for many products in accelerating our work and daily lives, is machine learning.

I'm sure that 'machine learning' is a term that you have heard at some point in your life, either in the midst of your academics or on the news proposing some futuristic (yet very viable) ideals like the 'AI job takeover'. On the surface, machine learning is the study of building mathematical models that 'learn' from data (which may contain multiple attributes) in order to predict some other attributes. This subject bodes high relevance to almost every field that comes to mind; for example, machine learning is used in cancer screening, where it uses the data it has on a patient obtained from clinical tests to give a proper diagnosis of cancer. You may not even realize it, but you are probably using it in some magnitude for your studies, career, or business (maybe even to view this blog post).

While machine learning is heavily employed today, its foundations are heavily underappreciated by many. Many are satisfied with just understanding how it 'does what it does' for our tasks; as a result, they fail to realize the beauty of mathematics behind its implementation. In fact, from my experience the majority of those who do possess some level of machine learning knowledge were unbothered to even learn the basics of pure math itself, such as linear algebra or optimization. Why should we bother to learn the math behind machine learning? Two driving motives come to mind.

Firstly, many hold the perspective that one 'does not need to learn math' to be good at machine learning, an opinion formed upon seeing large-scale, furnished models like Gemini that just seem to 'do the work for you'. Unfortunately, the reality is that the process taken to reach such levels of model robustness requires a lot of mathematics in itself. For example, when working with the messy, unfathomable block of cryptic information that we call data, the preprocessing steps that need to be taken (cleaning the data, identifying the best-fit distributions, choosing batch sizes or model types, e.g.) all require some base knowledge of math for proper execution. In other words, those who take the time to understand some sum of the mathematics behind machine learning are usually the ones who are the most efficient in creating the building blocks for model creation, especially those who are actively involved in the development of models.

Even for those who aren't as affiliated with the theoretical side of ML and are solely involved in using products that encapsulate these models, it is good to at least learn a bit of math behind the concept. Nowadays anyone can just take a model from some literature and deploy it for their purposes. Knowing some level of mathematics (especially if you are orthogonal to the subject) makes you stand out, which may be useful in an interview or job setting where you compete against a mass of applicants. Being able to not only understand 'what' ML does a task for you, but also 'how' it does this task on a higher level not only gives you a mathematical foundation of the 'now' empowering many industries today, but also showcases your ability to learn on a more abstract level. This will pay off for you in many ways. As a math major, I at least have bragging rights when I can explain the math behind stochastic gradient descent to a group of relatively uneducated math majors.

It would be wise to not take the 'magic' of machine learning for granted. The next time you decide to let ChatGPT do your calculus homework for you (everyone has done this, including me), perhaps you should pause and try to understand how such a tool is able to 'magically' give you the right answer within seconds.