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Vincent Chen is a student at Stanford University studying Computer Science. He is also a Research Assistant at the Stanford AI Lab.
It’s not entirely clear what level of mathematics is necessary to get started in machine learning, especially for those who didn’t study math or statistics in school.
In this piece, my goal is to suggest the mathematical background necessary to build products or conduct academic research in machine learning. These suggestions are derived from conversations with machine learning engineers, researchers, and educators, as well as my own experiences in both machine learning research and industry roles.
To frame the math prerequisites, I first propose different mindsets and strategies for approaching your math education outside of traditional classroom settings. Then, I outline the specific backgrounds necessary for different kinds of machine learning work, as these subjects range from high school-level statistics and calculus to the latest developments in probabilistic graphical models (PGMs). By the end of the post, my hope is that you’ll have a sense of the math education you’ll need to be effective in your machine learning work, whatever that may be!
To preface the piece, I acknowledge that learning styles/frameworks/resources are unique to a learner’s personal needs/goals— your opinions would be appreciated in the discussion on HN!
A Note on Math Anxiety
It turns out that a lot of people — including engineers — are scared of math. To begin, I want to address the myth of “being good at math.”
The truth is, people who are good at math have lots of practice doing math. As a result, they’re comfortable being stuck while doing math. A student’s mindset, as opposed to innate ability, is the primary predictor of one’s ability to learn math (as shown by recent studies).
To be clear, it will take time and effort to achieve this state of comfort, but it’s certainly not something you’re born with. The rest of this post will help you figure out what level of mathematical foundation you need and outline strategies for building it.
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As soft prerequisites, we assume basic comfortability with linear algebra/matrix calculus (so you don’t get stuck on notation) and introductory probability. We also encourage basic programming competency, which we support as a tool to learn math in context. Afterwards, you can fine-tune your focus based on the kind of work you’re excited about.
How to Learn Math Outside of School I believe the best way to learn math is as a full-time job (i.e. as a student). Outside of that environment, it’s likely that you won’t have the structure, (positive) peer pressure, and resources available in the academic classroom.
To learn math outside of school, I’d recommend study groups or lunch and learn seminars as great resources for committed study. In research labs, this might come in the form of a reading group. Structure-wise, your group might walk through textbook chapters and discuss lectures on a regular basis while dedicating a Slack channel to asynchronous Q&A.
Culture plays a large role here — this kind of “additional” study should be encouraged and incentivized by management so that it doesn’t feel like it takes away from day-to-day deliverables. In fact, investing in peer-driven learning environments can make your long-term work more effective, despite short-term costs in time.
Math and Code
Math and code are highly intertwined in machine learning workflows. Code is often built directly from mathematical intuition, and it even shares the syntax of mathematical notation. In fact, modern data science frameworks (e.g. NumPy) make it intuitive and efficient to translate mathematical operations (e.g. matrix/vector products) to readable code.
I encourage you to embrace code as a way to solidify your learning. Both math and code depend on precision in understanding and notation. For instance, practicing the manual implementation of loss functions or optimization algorithms can be a great way to truly understanding the underlying concepts.
As an example of learning math through code, let’s consider a practical example: implementing backpropagation for the ReLU activation in your neural network (yes, even if Tensorflow/PyTorch can do this for you!). As a brief primer, backpropagation is a technique that relies on the chain rule from calculus to efficiently compute gradients. To utilize the chain rule in this setting, we multiply upstream derivatives by the gradient of ReLU.
To begin, we visualize the ReLU activation, defined:
To compute the gradient (intuitively, the slope), you might visualize a piecewise function, denoted by the indicator function as follows:
NumPy lends us helpful, intuitive syntax here— our activation function (blue curve) is interpretable in code, where x is our input and relu is our output:
relu = np.maximum(x, 0)
The gradient (red curve) follows, where grad describes the upstream gradient:
grad[x < 0] = 0
Without first deriving the gradient yourself, this line of code might not be self-explanatory. In our line of code, set set all values in the upstream gradient
(grad) to 0 for all elements that satisfy the condition,