Coming back to this thread after checking some old notes from a similar contract. Before diving into the deep end, what kind of custom logic are you implementing in those PyTorch layers? Are we talking custom activation functions, or something more complex like a non-standard attention mechanism? The specific math you need usually depends on where the gradient flow is breaking. If the gradients are acting weird, you are likely hitting issues with Jacobian-vector products. Most folks skip over the chain rule for vectors and matrices, but it is basically the backbone of how autograd works under the hood. I would highly recommend picking up Mathematics for Machine Learning by Marc Peter Deisenroth Hardcover. It usually retails for about $55 and it bridges the gap between abstract theory and actual ML implementation better than most books. It covers the specific linear algebra and optimization concepts you are probably missing right now. Also, if you are working with high-dimensional data, you gotta understand eigendecomposition and singular value decomposition. If weights arent initialized correctly or operations are poorly scaled, gradients will either vanish or explode. It is worth checking out Introduction to Linear Algebra by Gilbert Strang 6th Edition too. It is a bit of an investment at nearly $90 but it is the gold standard for foundations. Knowing the why behind matrix operations makes debugging custom backprop way less of a headache.

@Reply #1 - good point! Calculus is vital, but I'm satisfied with how MIT Press Deep Learning Hardcover cleared up matrix ops for me. It saved my custom layers tbh.

Honestly, jumping into custom layers without a firm grasp of multivariable calculus is a recipe for silent errors. Ive seen many developers struggle with this when the gradients start acting up. If your foundations arent solid, the model becomes totally unreliable. Methodical verification is basically the only way to ensure the system actually works.

• Vector and matrix calculus, specifically the Jacobian
• The multi-dimensional Chain Rule for backpropagation
• Numerical stability and precision issues
• Probability and loss function derivations One thing to watch out for is broadcasting. PyTorch allows operations on mismatched shapes that might pass technically but are mathematically invalid for your specific layer logic. This leads to silent failures where the model trains but never converges properly. Tbh, I always sketch my tensor transformations on paper first. Its a boring safety step but prevents a lot of disaster later on...