Build Nesterov Param Update as one small, testable piece of Outer Optimizer with Nesterov Momentum.
By the end, you will know what this function owns, why the larger system needs it, how its mathematics works, and how to prove your code is correct.
Understand the idea first
Nesterov Param Update has one clear responsibility. Think of it as a small tool on a workbench: it should accept a well-defined input, perform exactly one job, and return an output the next step can trust. Keeping this boundary small is what lets a large DiLoCo Distributed Training system remain understandable.
This step exists because implement the server-side outer optimizer: state init, momentum buffer updates, nesterov parameter updates, and computing the outer pseudo-gradient from worker deltas. Later lessons assume this behavior already works, so correctness here removes uncertainty from everything downstream.
Build a mental picture
Imagine data moving through a row of small, labelled boxes. This lesson builds exactly one box. The label tells us what may enter, the implementation tells us what happens inside, and the return value tells the next box what it may safely expect.
Derive it carefully
The symbols below express the core relationship used in this part of the project. MathJax renders the equation so fractions, matrices, superscripts, and alignment remain readable.
- 1
Name every input and write down its shape, dtype, and legal range.
- 2
Express the transformation independently of the surrounding project.
- 3
Check the smallest normal case, an edge case, and an invalid case.
- 4
Only after those checks pass, connect the function to the next lesson.
Do not memorize the symbols. Ask what each symbol represents in code, what shape it has, and which axis is reduced.
Turn the idea into code
Start with the contract, implement the smallest correct behavior, and use tests before optimizing. Reveal the reference only after making a real attempt.
def nesterov_param_update(*args, **kwargs):
"""Implement Nesterov Param Update."""
# TODO: follow the contracts and checks on this page.
raise NotImplementedError
Prove it works
Use the smallest representative input and compare exact values.
Try empty, full, boundary, masked, or single-item input.
Confirm whether inputs should stay unchanged and repeated calls are independent.
Common mistakes
- Changing the input in place when later code expects it to remain unchanged.
- Returning the right values with the wrong shape or dtype.
- Testing only the happy path and missing empty, full, masked, or boundary inputs.