Gradients on Programmer.ie: Modern AI programming

Models From First Principles 07: PACS — Building an Optimizer From Gradient Statistics

Sat, 08 Aug 2026 15:05:00 +0100

So far in Models From First Principles, every post has asked some version of the same question:

What should the model compute?

MR.Q gave us a scalar quality estimate.

EBT split one shared representation into Q, V and Policy.

SICQL made those heads explicit, replaceable components.

HRM introduced repeated computation over fast and slow latent states.

Tiny compressed iterative refinement into one recursive latent state.

Sat, 08 Aug 2026 13:51:00 +0100

Your model runs.

The loss is finite.

Nothing crashes.

And it still does not learn.

This is one of the most frustrating states in machine learning because there is no stack trace telling you what is wrong.

The program is valid Python.

The tensors have legal shapes.

The GPU is busy.

The optimizer is stepping.

And the model is useless.

This post is a systematic way to debug that situation.

Sat, 08 Aug 2026 12:55:00 +0100

In the previous post we treated tensor shapes as a debugging problem rather than a mathematical vocabulary exercise.

We are going to do the same thing with autograd.

If you use PyTorch for any serious amount of time, you eventually see errors like:

RuntimeError: element 0 of tensors does not require grad and does not have a grad_fn

or:

RuntimeError: Trying to backward through the graph a second time...

or worse: