Definition
Plain language
A math result that lets you compute how an answer would shift without re-running all the steps that produced it.
As stated in the literature
A calculus theorem providing conditions under which an implicit equation defines a function locally; in iterative inference, it allows gradient computation through a fixed point without backpropagating through the iterations.
Also called: implicit differentiation, implicit gradient, implicit gradients
Why it matters: It's the mathematical foundation that makes deep equilibrium and other fixed-point models trainable with reasonable memory cost.
For example, if a fixed-point equation defines y as a function of x, the theorem tells you the derivative dy/dx in terms of the equation's partial derivatives, without ever solving for y explicitly.
Heard on the show
“The implicit function theorem gives you another option.”Episode 041 — When the Iteration Teaches the Model to Skip the Iteration