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
“And without strong convexity, the inverse Hessian is undefined, the implicit function theorem doesn't apply, and the unique best response that the Stackelberg framing depends on doesn't exist.”Episode 025 — The Missing Gradient Term That Predicts Sycophancy in RLHF