Martens & Grosse's 2015 canonical K-FAC paper defines the Fisher it approximates as an expectation over the model's own predictive distribution — the true-Fisher convention, not real training labels

K-FAC's originating paper defines the quantity it approximates as "F = E[DθDθᵀ]" and specifies exactly which expectation this is: "the expectation is taken with respect to the data distribution Qₓ over inputs x, and the model's predictive distribution P_{y|x}(θ) over y." (The next sentence substitutes the training distribution Q̂ₓ for Qₓ in practice — a substitution over inputs x only; y stays model-sampled.) In the vocabulary Kunstner, Balles & Hennig (2019) later formalize, that is the true-Fisher convention: y is drawn from the model's own output distribution, not taken from the training set. K-FAC's name-giving paper defines the object it approximates as requiring model-sampled y — not real labels.

Kunstner et al. corroborate this reading from the other side, noting in passing that "KFAC [Martens and Grosse, 2015]... [has] been re-implemented by third parties using the empirical Fisher" — implying the original K-FAC formulation used the true Fisher, and that empirical-Fisher K-FAC variants are a later substitution made by other authors, not part of Martens & Grosse's own construction.

This convention is the standard Amari's natural gradient and canonical K-FAC both share, and it is the baseline claim-gift-g-factor-matches-empirical-fisher-not-true-fisher-convention checks GIFT's own G = E[δδ⊤] factor against.