GIFT uses the Fisher/K-FAC object only to define communication coordinates and explicitly, repeatedly denies being a natural-gradient or K-FAC optimizer

Although GIFT builds its gradient transform on the same Fisher/K-FAC object as Amari's natural gradient (claim-gift-isotropy-transform-derived-from-fisher-kfac), the paper marks a sharp boundary against being read as a natural-gradient or K-FAC optimizer, and does so more than once. In its background section: "This paper does not use K-FAC as an optimizer and does not apply the natural-gradient update. We use the K-FAC block only as a tractable local metric for defining communication coordinates. The optimizer, model architecture, and distributed training structure remain unchanged; only the coordinate system used during low-precision gradient communication is modified." In related work, positioning itself against the natural-gradient literature: "GIFT does not change the optimizer into a second-order method; instead, it uses local geometry to transform gradients into communication coordinates that are more amenable to FP8 quantization... GIFT is complementary to both prior communication-compression methods and geometry-aware optimization methods." And plainly: "GIFT is not a K-FAC optimizer or a natural-gradient method; it uses K-FAC-style local geometry only to improve the fidelity of low-precision gradient communication."

So the relationship is precise, not loose. GIFT and Amari share the identical underlying curvature object (the Fisher information matrix, K-FAC-approximated), but deploy it for two different operations: Amari redirects the optimization step (natural gradient descent), while GIFT only re-coordinates a gradient vector before lossy compression, leaving the optimizer's actual update untouched. This boundary is what keeps observation-gradient-geometry-shared-object-across-two-of-three from overclaiming "GIFT does natural gradient."