The Affine Divergence: Aligning Activation Updates Beyond Normalisation

A systematic mismatch exists between mathematically ideal and effective activation updates during gradient descent. As intended, parameters update in their direction of steepest descent. However, activations are argued to constitute a more directly impactful quantity to prioritise in optimisation, as they are closer to the loss in the computational graph and carry sample-dependent information through the network. Yet their propagated updates do not take the optimal steepest-descent step. These quantities exhibit non-ideal sample-wise scaling across affine, convolutional, and attention layers. Solutions to correct for this are trivial and, entirely incidentally, derive normalisation from first principles despite motivational independence. Consequently, such considerations offer a fresh and conceptual reframe of normalisation's action, with auxiliary experiments bolstering this mechanistically. Moreover, this analysis makes clear a second possibility: a solution that is functionally distinct from modern normalisations, without scale-invariance, yet remains empirically successful, outperforming conventional normalisers across several tests. This is presented as an alternative to the affine map. This generalises to convolution via a new functional form, "PatchNorm", a compositionally inseparable normaliser. Together, these provide an alternative mechanistic framework that adds to, and counters some of, the discussion of normalisation. Further, it is argued that normalisers are better decomposed into activation-function-like maps with parameterised scaling, thereby aiding the prioritisation of representations during optimisation. Overall, this constitutes a theoretical-principled approach that yields several new functions that are empirically validated and raises questions about the affine + nonlinear approach to model creation.