Is Adaptation of Large Models Linear? Specialization and Local Geometry of Fine-Tuned Transformers

LoRA, task arithmetic, activation steering, random search around pretrained weights — all these methods rest on one tacit assumption: around a trained checkpoint, the model’s behavior is almost linear, and one can successfully adapt such a model simply by moving in the right direction. We asked ourselves which of these linear structures in networks actually exist, at what scale, and how long they live.

The answer turned out to be twofold. Along a single direction everything works well: the change in the loss function is essentially the projection of the perturbation onto the gradient, and that is precisely why random search works even in very high dimensions (we have a theorem for this). But take two updates, and the order in which they are applied starts to affect the result, while the useful basis in weight space rotates beyond recognition rather quickly.

I will talk about two of our recent works, show in which cases full fine-tuning can be replaced by small interventions, and why even a solution that already exists in the model cannot always be found by simple methods.