Parameter-efficient post-training reduces the number of trainable parameters, but still requires repeated end-to-end backpropagation through the frozen backbone. Every adaptation step therefore needs backward-capable hardware and must store or recompute activations.

We ask whether this repeated backward chain can be replaced by a one-time calibration. We introduce Local Credit Assignment (LoCA), a two-stage method for small-shift adaptation.

One probe backward pass fits a low-rank map at each transformer block from the final prediction error to a local hidden-state correction. LoCA then reuses these maps to form blockwise regression targets from forward activations and fits low-rank adapters with closed-form ridge solves.

No further backbone backward pass is required. We evaluate LoCA on five discriminative benchmarks with Qwen2.5 models from 0.5B to 14B.

In 16 of 25 reported task--scale comparisons, LoCA yields lower evaluation cross-entropy than the corresponding LoRA run. Its measured full-run GPU peak, including calibration, is 26--29% lower than LoRA's.

After calibration, its CPU steady-state memory is 36--52% lower and its per-pass time is 43--48% lower. A shared scale-normalized candidate set is reused across all tested Qwen2.5 sizes and on SmolLM2-1.7B.

LoCA thus amortizes global credit assignment into one calibration and enables later forward-only tuning when repeated backpropagation is impractical. The code associated with this paper is available here.