Shortcuts in the Tail: Debiasing via Post-Hoc Spectral Compression of Fine-Tuning Updates

Fine-tuning a language model for a target task routinely degrades capabilities the training data never explicitly threatened. We study this phenomenon, known as catastrophic forgetting, and propose a post-hoc repair solution that uses only the pretrained checkpoint $W_{\mathrm{base}}$ and its fine-tuned descendant $W_{\mathrm{ft}}$. The goal is not merely to revert the model toward the base checkpoint, but to recover capabilities damaged by fine-tuning while preserving both the target-task gains and any beneficial held-out improvements. We introduce DG-Hard, a checkpoint-only spectral repair method for the fine-tuning update $Δ= W_{\mathrm{ft}} - W_{\mathrm{base}}$. DG-Hard treats $Δ$ as a low-rank task-aligned signal embedded in an IID-like noise residual that gradient descent has no incentive to remove, and applies the Donoho-Gavish hard singular-value threshold to each weight-delta matrix, keeping the structured high-energy part of the update and removing the spectral bulk. This reduces repair to a closed-form SVD filtering step requiring no data-dependent tuning. A central difficulty is evaluation: average accuracy hides per-benchmark failures, while naive recovery scores reward models that simply revert toward the base. We therefore introduce a partition-conditional metric that separately tracks healing, preservation, non-damage, and target-task retention. Across $14$ (model, task) settings and nine cross-domain held-out benchmarks, DG-Hard achieves the strongest balanced repair among post-hoc baselines. DG-Hard also restores safety alignment degraded by benign fine-tuning on three independent safety axes, despite using no alignment data. These results suggest that part of fine-tuning-induced capability loss is not an unavoidable consequence of specialization, but a removable spectral residue in the weight update itself.

Parameter-efficient finetuning methods based on spectral decomposition have enabled progress in Continual Learning. In this paper we introduce TailLoR, which utilizes the singular bases U and V of the pre-trained weights as a fixed reference frame to learn a low-rank update applied to the singular value matrix. A soft spectral penalty discourages updates aligned with dominant singular directions, reducing interference while routing fine-grained adaptation into the highly flexible, long-tail spectral coordinates.

Every supervised fine-tuning run forces the same chain of decisions, such as learning rate, batch size, LoRA or full fine-tuning, how many epochs, which optimiser, and what data to feed the model. Each of these is typically rediscovered from scratch for every new model and dataset. Here we measure them under one instrument: a sweep that varies one lever at a time, and spans dense and mixture-of-experts models in two families (Qwen3 and Llama), on four real-world customer SFT datasets, for both LoRA and full fine-tuning. These datasets give a controlled testbed: each task carries an evaluation built with the customer, and its training data is produced by iterative supervised fine-tuning that refines model outputs until they pass that evaluation, so the supervised target is internally consistent and the task judge we report against is the criterion the data was built to satisfy. We ask how the optimal learning rate and batch size move with model scale, family, and data, and whether one selection rule transfers across them; what LoRA trades against full fine-tuning, and how its rank and alpha set what the adapter can learn; whether validation loss (or other metrics, such as loss landscape flatness) faithfully ranks downstream quality; whether post-training gains scale with model size and data volume, on a model ladder extended through mixtures-of-experts to 235B parameters; how many epochs to train before general instruction-following erodes; and whether a geometry-aware optimiser improves on AdamW. Each recommendation is paired with a measure of its uncertainty.