Breaking the Scale Barrier: One-Shot Knowledge Transfer via Frequency Transform

Transferring pre-trained knowledge from a source model to a target model of a different architectural size is a key challenge for flexible and efficient model scaling. However, current parameter-space methods treat Small-to-Large (S2L) and Large-to-Small (L2S) scaling as separate, incompatible problems, focusing on parameter synthesis and selection, respectively. This fragmented perspective has resulted in specialized tools, hindering a unified, bidirectional framework. In this paper, we propose BoT (Bidirectional knowledge Transfer), the first size-agnostic framework to unify S2L and L2S scaling. Our core insight is to treat model weights as continuous signals, where models of different sizes represent distinct discretizations of the transferable knowledge. This multi-resolution perspective directly casts S2L and L2S scaling as the signal processing operations of upsampling and downsampling, naturally leading to the adoption of the Discrete Wavelet Transform (DWT) and its Inverse (IDWT). BoT leverages the recursive nature of wavelets, using the decomposition level as a dynamic scaling factor to bridge disparate model sizes in a parameter-free and computationally efficient manner. Extensive experiments on DeiT, BERT, and GPT demonstrate significant pre-training FLOPs savings (up to 67.1% for S2L, 52.8% for L2S) and state-of-the-art performance on benchmarks like GLUE and SQuAD.

Finetuning pretrained models occurs in a low-dimensional subspace of the full parameter space. Prior work has focused on characterizing this optimization subspace, but largely ignored the complementary question: why do certain directions remain unexplored during finetuning? Are these stable directions irrelevant to downstream tasks, or do they already encode task-relevant structure that requires no further adjustment? Answering this question is central to understanding how pretrained knowledge transfers. Through systematic spectral analysis across vision and language models, we show that the leading singular vectors of pretrained weight matrices remain highly stable under finetuning and are shared across unrelated downstream tasks, revealing that pretraining establishes a reusable spectral coordinate system. Models pretrained on larger datasets exhibit greater spectral stability under distribution shift or task change, directly linking pretraining scale to geometric transferability. Motivated by these findings, we propose a parameter-efficient method that freezes pretrained singular vectors and optimizes only leading spectral coefficients, achieving competitive performance on GLUE with 0.2% trainable parameters. Our results reveal that the stable directions encode transferable structure rather than irrelevant noise: successful pretraining discovers spectral bases that downstream tasks inherit and operate within.

Model families are trained size by size. Can a pretrained large model instead be converted into a smaller sibling? We study the 1.4B->410M conversion in Pythia end to end. Representations align strongly across sizes (ridge R^2=0.84); parameters align weakly. Dense weight projection is destructive; a bit-exact control places the fault in basis mixing, which breaks rotary, per-head, GELU, and LayerNorm structure. Residuals after the best-fit linear operator carry no learnable or transferable signal under shuffle controls, so conversion value lives in initialization. Matched-budget continued pre-training separates two independent levers: least-squares compensation (function lever, best zero-shot) and variance-preserving rescale (dynamics lever, best endpoints). Placement follows the architecture: compensation is well-posed exactly where no normalization sits between cut and read; norm-fronted paths take rescale. Compensation is a low-budget, token-efficiency win, not a universal one. At 30M tokens it beats the best subcloning variant on a width-reduced pair (84.0+-1.8 vs. 89.7+-3.7, 3/3 seeds) and a held-out depth-reduced pair (109.3 vs. 117.9, 3/3 seeds). Selection given the same activation statistics recovers under half of that gap (3/3 seeds): the gain is the re-fit, not the information. At 33x the budget the two reach parity (40.3+-0.3 vs. 40.3+-0.5, 3 seeds), both far ahead of from-scratch, which transfer always beats (up to 18x at low budget, narrowing at convergence and at the largest scale). At ~5x the donor scale (6.9B->1.4B) stacking both levers over-corrects, consistent with an ill-conditioned compensation solve at large width, pointing to dimension-aware regularization as a fix. The init also beats structured pruning plus distillation, the standard pipeline, at matched budget, and improves further combined with it. Code, checkpoints, and the frozen eval corpus are released.