BoHA: Blockwise Hadamard Product Adaptation for Parameter-Efficient Fine-Tuning

Parameter-efficient fine-tuning (PEFT) of large language models trains a small task-specific parameter set while keeping the pretrained model frozen. The dominant Low-Rank Adaptation (LoRA) family makes this trade-off practical; however, evaluations under the same parameter budget assess single-task accuracy. In sequential adaptation settings, such evaluations should also measure how well performance on the first-stage task is retained after subsequent fine-tuning. To address this gap, we introduce BoHA, a blockwise $W_0$-coupled Hadamard product adapter that treats spatial support as an explicit design axis. BoHA partitions the frozen weight $W_0$ into a $b{\times}b$ grid and learns an independent low-rank Hadamard product factor in each block, preserving a matched LoRA-equivalent total rank with adapter-free merged inference. On a synthetic target, BoHA at per-block rank $r_b{=}1$ exactly reconstructs an update that requires rank $b^2$ under the global $W_0$-coupled Hadamard parameterization. Across Llama-3.2-1B/3B, Mistral-7B, and Gemma-2-9B on commonsense and arithmetic reasoning tasks, BoHA outperforms LoRA across all matched-budget single-task averages and remains competitive with the strongest Hadamard baseline. On a Llama-3.2-3B commonsense $\to$ arithmetic continual-learning diagnostic, BoHA retains $57.66\%$ first-stage accuracy and exceeds the $W_0$-free additive-control mean by $15.23\%$ under matched second-stage plasticity. These results demonstrate that blockwise $W_0$-coupled Hadamard adaptation is a competitive PEFT design choice when retention under sequential adaptation is part of the objective.