An Optimal Control View of LoRA and Binary Controller Design for Vision Transformers

While recent advancements in model fine-tuning predominantly emphasize the utilization of low-rank adaptation (LoRA), we propose an alternative approach centered on reducing the precision of adaptation matrices. In particular, we depart from the common viewpoint that considers adaptation matrices solely as weight differences, and reinterpret them as "control variables'' to perturb pre-trained ViT systems. This new perspective enables the establishment of a control-oriented framework, facilitating the exploration of optimal controls guided by the Pontryagin Maximum Principle. Furthermore, we demonstrate that for bounded control sets such as hypercubes, the optimal controls often take on boundary values, leading naturally to a binary controller design. Theoretical analysis reveals that employing a binary control strategy achieves the same reachable state as its full-precision counterpart in the continuous idealisation of deep residual structures, a finding corroborated by later empirical investigations. Our studies further indicate that the controller's rank holds greater significance than its precision. As such, opting for low-precision yet high-rank controls is demonstrated to obtain better performance for practical vision tasks.