The LoRA method has achieved notable success in reducing GPU memory usage by applying low-rank updates to weight matrices. Yet, one simple question remains: can we push this reduction even further? Furthermore, is it possible to achieve this while improving performance and reducing computation time? Answering these questions requires moving beyond the conventional weight-centric approach. In this paper, we present a state-based fine-tuning framework that shifts the focus from weight adaptation to optimizing forward states, with LoRA acting as a special example. Specifically, state-based tuning introduces parameterized perturbations to the states within the computational graph, allowing us to control states across an entire residual block. A key advantage of this approach is the potential to avoid storing large intermediate states in models like transformers. Empirical results across multiple architectures—including ViT, RoBERTa, LLaMA2-7B, and LLaMA3-8B—show that our method further reduces memory consumption and computation time while simultaneously improving performance. Moreover, as a result of memory reduction, we explore the feasibility to train 7B/8B models on consumer-level GPUs like Nvidia 3090, without model quantization. The code is available at an anonymous GitHub repository

We investigate the universal approximation property (UAP) of transformer-type architectures, providing a unified theoretical framework that extends prior results on residual networks to models incorporating attention mechanisms. Our work identifies token distinguishability as a fundamental requirement for UAP and introduces a general sufficient condition that applies to a broad class of architectures. Leveraging an analyticity assumption on the attention layer, we can significantly simplify the verification of this condition, providing a non-constructive approach in establishing UAP for such architectures. We demonstrate the applicability of our framework by proving UAP for transformers with various attention mechanisms, including kernel-based and sparse ones. The corollaries of our results either generalize prior works or establish UAP for architectures not previously covered. Furthermore, our framework offers a principled foundation for designing novel transformer architectures with inherent UAP guarantees, including those with specific functional symmetries. We propose examples to illustrate these insights.

In contrast to the prevailing interpretation of Low-Rank Adaptation (LoRA) as a means of simulating weight changes in model adaptation, this paper introduces an alternative perspective by framing it as a control process. Specifically, we conceptualize lightweight matrices in LoRA as control modules tasked with perturbing the original, complex, yet frozen blocks on downstream tasks. Building upon this new understanding, we conduct a thorough analysis on the controllability of these modules, where we identify and establish sufficient conditions that facilitate their effective integration into downstream controls. Moreover, the control modules are redesigned by incorporating nonlinearities through a parameter-free attention mechanism. This modification allows for the intermingling of tokens within the controllers, enhancing the adaptability and performance of the system. Empirical findings substantiate that, without introducing any additional parameters, this approach surpasses the existing LoRA algorithms across all assessed datasets and rank configurations.