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With the increase in the number of parameters in large language models, the training process increasingly demands larger volumes of GPU memory. A significant portion of this memory is typically consumed by the optimizer state. To overcome this challenge, recent approaches such as low-rank adaptation (LoRA), low-rank gradient projection (GaLore), and blockwise optimization (BAdam) have been proposed. However, in all these algorithms, the *effective rank of the weight updates remains low-rank*, which can lead to a substantial loss of information from the gradient. This loss can be critically important, especially during the pre-training stage. In this paper, we introduce FRUGAL (**F**ull-**R**ank **U**pdates with **G**r**A**dient sp**L**itting), a new memory-efficient optimization framework. FRUGAL leverages gradient splitting to perform low-dimensional updates using advanced algorithms (such as Adam), while updates along the remaining directions are executed via state-free methods like SGD or signSGD. Our framework can be integrated with various low-rank update selection techniques, including GaLore and BAdam. We provide theoretical convergence guarantees for our framework when using SGDM for low-dimensional updates and SGD for state-free updates. Additionally, our method consistently outperforms concurrent approaches, achieving state-of-the-art results in pre-training and fine-tuning tasks while balancing memory efficiency and performance metrics.