Large language models (LLMs) are revolutionizing how we interact with technology, but their massive memory requirements present a significant hurdle for training and deployment. Imagine trying to cram the entire Library of Congress onto a single bookshelf – that's the scale of the problem. Existing solutions like data and model parallelism, gradient checkpointing, and offloading help, but often fall short, especially with limited hardware. Parameter-Efficient Fine-Tuning (PEFT) techniques like LoRA offer some relief by focusing on adapting pre-trained models, but they often require a full-rank warm-up and can’t match the performance of full fine-tuning. GaLore takes a different approach: approximating optimizer states instead of model parameters. This allows full-parameter learning with better memory efficiency. However, GaLore alone hasn’t quite caught up to the performance of traditional optimizers. Enter Natural GaLore, a new technique that builds on GaLore by incorporating second-order information about the loss landscape. Think of it as giving the optimizer a topographic map of the terrain it needs to navigate, rather than just a compass. This 'map,' derived from the Empirical Fisher Information Matrix, helps Natural GaLore converge faster and achieve better performance without any extra memory overhead. In tests with LLaMA models ranging from 60 million to 1.1 billion parameters, Natural GaLore consistently outperformed standard GaLore, achieving lower perplexity scores – a key measure of language model accuracy. Furthermore, when fine-tuning the TinyLlama 1.1B model for complex function-calling tasks within the TinyAgent framework, Natural GaLore achieved a remarkable 83.09% accuracy, beating 16-bit LoRA and even surpassing GPT-4-Turbo by 4%, all while using 30% less memory. This is a huge leap forward in making LLMs more accessible and sustainable. By optimizing how these models learn, Natural GaLore paves the way for even more powerful and efficient AI on less powerful hardware, opening doors for wider adoption and innovation across various fields. This is just the beginning. Future research into low-memory projection matrices and broader applications of Natural GaLore promises even greater gains in memory efficiency, making powerful AI more accessible than ever before.