Large Language Models (LLMs) can perform many NLP tasks well, but fully fine-tuning them is expensive and requires a lot of memory. Parameter-Efficient Fine-Tuning (PEFT) methods such as LoRA reduce this cost by adding small low-rank updates to frozen model weights. However, these methods restrict the training to a limited subspace, which can sometimes reduce performance. For Small Language Models (SLMs), where efficiency gains matter even more, we introduce AdaGradSelect, an adaptive method that selects which transformer blocks to update based on gradients. Early observations showed that updating only the transformer blocks with the highest gradient norms can achieve performance close to full fine-tuning. Building on this insight, AdaGradSelect adaptively chooses which blocks to train. It uses a combination of Dirichlet-based sampling, which depends on how frequently blocks were updated in the past, and an epsilon-greedy exploration strategy. This lets the method explore different blocks in early training and gradually focus on the most important ones in later epochs. Experiments show that AdaGradSelect trains about 12 percent faster and uses 35 percent less GPU memory while delivering performance very close to full fine-tuning. On the GSM8K dataset, it outperforms LoRA (rank 256) by about 3 percent on average across models such as Qwen2.5-0.5B, LLaMA3.2-1B, and Phi4-mini-3.8B. It also achieves similar accuracy on the MATH dataset. Overall, AdaGradSelect provides a more effective and resource-efficient alternative to traditional fine-tuning methods.

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