Plasticity vs. Rigidity: The Impact of Low-Rank Adapters on Reasoning on a Micro-Budget

Recent advances in mathematical reasoning typically rely on massive scale, yet the question remains: can strong reasoning capabilities be induced in small language models ($\leq1.5\text{B}$) under extreme constraints? We investigate this by training models on a single A40 GPU (48GB) for under 24 hours using Reinforcement Learning with Verifiable Rewards (RLVR) and Low-Rank Adaptation (LoRA). We find that the success of this ``micro-budget" regime depends critically on the interplay between adapter capacity and model initialization. While low-rank adapters ($r=8$) consistently fail to capture the complex optimization dynamics of reasoning, high-rank adapters ($r=256$) unlock significant plasticity in standard instruction-tuned models. Our best result achieved an impressive 40.0\% Pass@1 on AIME 24 (an 11.1\% absolute improvement over baseline) and pushed Pass@16 to 70.0\%, demonstrating robust exploration capabilities. However, this plasticity is not universal: while instruction-tuned models utilized the budget to elongate their chain-of-thought and maximize reward, heavily math-aligned models suffered performance collapse, suggesting that noisy, low-budget RL updates can act as destructive interference for models already residing near a task-specific optimum.

Beyond Uniform Query Distribution: Key-Driven Grouped Query Attention

The Transformer architecture has revolutionized deep learning through its Self-Attention mechanism, which effectively captures contextual information. However, the memory footprint of Self-Attention presents significant challenges for long-sequence tasks. Grouped Query Attention (GQA) addresses this issue by grouping queries and mean-pooling the corresponding key-value heads - reducing the number of overall parameters and memory requirements in a flexible manner without adversely compromising model accuracy. In this work, we introduce enhancements to GQA, focusing on two novel approaches that deviate from the static nature of grouping: Key-Distributed GQA (KDGQA) and Dynamic Key-Distributed GQA (DGQA), which leverage information from the norms of the key heads to inform query allocation. Specifically, KDGQA looks at the ratios of the norms of the key heads during each forward pass, while DGQA examines the ratios of the norms as they evolve through training. Additionally, we present Perturbed GQA (PGQA) as a case-study, which introduces variability in (static) group formation via subtracting noise from the attention maps. Our experiments with up-trained Vision Transformers, for Image Classification on datasets such as CIFAR-10, CIFAR-100, Food101, and Tiny ImageNet, demonstrate the promise of these variants in improving upon the original GQA through more informed and adaptive grouping mechanisms: specifically ViT-L experiences accuracy gains of up to 8% when utilizing DGQA in comparison to GQA and other variants. We further analyze the impact of the number of Key-Value Heads on performance, underscoring the importance of utilizing query-key affinities.