Summary: Recent studies indicate that language models can develop reasoning abilities, typically through reinforcement learning. While some approaches employ low-rank parameterizations for reasoning, standard LoRA cannot reduce below the model's dimension. We investigate whether rank=1 LoRA is essential for reasoning acquisition and introduce TinyLoRA, a technique for shrinking low-rank adapters down to a single parameter. Using this novel parameterization, we successfully train the 8B parameter Qwen2.5 model to achieve 91% accuracy on GSM8K with just 13 parameters in bf16 format (totaling 26 bytes). This pattern proves consistent: we regain 90% of performance gains while utilizing 1000 times fewer parameters across more challenging reasoning benchmarks like AIME, AMC, and MATH500. Crucially, such high performance is attainable only with reinforcement learning; supervised fine-tuning demands 100-1000 times larger updates for comparable results.
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