The Blessing of Dimensionality in LLM Fine-tuning: A Variance-Curvature Perspective

• URL: http://arxiv.org/abs/2602.00170v1
• Date: Fri, 30 Jan 2026 00:26:35 GMT
• Title: The Blessing of Dimensionality in LLM Fine-tuning: A Variance-Curvature Perspective
• Authors: Qiyao Liang, Jinyeop Song, Yizhou Liu, Jeff Gore, Ila Fiete, Risto Miikkulainen, Xin Qiu,
• Abstract summary: We show that weight-perturbation evolution strategies can fine-tune language models with surprisingly small populations.<n>We also observe that fine-tuning reward often rises, peaks, and then degrades in both ES and GRPO.
• Score: 19.4447760660162
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Weight-perturbation evolution strategies (ES) can fine-tune billion-parameter language models with surprisingly small populations (e.g., $N\!\approx\!30$), contradicting classical zeroth-order curse-of-dimensionality intuition. We also observe a second seemingly separate phenomenon: under fixed hyperparameters, the stochastic fine-tuning reward often rises, peaks, and then degrades in both ES and GRPO. We argue that both effects reflect a shared geometric property of fine-tuning landscapes: they are low-dimensional in curvature. A small set of high-curvature dimensions dominates improvement, producing (i) heterogeneous time scales that yield rise-then-decay under fixed stochasticity, as captured by a minimal quadratic stochastic-ascent model, and (ii) degenerate improving updates, where many random perturbations share similar components along these directions. Using ES as a geometric probe on fine-tuning reward landscapes of GSM8K, ARC-C, and WinoGrande across Qwen2.5-Instruct models (0.5B--7B), we show that reward-improving perturbations remain empirically accessible with small populations across scales. Together, these results reconcile ES scalability with non-monotonic training dynamics and suggest that high-dimensional fine-tuning may admit a broader class of viable optimization methods than worst-case theory implies.

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