Weight-perturbation evolution strategies (ES) can fine-tune billion-parameter language models with surprisingly small populations (e.g., N≈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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arXiv:2602.00170, 2026.

@article{liang:arxiv26, title={The Blessing of Dimensionality in LLM Fine-tuning: A Variance-Curvature Perspective}, author={Qiyao Liang and Jinyeop Song and Yizhou Liu and Jeff Gore and Ila Fiete and Risto Miikkulainen and Xin Qiu}, journal={arXiv:2602.00170}, month={ }, url="http://nn.cs.utexas.edu/?liang:arxiv26", year={2026} }