Privacy is a central concern when fine-tuning large language models (LLMs) on sensitive data, and differentially private stochastic gradient descent (DP-SGD) provides formal privacy guarantees. When performance at a fixed privacy level largely saturates on established benchmarks such as GLUE, we ask whether the same performance can be reached with less computation. Both theory and practice show that lower-rank models are better suited to DP training, a property especially relevant for LLMs, whose fine-tuning gradients exhibit a strong low-rank structure. Methods such as DP-LoRA exploit this by expressing each layer's weight update as a sum of only a few non-zero rank-one components. However, we argue that while having few non-zero components is important, the isotropic noise injected by DP-SGD inflates the singular values of the gradient matrix, disrupting their naturally fast decay. We investigate whether this noise-induced spectral distortion slows optimization. Experiments on language classification (GLUE with RoBERTa) and table-to-text generation (E2E and DART with Qwen and Llama models up to 4B parameters) show that partially restoring the original singular-value profile significantly accelerates DP-SGD convergence without compromising privacy guarantees. Our code is available at https://github.com/alidadsetan/re-exam-private-low-rank.

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