ENGINE: Endogenous Variational MultiScale Optimization for Zeroth-Order LLM Fine-Tuning

Zhuoli Ouyang, Changxi Chi, Siyuan Li, Tailin Wu

Zeroth-order (ZO) optimization enables memory-efficient large language model (LLM) fine-tuning, and recent methods incorporate curvature-aware strategies. However, in the high-dimensional parameter space of LLMs, these methods inherently suffer from exploding gradient variance and prohibitive costs. Conversely, confining updates to a low-dimensional subspace reduces variance but discards critical high-dimensional information, degrading performance. This creates a fundamental dilemma: full-space optimization is informative but has high variance, while subspace optimization has low variance but loses information. To break this trade-off, we propose ENGINE (Endogenous Variational MultiScale). Inspired by computational mechanics, we adapt static Variational MultiScale (VMS) theory to dynamic optimization, treating low- and high-dimensional spaces as resolved coarse and unresolved fine scales. Through a multigrid-style V-cycle, ENGINE alternates between spaces, dynamically probing fine-grained information in the high-dimensional space and injecting it to correct the efficient low-dimensional sampling process. Under appropriate assumptions, statistical analysis characterizes variance…