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

Zeroth-order (ZO) optimization enables memory-efficient large language model (LLM) fine-tuning, with recent methods employing curvature-aware strategies. However, in the massive LLM parameter space, 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 high-variance, while subspace optimization is low-variance but blind. To break this trade-off, we propose \textbf{\Engine} (\textbf{En}do\textbf{g}enous Var\textbf{i}atio\textbf{n}al MultiScal\textbf{e}). Inspired by computational mechanics, we adapt static Variational MultiScale 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. Evaluations on standard LLM benchmarks show that \Engine\ achieves higher accuracy and faster convergence than existing ZO methods, achieving up to a $\mathbf{2.61\times}$ \textbf{speed-up in forward passes} with zero additional peak memory overhead. Code is available at \url{https://anonymous.4open.science/r/ENGINE-D0A4}.