LoRA Training Provably Converges to a Low-Rank Global Minimum Or It Fails Loudly (But it Probably Won't Fail)

Low-rank adaptation (LoRA) has become a standard approach for fine-tuning large foundation models. However, our theoretical understanding of LoRA remains limited as prior analyses of LoRA's training dynamics either rely on linearization arguments or consider highly simplified setups. In this work, we analyze the LoRA loss landscape without such restrictive assumptions. We define two regimes: a "special regime", which includes idealized setups where linearization arguments hold, and a "generic regime" representing more realistic setups where linearization arguments do not hold. In the generic regime, we show that LoRA training converges to a global minimizer with low rank and small magnitude, or a qualitatively distinct solution with high rank and large magnitude. Finally, we argue that the zero-initialization and weight decay in LoRA training induce an implicit bias toward the low-rank, small-magnitude region of the parameter space—where global minima lie—thus shedding light on why LoRA training usually succeeds in finding global minima.

Most modern AI models are developed in two stages. First, a model learns general knowledge about a broad range of topics. Then, it is slightly adjusted to specialize in a specific task or domain. The most commonly used method for efficiently doing this second-stage adjustment is called LoRA (Low-Rank Adaptation).These two learning stages are both essential, yet they differ in significant ways. While there is extensive research on algorithms for the initial training phase, the second, task-specific adaptation stage has received comparatively little theoretical attention.Our research addresses this gap by mathematically proving the effectiveness of training algorithms used in the second stage. Importantly, unlike most prior studies, we avoid relying on oversimplified assumptions about the complex architecture of AI models. This improved theoretical understanding helps ensure more reliable and efficient use of AI across various specialized tasks.