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

• URL: http://arxiv.org/abs/2502.09376v2
• Date: Fri, 14 Feb 2025 02:39:04 GMT
• Title: LoRA Training Provably Converges to a Low-Rank Global Minimum or It Fails Loudly (But it Probably Won't Fail)
• Authors: Junsu Kim, Jaeyeon Kim, Ernest K. Ryu,
• Abstract summary: Low-rank adaptation (LoRA) has become a standard approach for fine-tuning large foundation models. We show that LoRA training converges to a global minimizer with low rank and small magnitude. We argue that the zero-initialization and weight decay in LoRA training induce an implicit bias toward the low-rank, small-magnitude region.
• Score: 15.381439594872898
• License:
• Abstract: 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.

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