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Fine-tuning large language models (LLMs) using low-rank adaptation (LoRA) has become a highly efficient approach for downstream tasks, particularly in scenarios with limited computational resources. However, applying LoRA techniques to quantized LLMs poses unique challenges due to the reduced representational precision of quantized weights. In this paper, we introduce CLoQ (Calibrated LoRA initialization for Quantized LLMs), a simplistic initialization strategy designed to overcome these challenges. Our approach focuses on minimizing the layer-wise discrepancy between the original LLM and its quantized counterpart with LoRA components during initialization. By leveraging a small calibration dataset, CLoQ quantizes a pre-trained LLM and determines the optimal LoRA components for each layer, ensuring a strong foundation for subsequent fine-tuning. A key contribution of this work is a novel theoretical result that enables the accurate and closed-form construction of these optimal LoRA components. We validate the efficacy of CLoQ across multiple tasks such as language generation, arithmetic reasoning, and commonsense reasoning, demonstrating that it consistently outperforms existing LoRA fine-tuning methods for quantized LLMs, especially at 2-bit. 

Abstract Quantized or low-bit neural networks are attractive due to their inference efficiency. However, training deep neural networks with quantized activations involves minimizing a discontinuous and piecewise constant loss function. Such a loss function has zero gradient almost everywhere (a.e.), which makes the conventional gradient-based algorithms inapplicable. To this end, we study a novel class of biased first-order oracle, termed coarse gradient, for overcoming the vanished gradient issue. A coarse gradient is generated by replacing the a.e. zero derivative of quantized (i.e., staircase) ReLU activation composited in the chain rule with some heuristic proxy derivative called straight-through estimator (STE). Although having been widely used in training quantized networks empirically, fundamental questions like when and why the ad hoc STE trick works, still lack theoretical understanding. In this paper, we propose a class of STEs with certain monotonicity and consider their applications to the training of a two-linear-layer network with quantized activation functions for nonlinear multi-category classification. We establish performance guarantees for the proposed STEs by showing that the corresponding coarse gradient methods converge to the global minimum, which leads to a perfect classification. Lastly, we present experimental results on synthetic data as well as MNIST dataset to verify our theoretical findings and demonstrate the effectiveness of our proposed STEs.