LPCD: Unified Framework from Layer-Wise to Submodule Quantization

LPCD: Unified Framework from Layer-Wise to Submodule Quantization Yuma Ichikawa Fujitsu Limited RIKEN Center for AIP Yudai Fujimoto Fujitsu Limited Institute of Science Tokyo Akira Sakai Fujitsu Limited Tokai University Abstract Post-training quantization (PTQ) aims to pre- serve model-level behavior; however, most methods focus on individual linear layers. Even recent extensions, such as QEP and LoaQ, which mitigate error propagation or target spe- cific submodules, still rely on layer-wise formu- lations and fail to capture the behavior of larger submodules. We introduce Layer-Projected Coordinate Descent (LPCD), a unified frame- work that extends PTQ beyond layers by opti- mizing relaxed objectives across arbitrary sub- modules and projecting the solutions with stan- dard layer-wise quantizers. LPCD general- izes existing methods and provides a principled approach to quantizing complex submodules while maintaining the efficiency and compati- bility of layer-wise PTQ pipelines. Across di- verse LLM architectures and bit-widths, LPCD- based submodule quantization consistently en- hances both layer-wise PTQ methods and exist- ing submodule approaches. 1 Introduction Large-scale models achieve strong performance; however, they incur substantial memory and com- putational overhead, hindering practical deploy- ment (Chen et al., 2023). These constraints are particularly stringent for edge devices. To bridge the gap between accuracy and deployability, prior work has explored compression techniques such as quantization (Lang et al., 2024; Gong et al., 2024), pruning (Wang et al., 2024; Cheng et al., 2024), low-rank adaptation (Yang et al., 2024; Hu et al., 2022), and knowledge distillation (Xu et al., 2024a). Among these techniques, layer-wise post- training quantization (PTQ) is one of the most practical and widely adopted methods for large- scale LLMs (Frantar et al., 2022; Lin et al., 2024; Yao et al., 2022; Chee et al., 2023). By focus- ing on individual linear layers, layer-wise PTQ simplifies the problem to least-squares estimation of linear transformations, which enables for effi- cient solvers and straightforward implementations. Despite its simplicity, layer-wise PTQ delivers strong empirical performance and is used both as a standalone method and as an initialization for more complex block-wise or global PTQ frame- works (Malinovskii et al., 2024; Guan et al., 2024). However, the layer-wise structure imposes sig- nificant limitations. Classical layer-wise PTQ methods, such as GPTQ (Frantar et al., 2022), AWQ (Lin et al., 2024), and QuIP (Chee et al., 2023), optimize each linear layer independently by utilizing input activations. Consequently, these methods are confined to activation-aware weight approximation and indirectly affect model-level outputs; quantization error may become significant. Recent work, including QEP (Arai and Ichikawa, 2025) and GPTAQ (Li et al., 2025), relaxes this constraint by modifying the layer-wise objective to address the mismatch between pre- and post- quantization activations. This modification enables controlled error propagation across layers, preserv- ing the sequential pipeline and reducing the accu- mulation of quantization errors. LoaQ (Lin and Wan, 2025) further extends the target from linear layers to specific submodules by explicitly approx- imating the outputs of residual connections and RMSNorm. This approach aligns layer-wise PTQ more closely with the behavior at the model-level. Nevertheless, these advances remain specialized: QEP is still anchored in individual linear layers, and LoaQ is limited to specific submodules; they do not provide a unified treatment of more general submodules, activation, or KV-cache quantization. This study goes beyond traditional layer-wise formulations by introducing a framework for sub- module quantization that preserves the standard layer-wise pipeline. We propose Layer-Projected Coordinate Descent (LPCD), which optimizes ar- bitrary submodules in the output space and projects the relaxed solutions back into the quantization do- 1 arXiv:2512.01546v1 [stat.ML] 1 Dec 2025 0 10 20 30 Layer index 0.00 1.00 2.00 Layer Output MSE 1e-4 QEP LoaQ Ours (a) 4-bit weight 0 10 20 30 Layer index 0.00 2.00 4.00 6.00 Layer Output MSE 1e-4 (b) 3-bit weight 0 10 20 30 Layer index 0.00 0.50 1.00 1.50 Layer Output MSE 1e-2 (c) 2-bit weight Figure 1: Output MSE across Transformer blocks in Llama 3 8B with 4, 3, and 2 bit weight quantization. LPCD consistently yields lower quantization error than QEP and LoaQ. main using existing layer-wise PTQ algorithms as layer projectors. This perspective unifies classical layer-wise PTQ, QEP, and LoaQ as special cases while extending naturally to more general submod- ules, activations, and KV caches. We instantiate LPCD on grouped-query KV, VO aggregation, and MLP up-down blocks, demonstrating that LPCD significantly reduces quantization error compared to QEP and LoaQ, as shown in Figure 1. Extensive experimen

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