Compress Large Language Models via Collaboration Between Learning and Matrix Approximation

Sparse and low-rank matrix composite approximation has emerged as a promising paradigm for compressing large language models (LLMs), offering a more flexible pruning structure than conventional methods based solely on sparse matrices. The significant variation in weight redundancy across layers, along with the differing rank and sparsity structures of weight matrices, makes identifying the globally optimal pruning structure extremely challenging. Existing methods often depend on uniform or manually designed heuristic rules to allocate weight sparsity across layers, subsequently compressing each matrix using matrix approximation techniques. Given the above theoretical difficulty in global compression of LLMs and the limited computational and data resources available compared to the training phase, we argue that a collaboration between learning and matrix approximation is essential for effective compression. In this paper, we propose a novel LLM compression framework based on generalized bilevel optimization that naturally formulates an effective collaborative mechanism. Specifically, the outer loop frames the weight allocation task as a probabilistic optimization problem, enabling the automatic learning of both layer-wise sparsities and matrix-wise retained ranks, while the inner loop solves the corresponding sparsity and rank-constrained model compression problem via matrix approximation. Our main technical contributions include two key innovations for efficiently solving this bilevel optimization problem. First, we introduce a truncated Gaussian prior-based probabilistic parameterization integrated with a policy gradient estimator, which avoids expensive backpropagation and stabilizes the optimization process. Second, we design an adapted QR-based matrix approximation algorithm that significantly accelerates inner loop computations. Extensive experiments on Phi-3 and the LLama-2/3 family demonstrate the effectiveness of our method. Notably, it maintains over 95\% zero-shot accuracy under 50\% sparsity and achieves up to 2× inference speedup.