{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/searching-for-efficient-linear-layers-over-a","title":"Searching for Efficient Linear Layers over a Continuous Space of Structured Matrices","arxiv_id":"2410.02117","date":"2024-10-03","proceeding":null,"authors":["Andres Potapczynski","Shikai Qiu","Marc Finzi","Christopher Ferri","Zixi Chen","Micah Goldblum","Bayan Bruss","Christopher De Sa","Andrew Gordon Wilson"],"abstract":"Dense linear layers are the dominant computational bottleneck in large neural networks, presenting a critical need for more efficient alternatives. Previous efforts focused on a small number of hand-crafted structured matrices and neglected to investigate whether these structures can surpass dense layers in terms of compute-optimal scaling laws when both the model size and training examples are optimally allocated. In this work, we present a unifying framework that enables searching among all linear operators expressible via an Einstein summation. This framework encompasses many previously proposed structures, such as low-rank, Kronecker, Tensor-Train, Block Tensor-Train (BTT), and Monarch, along with many novel structures. To analyze the framework, we develop a taxonomy of all such operators based on their computational and algebraic properties and show that differences in the compute-optimal scaling laws are mostly governed by a small number of variables that we introduce. Namely, a small $\\omega$ (which measures parameter sharing) and large $\\psi$ (which measures the rank) reliably led to better scaling laws. Guided by the insight that full-rank structures that maximize parameters per unit of compute perform the best, we propose BTT-MoE, a novel Mixture-of-Experts (MoE) architecture obtained by sparsifying computation in the BTT structure. In contrast to the standard sparse MoE for each entire feed-forward network, BTT-MoE learns an MoE in every single linear layer of the model, including the projection matrices in the attention blocks. We find BTT-MoE provides a substantial compute-efficiency gain over dense layers and standard MoE.","url_abs":"https://arxiv.org/abs/2410.02117v2","url_pdf":"https://arxiv.org/pdf/2410.02117v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"searching-for-efficient-linear-layers-over-a","repo_url":"https://github.com/andpotap/einsum-search","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"mixture-of-experts","task_name":"Mixture-of-Experts"}],"methods":[{"method_slug":"attention","method_name":"Attention"},{"method_slug":"linear-layer","method_name":"Linear Layer"},{"method_slug":"moe","method_name":"MoE"},{"method_slug":"softmax","method_name":"Softmax"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2410.02117","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2410.02117"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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