Searching for Efficient Linear Layers over a Continuous Space of Structured Matrices (2410.02117v2)

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.

• The paper proposes a novel continuous parameterization framework for structured matrices, enabling optimal replacements for dense linear layers.
• Using a taxonomy based on rank, compute intensity, and parameter sharing, experiments on GPT-2 show comparable or improved performance with lower computational costs.
• The study introduces a structured Mixture-of-Experts that sparsifies every linear layer, promising significant compute savings in scaling large neural network models.

Efficient Linear Layers through Structured Matrices

The paper "Searching for Efficient Linear Layers over a Continuous Space of Structured Matrices" by Potapczynski et al. introduces an innovative approach to optimize the computational efficiency of neural networks by substituting dense linear layers with structured matrices. This work addresses the computational bottleneck presented by dense linear layers in large neural networks, particularly in prominent models like transformers, by providing a framework that explores a wide range of potential structures.

Framework for Structured Matrices

The authors propose a unifying framework that encompasses a continuous parameterization method for linear operators expressible via an Einstein summation, which allows for the exploration of various structured matrices. This includes previously known structures such as low-rank, Kronecker, Tensor-Train, Block Tensor-Train (BTT), and Monarch, as well as the introduction of novel configurations that improve computational scalability and efficiency.

The core idea of this parameterization is to navigate a space defined by specific variables ($\theta$), influencing the dimensions and properties of these structures. By adjusting these variables, the framework enables efficient searches for the most optimal structured matrices that can replace traditional dense matrices without sacrificing performance.

Taxonomy Development

The paper develops a taxonomy based on three key dimensions:

1. Rank Exponent ($\psi$): This relates to the rank of the structure, with $\psi=1$ being full-rank, indicating maximum expressivity.
2. Compute Intensity ($\nu$): Refers to the FLOPs per dimension, dictating the computational efficiency of the structure.
3. Parameter Sharing ($\omega$): Captures the extent of parameter sharing within the matrix, with a focus on achieving $\omega = 0$ for optimal resource utilization.

These dimensions facilitate identifying structures that offer computational cost reduction while maintaining full representational capacity.

Experimental Evaluation

The experiments involve training GPT-2 models on OpenWebText with several structured alternatives to dense layers. The paper finds that certain structures, particularly those with $\omega=0$ and $\psi=1$, closely mirror or outperform traditional dense layers in terms of efficiency. The findings suggest that harnessing full-rank and minimizing parameter sharing results in favorable scaling laws and computational efficiencies.

Furthermore, the investigation extends to autoregressive pixel modeling on CIFAR-5M and synthetic regression tasks, reinforcing the universality of the proposed taxonomy and parameterization.

Implications and Future Work

A key contribution of this work is the introduction of a structured Mixture-of-Experts (MoE) architecture. Unlike existing MoE approaches that apply sparsity at the feed-forward network level, the BTT-MoE framework applies a sparse mixture at every linear layer, demonstrating significant compute savings.

The paper's findings imply substantial potential for enhancing neural network architectures by replacing dense layers with carefully chosen structured matrices. This approach can significantly reduce computational costs, a crucial consideration for scaling AI models.

Future directions may explore the further refinement of these structures to optimize their application across various neural network tasks, as well as extending the taxonomy to accommodate more complex or multi-modal learning scenarios. Additionally, practical implementation and integration into existing machine learning frameworks could broaden the accessibility and impact of such efficient architectures.