How to Move Beyond Matrix Multiplication for Efficient LLMs 2024 Update

Building large language models traditionally involves extensive use of matrix multiplication, a fundamental operation in linear algebra that underpins many neural network architectures. However, there are innovative approaches to constructing these models that can bypass matrix multiplication altogether. This introduction explores the motivation, methodologies, and potential advantages of these alternative techniques.

Traditional Large Language Models (LLMs) and Matrix Multiplication

Traditional Large Language Models (LLMs) like GPT-3, BERT, and their successors rely heavily on matrix multiplication as a core computational operation. Understanding the role of matrix multiplication in these models is crucial to appreciating the computational complexity and the potential areas for innovation.

Understanding Matrix Multiplication in LLMs

Matrix multiplication is a fundamental operation in linear algebra, essential for neural network computations. In the context of LLMs, matrix multiplication is used primarily in the following areas:

  1. Linear Transformations: At the heart of neural networks are layers that apply linear transformations to the input data. These transformations are performed using weight matrices, which are multiplied by the input vectors or matrices to produce the output.
  2. Feedforward Neural Networks: Each layer in a feedforward neural network involves multiplying the input by a weight matrix and adding a bias vector. This operation is crucial for propagating data through the network.
  3. Attention Mechanisms: In models like Transformers, the attention mechanism involves computing the compatibility scores between different input elements. This requires multiplying query, key, and value matrices derived from the input data. The attention scores are then used to weigh the input elements, facilitating the model’s focus on relevant parts of the input.
  4. Training with Backpropagation: During training, the backpropagation algorithm computes gradients of the loss function with respect to the model’s parameters. This involves multiple matrix multiplications as gradients are propagated back through the layers.

Practical Implementation in LLMs

Initialization: Weight matrices are initialized (often randomly) and then iteratively updated during training to minimize the loss function.

Forward Pass: In each layer of the model, the input data is multiplied by the weight matrices, passed through activation functions, and transformed.

Attention Calculations: In Transformer-based models, attention scores are calculated by multiplying query matrices with key matrices, and the results are used to weight value matrices.

Gradient Descent: During training, the gradients of the loss with respect to each weight matrix are computed using matrix multiplications, and the weights are updated accordingly.

Challenges of Traditional Matrix Multiplication

Matrix multiplication is a fundamental operation in traditional large language models (LLMs) and neural networks. While it is crucial for the functionality of these models, it also presents several challenges that impact the efficiency, scalability, and sustainability of AI systems. Here, we delve into the specific challenges associated with matrix multiplication in the context of LLMs.

Computational Complexity

Matrix multiplication is computationally expensive, especially for large matrices. The time complexity of multiplying two matrices of size n×nn \times nn×n is O(n3)O(n^3)O(n3) using traditional algorithms, although optimized algorithms like Strassen’s can reduce this to approximately O(n2.81)O(n^{2.81})O(n2.81). However, even with optimizations, the computational demands remain high for large-scale models.

Time Complexity

Matrix multiplication has a time complexity of O(n3)O(n^3)O(n3) for two n×nn \times nn×n matrices using naive algorithms. While optimized algorithms like Strassen’s algorithm can reduce this to approximately O(n2.81)O(n^{2.81})O(n2.81), the complexity remains high for large matrices commonly used in LLMs.

Real-time Processing

For applications requiring real-time processing, such as conversational agents or interactive systems, the time taken for matrix multiplications can introduce significant latency, hindering performance.

Resource Requirements

Processing Power: High-performance GPUs or TPUs are typically used to handle the massive amount of parallel computations required for matrix multiplications in LLMs.

Memory: Storing large weight matrices and intermediate results during computation demands substantial memory. This is a significant consideration for both training and inference.

Energy Consumption: The intensive computations translate to high energy consumption, which is a growing concern for the sustainability of large-scale AI models.

Scalability Issues

Exponential Growth

As models grow in size and complexity, the number of parameters (and thus the size of the weight matrices) increases exponentially. This growth leads to a corresponding increase in computational demands, making it challenging to scale models efficiently.

Distributed Computing

Training very large models often requires distributed computing across multiple nodes or machines. Synchronizing matrix operations across distributed systems introduces additional complexity and potential inefficiencies, such as communication overhead and load-balancing issues.

Numerical Stability


Matrix multiplications can lead to numerical stability issues, especially when dealing with very large or very small numbers. Precision errors can accumulate over many operations, potentially degrading the performance of the model.

Overflows and Underflows

During matrix multiplications, especially with deep networks, intermediate values can become very large (overflow) or very small (underflow), causing computational inaccuracies. This can necessitate techniques like normalization and clipping, adding to the complexity.

Implementation Challenges

Algorithmic Complexity

Optimizing matrix multiplication operations for specific hardware architectures requires deep expertise in both algorithm design and hardware capabilities. This can limit the accessibility of advanced techniques to only well-funded or highly specialized research groups.

Software Optimization

Efficiently implementing matrix multiplications in software frameworks (such as TensorFlow or PyTorch) involves optimizing for both the underlying hardware and the specific characteristics of the model. This often requires extensive tuning and can be highly time-consuming.

Cost Implications

Hardware Costs

The need for high-performance computing resources drives up the cost of developing and deploying LLMs. This includes the cost of purchasing and maintaining GPUs/TPUs, as well as the electricity costs associated with running these resources at full capacity.

Operational Costs

The operational costs associated with training and deploying large models include not only hardware and electricity but also cooling systems to manage the heat generated by intensive computations.

Environmental Impact

Carbon Footprint

The energy consumption associated with large-scale matrix multiplications contributes to the carbon footprint of AI research. As the demand for larger and more powerful models increases, so does the environmental impact.

The New MatMul-Free LLM Architecture

Researchers have proposed a groundbreaking Large Language Model (LLM) architecture that eliminates the need for traditional matrix multiplication (MatMul), addressing the computational challenges associated with MatMul operations. This innovative architecture is realized through three key modifications:

MatMul-Free Dense Layers

Ternary Weights

The new architecture replaces traditional weight matrices with ternary weights, which are limited to the values -1, 0, and 1. This approach simplifies computations by transforming multiplication operations into addition and subtraction, which are computationally less expensive and more efficient to implement in hardware.

Fused BitLinear Layer

A hardware-efficient “Fused BitLinear Layer” is introduced to further optimize the processing of ternary weights. This layer combines multiple bit-level operations into a single, efficient computational step, significantly accelerating processing speed and reducing energy consumption. The Fused BitLinear Layer leverages the simplicity of ternary weights to perform linear transformations without the need for MatMul, providing a substantial performance boost.

MatMul-Free Token Mixer

Modified Gated Recurrent Unit (GRU) Architecture

The traditional self-attention mechanism, which relies heavily on MatMul, is replaced with a modified Gated Recurrent Unit (GRU) architecture. GRUs are a type of recurrent neural network that can handle sequences of data efficiently. The modification involves adapting GRUs to work without MatMul, using simpler and more efficient operations.

Ternary Weights for Simpler Calculations

Similar to the dense layers, the GRU-based token mixer uses ternary weights. This modification reduces the computational complexity of the token mixing process, enabling faster and more efficient calculations. The ternary weights allow for straightforward addition and subtraction operations, which are easier to implement and optimize in hardware.

MatMul-Free Channel Mixer

Gated Linear Units (GLUs)

The channel mixer component of the new architecture employs Gated Linear Units (GLUs) instead of traditional Feed-Forward Networks (FFNs). GLUs introduce gating mechanisms that control the flow of information, allowing the model to learn complex relationships without relying on MatMul. This approach enhances the model’s capacity to handle diverse inputs while maintaining computational efficiency.

Efficient Computations with Ternary Weights

GLUs in the new architecture also utilize ternary weights, streamlining computations and reducing the need for complex hardware. By limiting weights to -1, 0, and 1, the model can perform necessary operations using simple addition and subtraction, which are faster and more resource-efficient than multiplication.

Benefits of MatMul-Free LLMs

Eliminating matrix multiplication (MatMul) from large language models (LLMs) offers numerous benefits, ranging from computational efficiency to environmental sustainability. Here, we explore these benefits in detail:

1. Increased Computational Efficiency

Reduced Complexity

Matrix multiplication is a computationally expensive operation, particularly as the size of matrices increases. By replacing MatMul with simpler operations such as addition and subtraction, MatMul-free architectures reduce computational complexity. This results in faster processing times for both training and inference.

Hardware Optimization

MatMul-free architectures leverage hardware-efficient designs like the “Fused BitLinear Layer,” which combines multiple bit-level operations into a single computational step. This optimization enhances the overall speed and efficiency of the model.

2. Enhanced Scalability

Simplified Operations

The use of ternary weights (-1, 0, 1) simplifies computations, making it easier to scale models to handle larger datasets and more complex tasks. The simplified operations require fewer computational resources, allowing for the deployment of larger models without proportional increases in hardware requirements.

Efficient Resource Utilization

With reduced computational demands, MatMul-free LLMs make better use of available resources. This efficient resource utilization supports the scaling of models in both research and production environments, enabling the development of more sophisticated AI systems.

3. Lower Energy Consumption

Energy-Efficient Computations

Matrix multiplications are energy-intensive operations. By eliminating MatMul, the new architecture reduces energy consumption, making the training and deployment of LLMs more sustainable. This reduction in energy usage translates to lower operational costs and a smaller carbon footprint.

Environmental Sustainability

The decreased energy consumption aligns with global efforts to reduce the environmental impact of technology. MatMul-free LLMs contribute to more sustainable AI research and deployment, supporting initiatives to mitigate climate change and promote environmental responsibility.

4. Cost-Effectiveness

Reduced Hardware Costs

By eliminating the need for high-performance GPUs or TPUs specifically optimized for MatMul operations, MatMul-free LLMs can be implemented on less specialized, more cost-effective hardware. This reduction in hardware requirements lowers the overall cost of developing and deploying LLMs.

Lower Operational Costs

The decreased energy consumption and reduced need for cooling systems contribute to lower operational costs. Organizations can achieve significant cost savings in both the short and long term by adopting MatMul-free architectures.

5. Improved Numerical Stability

Precision and Stability

Matrix multiplications can lead to numerical stability issues, especially with very large or very small values. MatMul-free architectures using ternary weights mitigate these issues by simplifying the range of values used in computations. This simplification enhances the precision and stability of the model’s operations.

Reduced Risk of Overflows and Underflows

With simpler operations and a limited range of values, MatMul-free LLMs are less prone to numerical overflows and underflows. This reduction in numerical errors improves the reliability and accuracy of the models.

6. Innovation and Adaptability

New Research Opportunities

The development of MatMul-free LLMs opens up new avenues for research in neural network architectures and computational methods. Researchers can explore innovative techniques and algorithms that were previously constrained by the limitations of traditional matrix multiplication.

Adaptability to Specialized Applications

MatMul-free architectures can be adapted to specialized applications and use cases where traditional LLMs may face limitations. This adaptability enhances the versatility of LLMs, making them suitable for a broader range of industries and applications.

7. Accessibility and Inclusivity

Democratizing AI Development

By reducing the hardware and computational barriers to developing and deploying LLMs, MatMul-free architectures make advanced AI technology more accessible to a wider range of researchers, developers, and organizations. This democratization fosters inclusivity and encourages diverse contributions to the field of AI.

Support for Edge Computing

MatMul-free LLMs, with their reduced computational and energy requirements, are well-suited for deployment on edge devices. This capability supports the growing trend of edge computing, where AI models are run locally on devices rather than relying on centralized cloud resources.


The advent of MatMul-free Large Language Models (LLMs) marks a significant milestone in the evolution of artificial intelligence. By eliminating traditional matrix multiplication, these innovative architectures offer substantial improvements in computational efficiency, scalability, and energy consumption. They present a cost-effective alternative that not only reduces operational expenses but also aligns with environmental sustainability goals.

The simplification of operations through the use of ternary weights and hardware-efficient designs enhances numerical stability and opens new avenues for research and application. This adaptability makes MatMul-free LLMs suitable for a wide range of specialized applications, from edge computing to real-time systems, democratizing access to advanced AI capabilities.

As we continue to push the boundaries of AI technology, the MatMul-free approach provides a promising pathway to developing more efficient, sustainable, and accessible models. This shift not only benefits the AI community but also contributes to broader societal goals by making advanced AI tools available to a more diverse range of users and applications.