Multiplying Matrices Without Multiplying (2106.10860v1)

Abstract: Multiplying matrices is among the most fundamental and compute-intensive operations in machine learning. Consequently, there has been significant work on efficiently approximating matrix multiplies. We introduce a learning-based algorithm for this task that greatly outperforms existing methods. Experiments using hundreds of matrices from diverse domains show that it often runs $100\times$ faster than exact matrix products and $10\times$ faster than current approximate methods. In the common case that one matrix is known ahead of time, our method also has the interesting property that it requires zero multiply-adds. These results suggest that a mixture of hashing, averaging, and byte shuffling$-$the core operations of our method$-$could be a more promising building block for machine learning than the sparsified, factorized, and/or scalar quantized matrix products that have recently been the focus of substantial research and hardware investment.

• The paper presents Maddness, a novel algorithm that eliminates traditional multiply-add operations for efficient approximate matrix multiplication.
• It leverages learned hash functions and a fast integer summation method to achieve speeds up to 100 times faster than exact implementations on a single CPU thread.
• The study provides theoretical guarantees and robust empirical results on datasets like CIFAR-10, showcasing improved speed-accuracy tradeoffs and integration potential in AI applications.

Overview of "Multiplying Matrices Without Multiplying"

The paper introduces a novel approach to Approximate Matrix Multiplication (AMM) that significantly enhances computational efficiency. The methodology, termed "Maddness" (Multiply-ADDitioN-lESS), leverages hashing, averaging, and byte shuffling to perform matrix operations without relying on traditional multiply-add techniques. This contrasts the prevalent methods that often depend on either dense matrix products or sparsified, factorized approaches.

Technical Contributions

The authors propose a learning-based algorithm that dramatically improves the speed-accuracy tradeoff in matrix multiplication tasks. Key technical contributions include:

1. Efficient Vector Quantization Functions: The authors introduce a family of learned hash functions that encode large datasets with remarkable speed, achieving over 100GB/s on a single CPU thread. This capability reduces computation time substantially, making it particularly beneficial for scenarios typical in machine learning and data mining where matrices are tall and relatively dense.
2. Fast Integer Summation Algorithm: To further enhance performance, the paper presents an innovative algorithm for summing low-bitwidth integers without facing upcasting, saturation, or overflow issues. This is crucial for maintaining high-speed operations within the quantized framework.
3. Zero Multiply-Add Requirement: Notably, when one matrix is pre-known (such as pre-trained model weights in inference scenarios), Maddness operates without any multiply-adds, hence its nomenclature.
4. Theoretical Guarantees: The paper provides a formal generalization bound that broadens the theoretical understanding of matrix approximation errors in relation to singular value distributions.

Numerical Results

Empirical evaluations demonstrate the considerable gains achieved by the proposed method:

• Speed Improvements: The algorithm often runs 100 times faster than exact implementations and 10 times faster than the best-existing AMM methods.
• Robust Performance Across Datasets: Experiments on a variety of matrices from diverse real-world datasets—such as CIFAR-10 and CIFAR-100 image datasets—show that this approach maintains or improves the accuracy compared to state-of-the-art methods while drastically reducing computation time.

Implications and Future Developments

The implications of this research stretch across a broad spectrum of AI applications. With matrix multiplication being a fundamental operation in numerous algorithms and neural network layers, this method can potentially lead to significant advancements in real-time machine learning and data processing applications. Further, its compatibility with existing architectural paradigms suggests an easy integration pathway into current AI frameworks without the need for extensive hardware adjustments.

Future developments could explore extending Maddness to other domains, such as convolution operations or more complex deep learning architectures. Additionally, implementing the technique in hardware accelerators could yield substantial gains in energy efficiency given its reduced reliance on computationally expensive operations.

In summary, the paper presents a compelling, efficient alternative to conventional matrix multiplication techniques, promising noteworthy enhancements in both the theoretical landscape and practical execution of machine learning tasks.