How AI Discovered a Faster Matrix Multiplication Algorithm

Quanta Magazine
22 May 202313:00

TLDRAI has revolutionized matrix multiplication with a new algorithm that surpasses the longstanding Strassen's method. DeepMind's AlphaTensor, using reinforcement learning, discovered a more efficient way to multiply four by four matrices with binary elements. This breakthrough not only optimizes large-scale computations but also exemplifies the synergy between AI and mathematicians, opening new horizons in solving complex mathematical problems.

Takeaways

  • 🧠 Matrix multiplication is a fundamental mathematical operation with applications in various fields such as computer graphics, neural networks, and quantum physics.
  • 🔍 Researchers have been seeking more efficient matrix multiplication methods to solve larger problems that were previously considered too large to compute in a reasonable time.
  • 📚 The standard matrix multiplication algorithm involves a process that requires N-cubed steps, which becomes inefficient for large matrices.
  • 🇩🇪 Volker Strassen's algorithm, introduced in 1969, reduced the number of multiplication steps needed for 2x2 matrices from eight to seven, offering significant computational savings for larger matrices.
  • 🚀 In 1970, Shmuel Winograd proved that it's impossible to multiply two 2x2 matrices using six or fewer multiplications, establishing Strassen's algorithm as the best solution for small matrices.
  • 🤖 Google's DeepMind, known for AI achievements in games, applied machine learning techniques to tackle the problem of matrix multiplication, resulting in the development of AlphaTensor.
  • 🎲 AlphaTensor uses reinforcement learning, a method that involves strategic penalization and rewards to guide the AI towards an optimal solution.
  • 🔑 The AI program AlphaTensor discovered a new algorithm for multiplying 4x4 matrices with elements of zero or one, breaking the 50-year record set by Strassen's algorithm.
  • 🤝 The collaboration between AlphaTensor and mathematicians led to further refinements of the new matrix multiplication algorithms, demonstrating the potential of AI-assisted mathematical research.
  • 🌐 The discovery by AlphaTensor has sparked interest and further research in the field, with mathematicians using the AI's findings as a starting point for their own advancements.
  • 🛠️ The potential of human and artificial intelligence collaboration is vast and is only beginning to be fully explored, with AI serving as a tool to empower mathematicians rather than replace them.

Q & A

  • What is matrix multiplication and why is it significant in various fields?

    -Matrix multiplication is a fundamental mathematical operation used in fields such as computer graphics, neural networks, and quantum physics. It involves performing operations on a two-dimensional array of numbers, and it's significant because it is a core component in many computations in engineering and physics.

  • Why are researchers interested in finding more efficient matrix multiplication methods?

    -Researchers are interested in more efficient matrix multiplication methods because even a slight improvement can make larger, previously intractable problems computable within a reasonable time, expanding the scope of solvable issues.

  • What was the breakthrough in matrix multiplication achieved by Volker Strassen?

    -Volker Strassen discovered an algorithm in 1969 that reduced the number of multiplication steps required to multiply two 2x2 matrices from eight to seven, offering significant computational savings for larger matrices.

  • What is the significance of the algorithm discovered by Shmuel Winograd regarding matrix multiplication?

    -Shmuel Winograd proved that it is impossible to multiply two 2x2 matrices using six or fewer multiplications, thereby establishing Strassen's algorithm with seven multiplications as the optimal solution.

  • How did DeepMind's AI, AlphaTensor, contribute to the field of matrix multiplication?

    -AlphaTensor, developed by DeepMind, discovered a new algorithm for multiplying two 4x4 matrices with elements of zero or one, which required only 47 multiplication steps, breaking the 50-year-old record set by Strassen's algorithm.

  • What is reinforcement learning, and how does it relate to AlphaTensor's approach to discovering new algorithms?

    -Reinforcement learning is a type of machine learning where an AI system is rewarded or penalized based on its performance in achieving a task. AlphaTensor uses this approach to experiment with different ways to decompose a 3D tensor into rank-1 tensors, driving the program towards an optimal matrix multiplication algorithm.

  • What is a tensor and how does it relate to matrix multiplication?

    -A tensor is an array of numbers that can have any number of dimensions. In the context of matrix multiplication, the process of multiplying any two matrices of a given size can be described by a unique 3D tensor, where each element represents a multiplication step.

  • How does AlphaTensor use tensor decomposition to find more efficient matrix multiplication algorithms?

    -AlphaTensor decomposes a 3D tensor into rank-1 tensors, where each rank-1 tensor represents a multiplication step in an algorithm. By using the fewest possible rank-1 tensors to decompose the tensor, AlphaTensor finds algorithms that require fewer multiplication steps.

  • What is the impact of AlphaTensor's discovery on the collaboration between AI and mathematicians?

    -AlphaTensor's discovery has shown that AI can be a powerful tool to assist mathematicians in finding new results and guiding their intuition. It has sparked further research and collaboration between human mathematicians and AI, opening up new frontiers in problem-solving.

  • How did the mathematical community react to the use of computers in proving the Four Color Theorem in 1976?

    -Initially, the mathematical community was not prepared to cede logical reasoning to a machine. However, as the field of AI and machine learning advanced, the community has come to accept and embrace the role of computers in assisting with complex mathematical research.

  • What is the potential of AI in mathematical research, and how does AlphaTensor's success exemplify this?

    -AI has the potential to significantly enhance mathematical research by discovering new algorithms and providing insights that may not be immediately apparent to human mathematicians. AlphaTensor's success in finding a more efficient matrix multiplication algorithm exemplifies how AI can empower mathematicians to achieve more.

Outlines

00:00

🧠 Matrix Multiplication and Its Computational Challenge

The script introduces the fundamental yet complex operation of matrix multiplication, which is crucial in various fields such as computer graphics, neural networks, and quantum physics. It explains that while the operation is simple to understand, it poses a significant computational challenge due to its complexity. The standard algorithm for multiplying two N by N matrices requires a cubic number of steps, which becomes impractical for large matrices. The script also highlights the historical significance of Volker Strassen's algorithm, which reduced the number of multiplication steps for 2x2 matrices from eight to seven, offering substantial computational savings for larger matrices.

05:02

🤖 The Emergence of AI in Mathematical Research

This paragraph delves into the application of artificial intelligence in mathematical research, specifically focusing on matrix multiplication. It discusses the use of reinforcement learning by Google's DeepMind and their development of AlphaTensor, an algorithmic descendant of AlphaGo. AlphaTensor employs a unique approach by treating the process of matrix multiplication as a game, using tensor decomposition to simplify the search for more efficient algorithms. The script explains how AlphaTensor uses rank-1 tensors to represent multiplication steps and aims to decompose a 3D tensor with the fewest steps possible, thereby finding more efficient matrix multiplication methods.

10:03

🚀 AlphaTensor's Breakthrough and Collaboration with Mathematicians

The final paragraph discusses the breakthrough achieved by AlphaTensor, which discovered a new algorithm for multiplying two 4x4 matrices with elements of zero or one, breaking the 50-year-old record set by Strassen's algorithm. The script emphasizes the collaborative potential between AI and mathematicians, as evidenced by the work of Manuel Kauers and Jakob Moosbauer, who used AlphaTensor's findings to further refine matrix multiplication algorithms. It concludes by highlighting the empowering nature of AI in mathematical research, suggesting that such technology can assist rather than replace mathematicians.

Mindmap

Keywords

💡Matrix Multiplication

Matrix multiplication is a fundamental mathematical operation that combines two matrices to produce a new matrix. It is central to the video's theme as it is the focus of the research and the subject of the AI's discovery. The script describes it as a simple yet complex operation that is fundamental in various fields such as computer graphics and physics. The process involves multiplying elements from rows of one matrix with columns of another and summing the results, as illustrated by the standard algorithm for two 2x2 matrices.

💡Efficient Algorithms

Efficient algorithms are methods designed to perform tasks with minimal computational resources. In the context of the video, researchers are seeking more efficient matrix multiplication algorithms to handle larger problems within a reasonable time frame. The script mentions that even a small improvement in efficiency can make a significant difference when dealing with large matrices.

💡Volker Strassen

Volker Strassen is a German mathematician who is highlighted in the script for his groundbreaking work in matrix multiplication. In 1969, he developed an algorithm that reduced the number of multiplication steps needed to multiply two 2x2 matrices from eight to seven, which was a significant advancement in computational efficiency for larger matrices.

💡Shmuel Winograd

Shmuel Winograd is an IBM researcher mentioned in the script for proving the optimality of Strassen's algorithm. He demonstrated that it is impossible to multiply two 2x2 matrices using fewer than seven multiplications, thereby establishing Strassen's algorithm as the best solution for this task.

💡DeepMind

DeepMind is Google's artificial intelligence research lab that is featured in the script for discovering a new matrix multiplication algorithm. Known for training AI systems to master games, DeepMind applied its expertise to tackle the complex problem of finding more efficient ways to multiply matrices.

💡AlphaGo

AlphaGo is a DeepMind-developed AI program that gained recognition for defeating the top-ranked human Go player, Lee Sedol, in 2016. The script references AlphaGo to illustrate DeepMind's capabilities in pushing the boundaries of what AI can achieve, setting the stage for their work on matrix multiplication.

💡AlphaTensor

AlphaTensor is an algorithm descended from AlphaGo, specifically designed to tackle the problem of matrix multiplication. The script describes how AlphaTensor uses reinforcement learning to explore the vast search space of possible algorithms, ultimately discovering a more efficient method for multiplying 4x4 matrices with elements of zero or one.

💡Reinforcement Learning

Reinforcement learning is an area of machine learning where an agent learns to make decisions by receiving rewards or penalties for its actions. In the script, AlphaTensor uses reinforcement learning to experiment with different matrix multiplication strategies, with the goal of finding the most efficient algorithm.

💡Tensor

A tensor, as mentioned in the script, is a generalization of vectors and matrices to potentially higher dimensions. In the context of matrix multiplication, the process can be represented by a unique 3D tensor, where each element can be decomposed into rank-1 tensors, which correspond to multiplication steps in an algorithm.

💡Tensor Decomposition

Tensor decomposition is the process of breaking down a tensor into simpler components, often rank-1 tensors. The script explains how this process is used in AlphaTensor's search for efficient matrix multiplication algorithms, by decomposing the 3D tensor representing matrix multiplication into the fewest possible rank-1 tensors.

💡Machine Learning Techniques

Machine learning techniques are methods that enable computers to learn from data and improve at tasks without being explicitly programmed. The script discusses how DeepMind applied these techniques to the problem of matrix multiplication, allowing AlphaTensor to explore a vast search space and discover new algorithms.

Highlights

AI has discovered a faster matrix multiplication algorithm, breaking a 50-year-old record.

Matrix multiplication is fundamental in various fields like computer graphics, neural networks, and quantum physics.

Efficient matrix multiplication can make larger, previously intractable problems computable within a reasonable time.

The standard matrix multiplication algorithm requires N-cubed steps, which becomes unwieldy for large matrices.

Volker Strassen's algorithm reduced the number of multiplication steps for 2x2 matrices from eight to seven.

Strassen's algorithm offers significant computational savings for large matrices by breaking them into smaller ones.

Shmuel Winograd proved that no algorithm can use fewer than seven multiplications for 2x2 matrices, solidifying Strassen's status.

DeepMind's AI, AlphaTensor, was trained to discover new matrix multiplication algorithms using reinforcement learning.

AlphaTensor is based on the AlphaZero reinforcement learning algorithm, which has mastered various games.

AlphaTensor uses tensor decomposition to find efficient matrix multiplication algorithms.

The process involves decomposing a 3D tensor into rank-1 tensors, each representing a multiplication step.

AlphaTensor's training involved guessing rank-1 tensors to decompose the original tensor with minimal steps.

AlphaTensor rediscovered Strassen's algorithm and further improved upon it for 4x4 matrices with modulo-2 elements.

The new algorithm reduces the multiplication steps from 64 (standard) or 49 (Strassen's) to only 47 for 4x4 matrices.

AlphaTensor also discovered thousands of other fast algorithms for different matrix sizes in modulo-2.

Manuel Kauers and Jakob Moosbauer used AlphaTensor's findings to further optimize a 5x5 matrix multiplication algorithm.

The collaboration between AI and mathematicians is a new frontier, empowering people to achieve more.

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