# How AI Discovered a Faster Matrix Multiplication Algorithm

TLDRDeepMind's AI, AlphaTensor, has made a groundbreaking discovery in the field of matrix multiplication, a fundamental operation in mathematics with applications in various fields. By using reinforcement learning, AlphaTensor has developed a new algorithm that surpasses the 50-year-old Strassen's method, particularly for multiplying four by four matrices with binary elements. This achievement not only demonstrates the potential of AI in mathematical research but also highlights the collaborative future between technology and mathematicians in exploring new frontiers.

### Takeaways

- 🧠 Matrix multiplication is a fundamental mathematical operation used in various fields like computer graphics, neural networks, and quantum physics.
- 🔍 Researchers have been seeking more efficient matrix multiplication methods to solve larger problems within a reasonable time frame.
- 📚 The traditional method of matrix multiplication involves a straightforward algorithm that requires N-cubed steps for N by N matrices.
- 📉 The complexity of traditional matrix multiplication grows exponentially with the size of the matrices, limiting its applicability to very large matrices.
- 🇩🇪 Volker Strassen's algorithm, introduced in 1969, reduced the number of multiplication steps needed for 2x2 matrices from eight to seven.
- 🌟 Strassen's algorithm provides computational savings for larger matrices by breaking them down into smaller 2x2 matrices.
- 🚀 DeepMind's AlphaTensor, an AI system, discovered a new algorithm for multiplying 4x4 matrices with elements of zero or one, breaking Strassen's 50-year record.
- 🎲 AlphaTensor uses reinforcement learning, a technique that involves strategic penalties and rewards to guide the AI towards an optimal solution.
- 📊 The process of finding efficient matrix multiplication involves tensor decomposition, breaking down a 3D tensor into rank-1 tensors that represent multiplication steps.
- 🤖 AlphaTensor's discovery of a faster algorithm for 4x4 matrices with modulo-2 arithmetic demonstrates the potential of AI in mathematical research.
- 🤝 The collaboration between AlphaTensor and human mathematicians led to further refinements and insights, highlighting the synergy between AI and human intellect in problem-solving.

### 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 basic component in many computational processes in engineering and physics.

### Why is finding more efficient matrix multiplication methods important?

-Efficient matrix multiplication methods are important because they allow for the handling of larger problems that would otherwise be too time-consuming to compute. Faster algorithms make it possible to solve bigger problems within a reasonable timeframe.

### What was the breakthrough in matrix multiplication that occurred more than 50 years ago?

-The breakthrough was Volker Strassen's algorithm in 1969, which reduced the number of multiplication steps needed to multiply two 2x2 matrices from eight to seven, offering significant computational savings for larger matrices.

### How does Strassen's algorithm provide computational savings for larger matrices?

-Strassen's algorithm allows large matrices to be broken down into smaller 2x2 matrices. The savings in multiplication steps for these smaller matrices propagate, resulting in fewer overall steps needed for large matrix multiplications.

### What is the significance of the new algorithm discovered by DeepMind for multiplying four by four matrices with elements of zero or one?

-The new algorithm discovered by DeepMind allows for even faster multiplication of large matrices by breaking them down into four by four matrices instead of two by two, which is more efficient for matrices with elements of zero or one.

### What is DeepMind and how has it contributed to the field of AI?

-DeepMind is Google's artificial intelligence research lab known for training AI systems to master various games and achieve significant milestones, such as AlphaGo's victory over the top-ranked human Go player, Lee Sedol, in 2016.

### What is the basis of the algorithm developed by DeepMind's AlphaTensor for matrix multiplication?

-AlphaTensor is based on a reinforcement learning algorithm called AlphaZero. It uses a single-player game approach to decompose 3D tensors into rank-1 tensors, with the goal of using as few of these tensors as possible to achieve the matrix multiplication.

### How does the process of tensor decomposition relate to finding more efficient matrix multiplication algorithms?

-Tensor decomposition involves breaking down a 3D tensor into rank-1 tensors, which represent multiplication steps in a matrix multiplication algorithm. The fewer rank-1 tensors used in the decomposition, the fewer multiplication steps required, leading to a more efficient algorithm.

### What is the role of reinforcement learning in AlphaTensor's approach to discovering new algorithms?

-Reinforcement learning is used to train AlphaTensor to strategically penalize and reward itself as it experiments with different ways to achieve its task of decomposing a 3D tensor using the fewest rank-1 tensors possible, driving the program towards an optimal solution.

### How did the mathematical community respond to the use of computer programs in mathematical research in the past, and how has this changed?

-In the past, the mathematical community was not prepared to cede logical reasoning to machines, as seen in the initial skepticism towards the computer-assisted proof of the Four Color Theorem in 1976. However, the field has evolved, and now there is a greater acceptance and collaboration between AI and mathematicians, as seen with AlphaTensor's contributions.

### What potential does the collaboration between AI and mathematicians have, and how does AlphaTensor's discovery exemplify this?

-The collaboration between AI and mathematicians has the potential to push the frontiers of mathematical discovery. AlphaTensor's discovery exemplifies this by inspiring human mathematicians to further refine its algorithms, demonstrating a powerful partnership between technology and human intellect.

### Outlines

### 🧠 Matrix Multiplication: A Fundamental Yet Complex Operation

This paragraph introduces matrix multiplication as a crucial operation in various fields, including computer graphics, neural networks, and quantum physics. It is a fundamental concept in mathematics that is simple to understand but complex to master, with applications in engineering and physics. The importance of finding more efficient matrix multiplication methods is highlighted, as even a small improvement can make a significant difference in solving larger problems. The paragraph also discusses the traditional method of matrix multiplication, which involves a considerable number of steps, and introduces the challenge of finding faster algorithms, which has been a longstanding issue in the field.

### 🚀 Strassen's Algorithm and the Quest for Efficiency

This paragraph delves into the history of matrix multiplication algorithms, focusing on Volker Strassen's groundbreaking work. In 1969, Strassen introduced an algorithm that reduced the number of multiplication steps needed to multiply two 2x2 matrices from eight to seven. Although the reduction may seem minor, it offers substantial computational savings for larger matrices, as they can be broken down into smaller 2x2 matrices. The paragraph also mentions Shmuel Winograd's proof that no algorithm can perform the task with fewer than seven multiplications, establishing Strassen's algorithm as the optimal solution for a long time. The narrative then shifts to a new development in 2022, where an algorithm by Google's DeepMind broke Strassen's record for a specific case of 4x4 matrices with binary elements.

### 🤖 AlphaTensor: AI's Role in Advancing Mathematical Research

The final paragraph discusses the role of artificial intelligence in advancing mathematical research, specifically in the context of matrix multiplication. It introduces AlphaTensor, an AI system developed by DeepMind, which uses reinforcement learning techniques to search for more efficient matrix multiplication algorithms. The process involves representing matrix multiplication as a 3D tensor and decomposing it into rank-1 tensors, which correspond to multiplication steps. The fewer rank-1 tensors used, the fewer multiplication steps required. AlphaTensor's approach to learning involves guessing and subtracting rank-1 tensors to decompose the original tensor, with the goal of minimizing the number of steps. The paragraph concludes with the impact of AlphaTensor's discoveries on the field, including the rediscovery of Strassen's algorithm and the development of new, faster algorithms, and emphasizes the collaborative potential between AI and mathematicians.

### Mindmap

### Keywords

### 💡Matrix Multiplication

### 💡Efficient Algorithms

### 💡Volker Strassen

### 💡Shmuel Winograd

### 💡DeepMind

### 💡AlphaGo

### 💡AlphaTensor

### 💡Reinforcement Learning

### 💡Tensor

### 💡Rank-1 Tensors

### 💡Machine Learning Techniques

### Highlights

Matrix multiplication is a fundamental operation in mathematics with applications in various fields.

Efficient matrix multiplication can make large, previously intractable problems computable.

Traditional matrix multiplication algorithms are limited by their computational complexity.

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

Strassen's algorithm has significant computational savings for larger matrices due to its recursive nature.

Shmuel Winograd proved that six or fewer multiplications are impossible for 2x2 matrices, solidifying Strassen's algorithm as optimal.

DeepMind's AlphaTensor AI discovered a new algorithm for 4x4 matrices with elements of zero or one, breaking Strassen's record.

AlphaTensor uses reinforcement learning to optimize matrix multiplication algorithms.

AlphaTensor's discovery was a result of playing a single-player game designed to decompose 3D tensors.

The process of tensor decomposition is crucial for finding efficient matrix multiplication algorithms.

AlphaTensor's algorithm for 4x4 matrices with modulo-2 elements uses only 47 multiplications, improving on Strassen's.

AlphaTensor also discovered new algorithms for 5x5 matrices in modulo-2, expanding the horizons of matrix multiplication efficiency.

The collaboration between AlphaTensor and mathematicians led to further refinement of the 5x5 matrix multiplication algorithm.

AI-assisted mathematical research is a new frontier that combines the strengths of both humans and artificial intelligence.

DeepMind's work on matrix multiplication demonstrates the potential for AI to assist in complex mathematical discoveries.

The future of AI in mathematics is not about replacing mathematicians but empowering them with new tools and insights.