AlphaTensor, builds upon AlphaZero, an agent that has shown superhuman performance on board games, like chess, Go and shogi, and this work shows the journey of AlphaZero from playing games to tackling unsolved mathematical problems for the first time.
The ancient Egyptians created an algorithm to multiply two numbers without requiring a multiplication table, and Greek mathematician Euclid described an algorithm to compute the greatest common divisor, which is still in use today.
During the Islamic Golden Age, Persian mathematician Muhammad ibn Musa al-Khwarizmi designed new algorithms to solve linear and quadratic equations. In fact, al-Khwarizmi’s name, translated into Latin as Algoritmi, led to the term algorithm. But, despite the familiarity with algorithms today – used throughout society from classroom algebra to cutting edge scientific research – the process of discovering new algorithms is incredibly difficult, and an example of the amazing reasoning abilities of the human mind.
They published in Nature. AlphaTensor is the first artificial intelligence (AI) system for discovering novel, efficient, and provably correct algorithms for fundamental tasks such as matrix multiplication. This sheds light on a 50-year-old open question in mathematics about finding the fastest way to multiply two matrices.
Trained from scratch, AlphaTensor discovers matrix multiplication algorithms that are more efficient than existing human and computer-designed algorithms. Despite improving over known algorithms, they note that a limitation of AlphaTensor is the need to pre-define a set of potential factor entries F, which discretizes the search space but can possibly lead to missing out on efficient algorithms. An interesting direction for future research is to adapt AlphaTensor to search for F. One important strength of AlphaTensor is its flexibility to support complex stochastic and non-differentiable rewards (from the tensor rank to practical efficiency on specific hardware), in addition to finding algorithms for custom operations in a wide variety of spaces (such as finite fields). They believe this will spur applications of AlphaTensor towards designing algorithms that optimize metrics that we did not consider here, such as numerical stability or energy usage.
The discovery of matrix multiplication algorithms has far-reaching implications, as matrix multiplication sits at the core of many computational tasks, such as matrix inversion, computing the determinant and solving linear systems.
The process and progress of automating algorithmic discovery
First, they converted the problem of finding efficient algorithms for matrix multiplication into a single-player game. In this game, the board is a three-dimensional tensor (array of numbers), capturing how far from correct the current algorithm is. Through a set of allowed moves, corresponding to algorithm instructions, the player attempts to modify the tensor and zero out its entries. When the player manages to do so, this results in a provably correct matrix multiplication algorithm for any pair of matrices, and its efficiency is captured by the number of steps taken to zero out the tensor.
This game is incredibly challenging – the number of possible algorithms to consider is much greater than the number of atoms in the universe, even for small cases of matrix multiplication. Compared to the game of Go, which remained a challenge for AI for decades, the number of possible moves at each step of their game is 30 orders of magnitude larger (above 10^33 for one of the settings they consider).
Essentially, to play this game well, one needs to identify the tiniest of needles in a gigantic haystack of possibilities. To tackle the challenges of this domain, which significantly departs from traditional games, we developed multiple crucial components including a novel neural network architecture that incorporates problem-specific inductive biases, a procedure to generate useful synthetic data, and a recipe to leverage symmetries of the problem.
They then trained an AlphaTensor agent using reinforcement learning to play the game, starting without any knowledge about existing matrix multiplication algorithms. Through learning, AlphaTensor gradually improves over time, re-discovering historical fast matrix multiplication algorithms such as Strassen’s, eventually surpassing the realm of human intuition and discovering algorithms faster than previously known.
Exploring the impact on future research and applications
From a mathematical standpoint, their results can guide further research in complexity theory, which aims to determine the fastest algorithms for solving computational problems. By exploring the space of possible algorithms in a more effective way than previous approaches, AlphaTensor helps advance our understanding of the richness of matrix multiplication algorithms. Understanding this space may unlock new results for helping determine the asymptotic complexity of matrix multiplication, one of the most fundamental open problems in computer science.
Because matrix multiplication is a core component in many computational tasks, spanning computer graphics, digital communications, neural network training, and scientific computing, AlphaTensor-discovered algorithms could make computations in these fields significantly more efficient. AlphaTensor’s flexibility to consider any kind of objective could also spur new applications for designing algorithms that optimise metrics such as energy usage and numerical stability, helping prevent small rounding errors from snowballing as an algorithm works.
While they focused here on the particular problem of matrix multiplication, we hope that our paper will inspire others in using AI to guide algorithmic discovery for other fundamental computational tasks. Their research also shows that AlphaZero is a powerful algorithm that can be extended well beyond the domain of traditional games to help solve open problems in mathematics. Building upon our research, they hope to spur on a greater body of work – applying AI to help society solve some of the most important challenges in mathematics and across the sciences.
Improving the efficiency of algorithms for fundamental computations can have a widespread impact, as it can affect the overall speed of a large amount of computations. Matrix multiplication is one such primitive task, occurring in many systems—from neural networks to scientific computing routines. The automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. However, automating the algorithm discovery procedure is intricate, as the space of possible algorithms is enormous. Here we report a deep reinforcement learning approach based on AlphaZero1for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. Our agent, AlphaTensor, is trained to play a single-player game where the objective is finding tensor decompositions within a finite factor space. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago2. We further showcase the flexibility of AlphaTensor through different use-cases: algorithms with state-of-the-art complexity for structured matrix multiplication and improved practical efficiency by optimizing matrix multiplication for runtime on specific hardware. Our results highlight AlphaTensor’s ability to accelerate the process of algorithmic discovery on a range of problems, and to optimize for different criteria.
Brian Wang is a Futurist Thought Leader and a popular Science blogger with 1 million readers per month. His blog Nextbigfuture.com is ranked #1 Science News Blog. It covers many disruptive technology and trends including Space, Robotics, Artificial Intelligence, Medicine, Anti-aging Biotechnology, and Nanotechnology.
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