Discovering faster matrix multiplication algorithms with reinforcement learning

Abstract

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 AlphaZero1 for 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.

Inferring the fractional nature of Wu Baleanu trajectories

“…The development of mathematical proofs and arguments seems to be one of the most difficult challenges. Nevertheless, some barriers have already fallen with the discovery of new multiplication algorithms [5].…”

Fault-Tolerant Strassen-Like Matrix Multiplication

“…Finally, we have to note that our scheme is based on computer-aided efficient search procedures for local relation enumerations. However, we are also actively looking for systematic redundancy generation schemes for recursive matrix multiplication methods as well as deep learning techniques recently fueled by DeepMind initiatives for faster matrix multiplications [18].…”

Exploring explicit coarse-grained structure in artificial neural networks

“…In the past decade, machine learning has drawn great attention from almost all natural science and engineering communities, such as mathematics [1][2][3], physics [4][5][6][7][8][9][10], biology [11][12][13], and materials sciences [14][15][16], and has been widely used in various aspects of modern society, e.g., automatic driving systems, face recognition, fraud detection, expert recommendation system, speech enhancement, and natural language processing, etc. Especially, the deep learning techniques based on the artificial neural networks [17,18] have become the most popular and dominant machine learning approaches progressively, and their interactions with many-body physics have been intensively explored in recent years.…”

“…Similarly, the 2nd scale transformation is composed of two isometries U (2) s, and each U (2) maps {σ (1) } to variables {σ (2) } sitting on the red lines. U (3) constitutes the last scale transformation, and maps {σ (2) } to variables {σ (3) } sitting on the black lines. Eventually, the operator is represented in terms of {σ (3) }, and this completes the full RG process.…”