Computer Science student and professor comment on DeepMind AI solving math problems using games

Avanti Khare, Sci-Tech Editor

Researchers at DeepMind in London have shown that AI can complete fundamental mathematical calculations by turning the problem into a game, according to Nature News. This AI then leverages machine learning techniques that another AI by the same research lab was used to beat human players in strategy games such as Go and chess. 

Increasingly, scientists are turning AI back on itself, using machine learning to improve its own underlying algorithms. “There isn’t much of a fundamental difference between a game, a puzzle, or a math problem. You have some starting state, some way to manipulate that state, and some goal state,” said Fiona Shyne ‘23, who is applying AI to game-playing mechanics for her thesis. 

Each game is a one-player puzzle that starts with a three-dimensional tensor, a grid of numbers, filled in correctly. The AI, called AlphaTensor, aims to get all of the numbers in the tensor to zero in the fewest steps by selecting from a collection of valid moves. Each move represents a calculation that combines entries from the matrices being multiplied to be added to the output matrix. 

AlphaTensor was designed to perform matrix multiplication. This involves multiplying numbers arranged in grids called matrices. These numbers might represent sets of pixels in images or air conditions in a weather model. To multiply two matrices together, the mathematician must multiply individual numbers and add them in specific ways to produce a new matrix. 

“The standard measure for efficiency of matrix multiplication is Ω (“omega”). Ω is the smallest number for which we can calculate the product of n-by-n matrices using at most n^Ω multiplications and additions between numbers,” said Matthew Anderson, Assistant Professor of Computer Science, whose active research area is matrix multiplication. 

The main takeaway from this article is that it is possible to frame the problem of designing fast algorithms for matrix multiplication as a game, which a general-purpose game-playing AI, like DeepMind, can attempt to play. It serves as a proof of concept that AI can be used to solve complex mathematical problems.

— Professor Matthew Anderson, Computer Science Department

 “For a long time the best (fastest) algorithm was thought to have Ω = 3, but in 1969 Volker Strassen, a German mathematician and computer scientist, showed that Ω was less than 2.81…” Professor Anderson explained. 

“The result produced by the Nature article shows a faster way of multiplying two 5-by-5 matrices than was previously known, resulting in Ω < 2.77—an improvement over Strassen’s original algorithm, but substantially worse than what has been known in general since the 1980s,” Professor Anderson continued. 

DeepMind’s approach to matrix multiplication uses a form of machine learning called reinforcement learning in which an AI “agent”, often a neural network, learns to interact with its environment to achieve a multistep goal, such as winning a board game. If it reaches the desired outcome, the agent is reinforced. 

AlphaTensor also uses a game-playing method called tree search, where the AI explores the outcomes of branching possibilities while planning its next action. In choosing which paths to prioritize during tree search, AlphaTensor asks a neural network to predict the most promising actions at each step. While the agent is still learning, it uses the outcomes of its games as feedback to hone the neural network, which further improves the tree search, providing more successes to learn from.