Author | Ben Dickson
Translated by | Sambodhi
Planning | Ling Min
DeepMind researchers recently published a paper titled "Advancing Mathematics by Guiding Human Intuition with AI," arguing that deep learning can help uncover mathematical relationships that have been overlooked by human scientists. Soon, the paper attracted widespread attention in the tech media.
Some mathematicians and computer scientists have praised DeepMind's work and the results achieved in his papers, calling it groundbreaking. Others are skeptical, arguing that the paper and its coverage in the mass media may have exaggerated the use of deep learning in mathematics.
A mathematical discovery framework based on machine learning
In their paper, DeepMind scientists argue that AI could be used to "assist in the discovery of cutting-edge theorems and conjectures in mathematical research." They propose a framework for "enhancing the standard mathematician's toolkit through powerful pattern recognition and interpretation methods in machine learning."
A framework for using machine learning in mathematical discovery (courtesy of DeepMind)
Mathematicians first make assumptions about the relationship between two mathematical objects. To test this hypothesis, they used computer programs to generate data for both types of objects. Next, a supervised machine learning model algorithm computes these numbers and attempts to adjust their parameters to map one type of object to another.
"The most important contribution of machine learning in this regression process is that as long as there is enough data, a series of possible nonlinear functions can be learned," the researchers wrote. ”
If the trained model performs better than random guesses, it may indicate that there is indeed a discoverable relationship between the two mathematical objects. By using different machine learning techniques, researchers are able to discover data points that are more relevant to the problem, refine their assumptions, generate new data, and train new models. By repeating these steps, they can narrow down reasonable conjectures and speed up the final solution.
DeepMind's scientists describe the framework as a "testbed of intuition" that can quickly verify "whether intuition about the relationship between two quantities is worth pursuing" and provide guidance for the relationships they might have.
Using this framework, DeepMind's researchers used deep learning to come up with "two fundamental new discoveries, one in topology and the other in representation theory." ”
One of the interesting things about this work is that there is no need for huge hash rates, which have become a mainstay of DeepMind's research. According to the paper, the deep learning model used in both findings can be trained in a matter of hours "on a machine with only one graphics processing unit."
Knots and representations
A button knot is a closed curve in space that can be defined in a variety of ways. As the number of their intersections increases, they will become more complex. The researchers wanted to see if they could use machine learning to discover the mapping between algebraic invariants and hyperbolic invariants, two fundamentally different ways of defining knots.
The researchers wrote: "We hypothesized that there is an undiscovered relationship between a knotted hyperbolic invariant and an algebraic invariant. ”
Using the SnapPy package, researchers can generate "signatures," 1 algebraic invariants, and 12 promising hyperbolic invariants for 1.7 million knots with up to 16 intersections.
Next, they created a fully connected feed-forward neural network with three hidden layers, each with 300 units. They trained a deep learning model that maps hyperbolic invariants to signatures. Their initial model was able to predict signatures with 78% accuracy. Through further analytical studies, they found a small set of parameters in hyperbolic invariants that could predict signatures. The researchers refined their conjectures, generated new data, retrained their model, and came up with a final theorem.
The researchers describe the theorem as "one of the first results of algebraic and geometric invariants connecting knots, and it has many interesting applications." ”
"We expect that in low-dimensional topology, there will be many other applications to this newly discovered relationship between natural slope and signature." The researchers write: "It is incredible that such a simple and profound relationship has been overlooked in this field that has long been widely studied." ”
The second result of the paper is also a mapping of two different views of symmetry, which is far more complex than the knot.
In this example, they used a graph neural network (GNN) to find a relationship between the Bruhat interval graph and the Kazhdan-Lusztig (KL) polynomial. One of the benefits of graph neural networks is the ability to compute and learn from huge graphs that are difficult for the mind to process on its own. Deep learning takes an interval plot as input and tries to predict the corresponding KL polynomial.
Similarly, by generating data, training deep learning models, and reshooting processes, scientists are able to draw a provable conjecture.
Popular reactions to DeepMind's mathematical ARTIFICIAL intelligence
Speaking about DeepMind's findings in knot theory, Mark Brittenham, a knot theorist at the University of Nebraska-Lincoln, said in an interview with Nature: "The fact that the authors have used a very direct method to confirm that invariants are related tells us that there are many very basic things in this field that we do not yet fully understand. Brittenham added that DeepMind's technology is novel in finding surprising connections compared to other efforts to apply machine learning to Knot.
Adam Zsolt Wagner, a mathematician at Tel Aviv University in Israel, who also spoke to Nature, said that the method proposed by DeepMind could prove valuable for certain types of problems.
Wagner, who has experience applying machine learning to mathematics, says, "Without such tools, mathematicians might spend weeks or even months trying to prove a formula or theorem that would eventually prove wrong." But he also added that it's unclear how widespread its impact will be.
Reasons to be skeptical
Following the publication of DeepMind's findings in the journal Nature, Ernest Davis, a professor of computer science at New York University, published a paper of his own that raises important questions about DeepMind's framework for outcomes and the limitations of deep learning's application in general mathematics.
In the first result presented in DeepMind's paper, Davis observes that knot theory is not a typical problem in which deep learning is superior to other machine learning or statistical methods.
"The advantage of deep learning is that there are many low-level input features for each instance (image or text), it is difficult to reliably identify high-level features, and for anyone, the functions that relate the input features to the answer are complex, and no subset of the input features is completely decisive," Davis writes. ”
The button question has only 12 input features, of which only three are correlated. The mathematical relationship between the input feature and the target variable is simple.
"It's hard to understand why a neural network with 200,000 parameters would be the preferred method; simple, traditional statistical methods or support vector machines are more appropriate," Davis writes. ”
In the second project, the role of deep learning is even more important. "Unlike knot theory projects that use a universal deep learning architecture, neural networks are carefully designed to meet a deeper mathematical knowledge of the problem. In addition, deep learning works better on preprocessed data than on raw data, with an error rate of about 1/40. He wrote.
On the one hand, Davis says, these findings stand in stark contrast to those critical that incorporating domain knowledge into deep learning is very difficult. "On the other hand, deep learning enthusiasts, on the other hand, often praise deep learning as a 'plug and play' approach to learning that can solve any problem at hand with raw data; this runs counter to this accolade," he wrote. ”
In these tasks, successful application of deep learning may depend heavily on how the training data is generated and how the mathematical structure is encoded. This suggests that the framework may be suitable for a small class of mathematical problems.
"Finding the best way to generate and encode data involves a mix of theory, experience, art, and experimentation. The burden of it all falls on the experts of humanity," he wrote. "Deep learning can be a powerful tool, but it's not a panacea."
Davis cautions that in the current hype about deep learning, "there is an unusual motivation to focus on the role of deep learning in this study, not just DeepMind's machine learning experts, or even mathematicians." ”
Davis concludes that, as mentioned in this article, deep learning is best seen as "another analytical tool in the experimental math toolbox, rather than an entirely new mathematical approach." ”
Notably, the authors of the original paper also point out some limitations of their framework, such as "it requires the ability to generate large data sets represented by objects, and patterns that are detectable in computable examples." In addition, in some areas, it may be difficult to learn the function of interest in this paradigm. ”
Deep learning and intuition
One of the controversial themes is the paper's claim that deep learning is "guiding intuition." Davis describes this as "a very inaccurate description of what mathematicians get or expect to get when they use this kind of deep learning." ”
Intuition is one of the important differences between humans and artificial intelligence. It's a better decision-making ability than random guessing, and most of the time, it can steer you in the right direction. As the history of ARTIFICIAL intelligence to date shows, there are no predefined rules and patterns in massive amounts of data that capture intuition.
"In the mathematical world, the word 'intuition' means that a concept or proof can be based on deeply ingrained feelings about familiar domains such as numbers, space, time, or motion, or in some other way 'meaningful' or 'seems to be correct' without the need for explicit calculations or step-by-step reasoning." Davis wrote.
Davis argues that in order to gain an intuitive grasp of mathematical concepts, it is often necessary to use multiple specific examples, but this is not statistically relevant. In other words, you don't gain intuition by running millions of examples and observing the percentage of certain patterns that recur.
This means that it is not deep learning models that allow scientists to intuitively understand the concepts they define, the theorems they prove, and the conjectures they come up with.
"What deep learning does is give them some advice about which features of the problem seem important and which don't," Davis writes. This is not worth scoffing at, but it should not be exaggerated either. ”
About the Author:
Ben Dickson is a software engineer and founder of TechTalks. Write articles on technology, business, and politics.
https://bdtechtalks.com/2021/12/13/deepminds-machine-learning-mathematics/