Advances in artificial intelligence, especially in the branch of machine learning, are rampant. Among the long list of recent applications, the well-known one stands out ChatGPT either Geminiwhich in addition to text processes images, audio and video. To implement these models, large amounts of data are used, which are sometimes confidential. An example is the case of algorithms that help in the diagnosis and treatment of diseases, which use personal medical data. Therefore, it is essential to find ways to preserve the privacy of the data used. A recent approach makes use of mathematical concepts from quantum physics to address this challenge.
Specifically, it involves studying the symmetries that appear in the parameters of the model, that is, in the numbers that configure it. Artificial intelligence algorithms (or models) are, after all, complex functions that process the information received to make a prediction. These functions are defined by numbers called parameters—in the case of ChatGPT, 220 billion of them—which determine what portions of the information are processed and how intensely they are processed. For example, a model that anticipates the risk of suffering from a disease, r, based on the age e, height a, and weight p of each person, could be r = (xe + ya) / zp This model has three parameters, x, y, and z, that determine the intensity with which each of the input variables—age, height, and weight—influences the risk; By entering the data of a specific patient, the algorithm will make the prediction about the possibility that a person with those characteristics will develop the disease.
The value of the parameters is set using large, already resolved reference data sets, for which the result that should be obtained is known. This process is called training. In the example of the previous model, the training data would be medical data from a large number of patients, with their corresponding diagnosis. With them, the parameters are adjusted to maximize correct predictions in the reference.
It turns out, then, that the selection of parameters for a model depends on the data used to train it. And although, in theory, the model only learns patterns from the training data, in practice they learn much more. In fact, several scientists have warned that the parameters of these algorithms can indicate whether a specific piece of data was part of the data used to train itand even, in certain cases, they can extract the complete training data from them.
In these situations, a solution is to build another model, with other parameters and with other training data, but for each data entered it makes exactly the same prediction as the original. This idea of “different parameters describing the same model” corresponds to a precise mathematical entity: it is the concept of gauge symmetry.
This term is not only of mathematical interest, but is a fundamental element in several areas of physics, such as general relativity, particle physics, or quantum mechanics. Now, a recent job has shown that, indeed, if an algorithm has one of these gauge symmetries, it is possible to build another that will make the same predictions, whose parameters are not related to the data used to train the initial model. In this way, studying the parameters will not be able to reveal information about the training data.
So, the challenge is to find artificial intelligence algorithms that have gauge symmetries. This is not easy, because in artificial intelligence symmetries are seen as unwanted properties, which must be gotten rid of. However, in the field of quantum physics gauge symmetries are very present and have been widely studied. Specifically, the tensor networkswhich are used in the simulation of quantum systems made up of many particles, have this type of symmetry. In addition, these networks allow very complicated systems to be modeled, similar to artificial intelligence algorithms. This has made the Tensor networks have begun to be used as artificial intelligence algorithms a few years ago.
At the moment, the modeling carried out by tensor networks does not yet compete, in terms of overall quality, with that of other more popular modern algorithms – based on deep neural networks, for example. However, they have shown important advantages, such as the ability to understand what factors are driving a specific prediction. To these virtues another is now added, thanks to its gauge symmetries: the protection of the privacy of the data used during training. This positions tensor networks as very promising candidates for the development of artificial intelligence, and illustrates in a very clear way how fundamental ideas in mathematics and quantum physics can have an impact on everyday technologies.
Alejandro Pozas is a postdoctoral researcher at the University of Geneva, Switzerland.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing, translation and coordination: Agate Timón García-Longoria. She is coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT)
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