Profile of Yuval Peres

Yuval Peres received an early introduction to mathematics and probability. His mother studied physics and specialized in statistics, and his father was a sociologist who specialized in opinion polls. “At the breakfast table, we had discussions of chi-square tests and reliability of sampling, and I would be fascinated by these questions of what you can learn from surveys,” says Peres. Peres’s paternal grandfather also had great influence on him. “He did a PhD in physics in the 1920s with Max Planck in Berlin, and then he later had to escape Germany and, much later in life, returned to science, specifically to statistics,” says Peres. Peres was enthralled by his grandfather’s vivid explanations of various mathematical and scientific problems. “He brought out science and math as a human enterprise, showing how discoveries were made not in a vacuum but due to human interaction,” he says.

Yuval Peres. Image courtesy of Yuval Peres.

As a teenager, Peres inherited his grandfather’s books on probability and became fascinated with mathematics. During a distinguished career, Peres has continued to work on probability and ergodic theory and has tackled numerous other topics, including fractals, random walks, Brownian motion, game theory, and information theory. His research has contributed to fields as diverse as computer science, mathematical biology, and statistical physics. In his Inaugural Article (1), he describes a novel approach to tackling the problem of fair allocation, which involves using gravitational allocation to partition a sphere equally between different nodes. Peres is now a principal researcher at Microsoft Research and was elected a foreign associate of the National Academy of Sciences in 2016.

Finding Connections

When Peres was an undergraduate at Tel Aviv University, a friend’s book recommendation had an immediate impact on his career. The book was about ergodic theory and number theory and was written by Hillel (Harry) Furstenberg. “When I started reading this book, I was so transfixed that, while still clutching the book, I took the first bus to Jerusalem to meet Harry Furstenberg at Hebrew University,” says Peres. “I just waited until he had some time to meet, talked with him, and then he agreed to advise me,” he says.

Peres completed his PhD work with Furstenberg, and the experience would influence his approach to mathematics. “A theme of his work that has continued in my own is looking for connections between different fields,” says Peres. Peres began his first faculty job at the University of California, Berkeley in 1993, although he continued to alternate between Berkeley and Jerusalem until 2000. Both universities were excellent places to look for connections between different fields. “Berkeley is a very lively place with lots of visitors, and I really enjoyed talking to people with different backgrounds,” says Peres.

It was also a good time to be looking for connections between probability theory and other fields, particularly computer science. “The role of probability in science in general and computer science in particular has grown really dramatically over the last two decades,” says Peres. “My role is someone who’s come from probability but is very open to problems emerging from computer science, and that’s been a very fruitful juncture to be in,” he says.

Peres’s ability to find connections between probability and other fields is evidenced by his work on reconstruction on trees. “I first heard the question from two computer scientists, Claire Kenyon and Leonard Schulman, who were motivated by noisy computation,” he says. “When you’re trying to preserve some bit of information which is subject to corruption, one standard way this is done is by replication; so you have a bit, and you make three copies.” says Peres. “Now, suppose you repeat that and you keep going, and after you do this for many generations, you have a large number of bits, and each one is supposed to be like the original bit, but they have a lot of chance to be corrupted along the way,” says Peres. “The question is, what’s this error rate? Under what error rate do you still have some real useful information about the original bit, and what error rate is so high that these bits are kind of useless?”

As Peres tackled this computer science problem, he realized that the same problem had also cropped up in mathematical biology and statistical physics as simplified models of mutations and magnetism, respectively. Peres analyzed the problem by integrating insights from all three fields and, together with his colleagues, published an article that made this connection (2). “You had these three domains of applied math where the same problem arose and where people had made different advances with different tools, so I was able, with various coauthors, to first combine the insights of the different groups and then go further using probability theory,” says Peres.

Machine Learning and Optimization

In 2006, Peres joined Microsoft Research, where he is a principal researcher in a group of mathematicians and computer scientists working on machine learning and optimization. Peres has recently become fascinated by a topic called “learning with expert advice.”

“Imagine that you have several routes to drive to work, and you have to choose which route to use,” he says. “The problem is that there are two opposite tendencies: One is to keep doing what seems to be best now and stick with that; that’s called exploitation, exploiting the information you have. The other is exploration, or saying I’m not really sure what’s best, so let me just keep trying all the options until I acquire a lot of information and I’m really sure,” he says. “You have to strike the right balance, and this is the famous tradeoff of exploration and exploitation.”

Peres’s main contribution to this topic has been regarding the case where there are switching costs to switch from one option to the other (3). “For instance, if you have a robot in an industrial plant that can do several tasks, and you don’t know what the most useful thing is to do, but every time you reconfigure it, there’s a cost to that,” he says. “When you add switching costs to this learning task, it makes exploration more costly,” says Peres. “People weren’t sure how the optimal algorithm behaves in that case, and it turned out that a full understanding of that involved using quite sophisticated probability theory.”

The Overhang Problem

Peres has won numerous honors for his work, including the Rollo Davidson Prize in 1995, the Loève Prize in 2001, and the David P. Robbins Prize in 2011. The David P. Robbins Prize was awarded for work by Peres et al. (4) on a 150-y-old mathematical problem called the “overhang problem,” and built on previous work by Paterson and Zwick (5), who also won the prize.

The overhang problem has a simple premise: How do you stack blocks on a table to achieve the maximum possible overhang? “The basic problem is you have a bunch of blocks and you’re trying to put them at the edge of the table so they will extend as far as possible beyond the edge of the table,” says Peres. “There is a classical solution, which is already surprising; namely, with enough blocks, you can go as far as you like beyond the edge,” he says.

Paterson and Zwick (5) discovered a different pattern, wherein the blocks are placed so that they resemble an oil lamp, that goes much further for the same number of blocks. “What they didn’t know is whether this was really as far as you can go,” says Peres. “Working with Zwick, Paterson, and other collaborators Winkler and Thorup, we found a connection to random walks, which was not obvious at all,” he says.

“There’s nothing random about the initial problem, but it turns out that when you write the physics equations of the forces on the blocks, they naturally lend themselves to a probability interpretation, and then we were able to solve this,” says Peres. They showed that the pattern discovered by Paterson and Zwick (5) was indeed the optimal distance for a given number of blocks (4).

“This again captures the theme of collaboration with other scientists, notably computer scientists, and seeing sometimes surprising connections to probability and being able to use those to solve a problem,” says Peres.

New Approach to Sphere Allocation

In his Inaugural Article (1), Peres tackled another important mathematical problem: sphere allocation. Given a certain number of nodes on the surface of a sphere, how can the sphere’s area be partitioned equally among the nodes? “You want each territory to contain a relevant node, yet the territories should be as compact as possible, meaning the average distance within a territory should be small,” he says.

Peres realized that he could apply gravitational allocation to this problem, by treating the nodes as gravitational wells. “The basic idea there is that you have these nodes, and each one applies a gravitational force to the sphere, and it divides the sphere naturally into regions which are basins of attraction,” says Peres. Every point in the sphere lies in the basin of attraction of a particular node, and these basins of attraction result in partitions of equal area no matter how the nodes are distributed. “It’s not clear from this definition that you will get equal areas, but that’s kind of the beauty of it,” he says.

It turns out that the gravitational allocation method has applications for another widely studied problem: matching. “In the matching problem, you have n blue points and n red points, and you’re trying to match them up so that the average distance within matched pairs is small,” says Peres. The optimal algorithm for this type of matching was known, but it was quite complex. “It turns out that one can use this gravitational allocation method to get quite a fast method that matches the optimal up to a constant,” he says.

Tackling New Challenges

Peres continues to work on gravitational allocation and its applications to matching, as well as problems in machine learning and optimization. However, over the last two years, he says he has been particularly fascinated by a topic that arose independently in two distinct fields: communication theory and biology. The application of the topic from communication theory to biology is particularly relevant for phylogenetic reconstruction: how to reconstruct the DNA of an initial ancestor from the DNA of a lot of descendants, each with various deletions.

This turns out to be a really hard problem, and one Peres is still trying to crack. “We have made a lot of progress on figuring this out when the original sequence itself is random,” says Peres. “The progress involves some nice complex analysis, using unexpected tools, but we’re still very far from really understanding the answer,” he says (6).

Like so many other hard problems Peres has tackled throughout his career, the challenge is likely to spur him on. Just as with his many previous successes, whenever he figures it out, it is likely to involve unexpected insights from previously unseen connections.