One talk in particular caught my attention: Leonardo Rastelli‘s talk on “The Superconformal Bootstrap” who discussed the work of the Simons Bootstrap collaboration. I didn’t understand much of the talk (in fact, probably less than 10 percent) but the high level tidbits I got seemed fascinating, and so am posting here some of my understanding. Most, if not all, of what is written below is probably false or inaccurate, but I hope other people that understand this more will correct me.
So, it seems one way to think of a physical theory is as being a way to predict some observables. More formally, a theory maps a point in spacetime to the observable values. In fact apparently the right way to think about this map is to map a tuple of points to the correlation between these observables, something that is known as a “correlation function”.
The traditional view is that these observables arise out of some local interactions between particles. In a computer science way, we could think of modelling spacetime by a graph G such as the d-dimensional lattice. The state of the system corresponds to some assignment of values to the vertices of G, and there is some function that maps each state to its “energy” by summing up over the local interactions. Then the probability of obtaining a particular assignment is weighed by something exponentially small in its energy, and the predictions are obtained by sampling (or computing analytically) this distribution. Unfortunately, it seems (if I understand correctly) that there are some theories for which the physicists don’t know of (and even strongly suspect that there exists no) such local “Lagrangian” explanation for the theory. Moreover (as we can all relate to), even when such an explanation exists, computing the global predictions from the local information could be very hard.
Apparently however, if one posits certain symmetries on the theory, and particular conformal symmetry (which I believe means that the theory be scale free – the predictions are the same if we focus on a small region of space time as it would be in a large one, and is also invariant under rotations), then there are some global constraints on the form of these theories. However, figuring out what these constraints mean is not so simple, and I guess many physicists thought that even after doing all this work, probably they would not be able to derive much from these seemingly few global constraints.
However, it turns out that they can use semidefinite programming to calculate constraints on the allowed theories in “theoryspace” and in fact, using these semidefinite programs alone it might be possible to completely determine some properties of physical theories from first principles. This is not about using optimization to analyze data in applied physics but rather doing pure theoretical physics via semidefinite programming. (In fact, if I understand correctly, these conformal theories are about idealized universes which do not precisely match our own.)
This reminds me of course of Razborov’s Flag Algebra work of using semidefinite programming to derive inequalities in combinatorics. In fact, at least to me, some of the notation used in both cases look quite similar (one of the figures below is from a talk by Lovász on flag algebras, the other is from the Simons collaboration on the superconformal bootstrap)
One can almost imagine an updated version of David Hilbert’s famous quote
We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by semidefinite programming, for in mathematics there is no ignorabimus.
p.s. Scott Aaronson invokes a roughly contemporaneous quote of Roosevelt in discounting critics. While I agree with both of Scott’s sentiment and the examples he mentions (except perhaps that the particular “critics” he talks about are best left alone in the dark corners of the Internet), I can’t help but thinking that he is being a little hypocritical. After all, Scott himself keeps criticizing one person who constantly strives valiantly for the American people. A person that, unlike some of his critics, was never captured by the enemy, but overcame debilitating foot spurs to achieve great skill in golf, only to make the ultimate sacrifice and restrict his playing to the weekends for the good of the nation. A person who unites Americans of all stripes, from white nationalists to businessmen who want their taxes cut. Perhaps Scott can take some of his own medicine, and learn to appreciate greatness (and bigness) rather than criticize it.
This is not exactly related, but what prompted me to write this post was hearing from a friend (who is a non CS faculty in another part of the U.S.) whose children were kicked out from a private (non religious) school when the principal learned they come from an LGBT family. I was truly shocked that something like that can happen in a fairly large U.S. city. It got me thinking (again) of how easy it is to believe that such issues are a thing of the past, or happen in only the remotest parts of the world or the country, when you are not part of the affected population.
STOC 2018 will again be a Theory Fest and also celebrate the 50th anniversary of STOC. As part of the “retro” atmosphere, the STOC PC also asks submissions to be sent by mail in 20 printed copies at most 10 pages.
The Microsoft Research family is shocked and deeply saddened by the tragic loss of our colleague and friend Michael Cohen. Michael was a brilliant mathematician and a rising star in his field. He spent this past summer with us as a Microsoft Research Fellow, doing what he loved most. Over the summer, he made sweeping progress in online learning and online algorithms, two fields he had just recently become acquainted with. In addition to solving five open problems in these areas, he continued his substantial progress on the k-server problem, one of the most celebrated and notoriously difficult challenges in the space of adaptive algorithms.
Michael was a truly exceptional individual. We will remember Michael for his infectious smile and his larger-than-life personality. We will never forget his unrelenting curiosity, his thirst for knowledge, and his deep love for mathematics and theoretical foundations of computing. We are stunned by his loss. We will hold onto our memories of Michael, and know that his ideas and scientific accomplishments will continue on as important advances.
We extend our most sincere condolences to Michael’s family and friends.
Our Friday theory reading group will move to the CMSA center and feature many speakers from the special year.
In addition, the program will feature a series of public lectures by Noga Alon (Sept. 7), Jennifer Chayes (Nov. 2), Jacob Fox (Feb. 1), and Dan Spielman (March 20).
Combinatorics will also be featured in this year’s Ahlfors Lecture Series, given by Timothy Gowers (University of Cambridge) on October 11-12. Leslie Valiant will give the Ding Shum lecture on October 10.
Harvard will also host the Women In Theory workshop in the summer of 2018 (details forthcoming),
Participation: Researchers interested in participating in the special year through a short- or long-term visit to CMSA are encouraged to contact the organizers through the CMSA Administrative Coordinator, Sarah LaBauve (slabauve@math.harvard.edu). Each of the workshops is also open to participation by all interested researchers subject to capacity. Registration forms can be found on the webpages for the individual workshops.
Like many other new grad students, when I just started at the Weizmann Institute of Science, I was extremely anxious. While I loved (the little I knew about) theory, I was not sure if I’m smart enough to do original research in this area. The professors seemed like beings from another world, but one thing that helped was that at least I did share some background with them. Almost all of them were males (and in fact, this being Israel, almost all were, like me, Ashkenazi secular jews). So, it was not completely unthinkable to imagine that I would some day be like them.
Another thing that helped me a lot in my early time in Weizmann was my study group.
I quickly found three friends, all male, and we spent many hours together studying for exams, talking about research, but also playing video games.
As I progressed in the field, I’ve had a great many research interactions and collaborations. I’ve met people at all hours and various locations, including coffee shops, restaurants, and private homes. I’ve never had to worry about the potential for sexual harassment, nor that I would be judged by my looks or what I wear. I am the type of person that speaks their opinion loudly, perhaps sometimes a little too loudly. In my experience, women that act the same are judged by a different standard and often “rub people the wrong way” and get tagged as having problematic personalities. Also, I was never afraid to ask potentially stupid questions. I did not need to fear that asking such a question would confirm people’s bias about me that I don’t belong in this field. Asking potentially stupid questions is one of the most useful ways to learn and evolve as a scientist.
Last but not least, I would most definitely not be where I am if my wife did not quit her job in Israel to move with me across the Atlantic for my postdoc. Also, while I think of myself as an involved father, I have to admit that Ravit still does more of the childcare duties. (As I’m reading this out loud, my daughter is commenting that I am not such an involved father…) Again, I am not saying that all male scientists are like me, nor that there are no female scientists that greatly rely on the support of their partners, but I don’t think my situation is atypical.
On more general terms, one misconception I see about science in such discussions is that it is about “things” or “facts” and not people, and hence perhaps should be free of biases.
But in fact science is an extremely social enterprise. Since graduating, I have never written a solo paper. Much of my work is spent talking to people in front of a whiteboard or notepad, and I’ve learned a great deal from discussions with fellow scientists. This also explains why certain subfields of science can have outlying gender or ethnicities ratios. People just feel more comfortable working with others similar to them, and role models greatly influence the fields students choose to pursue. For example, there is nothing innately Hungarian about combinatorics. Similarly, there is nothing innately feminine about cryptography, but rather a few extremely strong female scientists have had a huge effect. The influence of people such as Shafi Goldwasser, Tal Rabin, and others has not been limited just to their (amazing) research results but also as role models and mentors to many female cryptographers (and many male cryptographers as well, myself included).
I don’t like the expression “leveling the playing field” because Science is not a game. This is not about competing but about collaborating with each other to uncover the mysteries of the universe. But us males (especially those that, like me, come from amply represented ethnicities), should be aware and grateful for the advantages we’ve had in our careers. We should not try to remove these advantages: access to role models, freedom from bias, ease of networking, are all good things. We should however strive for a world where everyone enjoys the same benefits. In such a world we’ll have more scientists that are more productive and happier, and that will only be better for science.
Early registration deadline for the conference is Sept 25th, 2017.
Speaking of workshops, there is one way to guarantee that FOCS has a workshop in an area you care about, and it is to organize such a workshop yourself!
Speaking from experience, organizing a workshop is non-zero but not unmanageable amount of work. It is a great way to connect with colleagues in your area, as well as expose some other people in the TCS community to it. The call for workshops is now up at http://focs17.simons.berkeley.edu/workshopAndTutorial.html
A proposal for a workshop can be quite minimal – the main thing you need is the names of the organizers and speakers. To submit a proposal, send an email to Aleskander Mądry and James Lee by September 3, 2017.
Of course there are results such as the time hierarchy theorem and Ladner’s theorem that tell us that “intermediate complexity” do exist, but the hope is that this doesn’t happen for “natural” problems.
(In particular, we can artificially create problems of complexity or by padding exponentially hard problems. This is one of the reasons why Impagliazzo Paturi and Zane argued that for NP problems, the parameter should be thought of the solution size and not the input size.)
One family of natural problems are constraint satisfaction problems (CSPs). For some finite alphabet and constant , a CSP is characterized by a set of predicates from .
The input for is a collection of functions , each of which applies some predicate in to a -subset of its coordinates.
The basic task in CSPs is exact satisfiability, where the goal is to tell whether there is some assignment for the inputs that such that .
The celebrated CSP dichotomy conjecture (which perhaps should now be called a theorem) implies (in its modern, algebraic form) that for every , either there is a polynomial-time (in fact I believe that at most time) algorithm for the exact satisfiability of or there is a constant-factor blowup reduction from 3SAT to , implying that (under the exponential-time hypothesis or ETH) its complexity is .
The approximation problem for is parameterized by two numbers (known as the soundness and completeness paramters) and asks to distinguish, given an input of , between the case that there is some assignment such that , and the case that for every . There is one sense in which understanding the complexity of the approximation problem is easier, since in the context of approximation, the pesky difference between 3SAT and 3XOR (bane of many failed NP vs P proofs, and the main reason the dichotomy conjecture is so hard to prove) disappears. On the other hand, in this context a problem such as BIPARTITNESS (or ), which was in the EASY column of exactly solving, become the NP-hard problem of Max-Cut.
One of the most fascinating consequences of the Unique Games Conjecture is that, if it is true, then there are in fact CSPs for which their approximation is neither EASY nor HARD. That is, the Unique Games Conjecture (plus the ETH) implies the following conjecture:
Intermediate Complexity Conjecture: For every , there exist some , and such that vs approximation of can be done in time but (the potentially easier) vs approximation cannot be done in time .
This is a consequence of the subexponential time algorithm for unique games, which in particular implies that for, say, , there is some that tends to zero with such that the vs problem for the unique games CSP can be solved in time .
On the other hand, the Unique Games Conjecture implies that for every , no matter how close is to zero, or is to , the vs problem for unique games is NP hard and hence (under the ETH) cannot be solved in time .
I find this conjecture very fascinating. A priori, we might expect a zero one law for CSP’s complexity, where, depending on the approximation quality, one could either solve the problem in time or require at least time. Indeed, we have good reasons to believe that predicates such as 3XOR and 3SAT satisfy such a zero/one law. But, if the intermediate complexity conjecture is true then for some predicates (such as unique games itself, and probably even max cut, which can hardly be called “unnatural”) we would get a non-trivial curve of the running time vs approximation quality, that looks something like this:
Figure 1: Conjectured time complexity of vs approximation of unique games as a function of assuming the unique games conjecture. The UGC characterizes the threshold when the time complexity becomes for some , while there is also a (not yet fully determined) threshold when there is a linear blowup reduction from 3SAT and hence (under the ETH) the complexity is .
(One other setting where people were not sure if the complexity of some finitely specified object must be either exponential or polynomial is group theory, where one can define the complexity of a finitely generated group as the number of distinct elements that can be generated by a word of length $n$. There it turned out there are groups of “intermediate complexity”.)
What is also quite interesting is that we might be able to prove the intermediate complexity conjecture without resolving the UGC. In an exciting recent work, Dinur, Khot, Kindler, Minzer and Safra gave a combinatorial hypothesis that, if true, would imply the NP hardness of vs approximation for unique games where , and can be arbitrarily close to . Thus, if their hypothesis is true then, combined with the subexponential time algorithm mentioned above, it would imply the intermediate complexity conjecture. (If the construction is improved to obtain completeness that is close to then of course it would prove the unique games conjecture as well.)
Through discussions with Pravesh Kothari and David Steurer, we showed that the DKKMS combinatorial hypothesis is equivalent to a fairly simple to state statement. Informally, it can be stated as follows: let be the graph on binary matrices, such that two matrices have an edge if has rank one over GF(2). (UGC/PCP experts might recognize this graph as the degree two short code graph.) This graph is not a small set expander in the sense that, if you think about it for few minutes, you see that there are sets matrices of measure such that if we pick a random and a random neighbor of , then with constant probability will also be in . The DKKMS hypothesis is equivalent to the conjecture that these sets that you think about after few minutes encompass all such examples.
Let me state this more formally: Lets say that a subset is a basic set if or where are column vectors. Note that each such set has measure among all matrices and that a random neighbor of an element in will be in with probability . For every constant , you can clearly construct a set where a random neighbor stays inside with probability by intersecting basic sets. You can also keep this probability constant even if you just subsampled a constant fraction of this intersection (and you can add a few random vertices as well). The DKKMS hypothesis is equivalent to the statement that this is all you can do. That is, it is equivalent to the following statement:
Inverse short code conjecture: For every there exists some and such that if is large enough and is the graph above with vertex set and if , then for every , if satisfies then there are basic sets such that satisfies .
I call this an “inverse conjecture” since it is aimed at characterizing the sets that have unusually small expansion in the short-code graphs in terms of their correlation with basic sets.
I find it a natural question, though have to admit that, despite thinking about it some time (and also talking about it, not just with David and Pravesh but also Irit Dinur, Shachar Lovett and Kaave Hoesseini) I still don’t have a strong intuition whether it’s false or true.
Given that the short code graph is a Cayley graph over the Boolean cube, this seems like a question that could be amenable to tools from Fourier analysis and additive combinatorics.
Some partial progress has been made by the DKKMS folks in their new manuscript.
In fact, the same observations show that their original hypothesis is also equivalent not just to the “inverse short code conjecture” but also to Hypothesis 1.7 in this report, which is an analogous “inverse conjecture” for the Grassmanian graph where, for , the vertices are dimension subspaces of , and two subspaces are neighbors if their intersection has dimension .
Nevertheless, at the moment this question is still wide open, and I hope this writeup encourages more people to work on it. Other than resolving this question, there are some other interesting research directions, including:
The underlying building block for DKKMS is a problem known as a smooth label cover. This is not precisely a CSP, but can we show that it has intermediate complexity? i.e., give a subexponential time algorithm for it? If the DKKMS hypothesis is correct then we know that such an algorithm should be an SDP hierarchy.
Suppose the combinatorial hypothesis is true, is there an inherent reason why its proof should not be a low degree SOS proof? After all, related isoperimetric results on the short code graph have been shown by low degree SOS proofs. Also, is the proof of the current best bounds coming from the DKKMS manuscript SOS-izable?
The quantiative bounds for the DKKMS reduction are terrible. I haven’t worked out all the details but it seems that the reduction from 3XOR to vs approximation of unique games would map an variable instance to an instance of at least (or maybe even ) size. Is this inherent?
Is there a general conjecture that would predict for (say even if is a single predicate) and what should be the time complexity of vs approximation for it? We have some approaches of trying to predict when the complexity should be by considering pairwise independent distributions. I feel like a conjecture trying to characterize what can be done in time would have something to do with defining a suitable notion of “$(1+\epsilon)$-wise independent distributions” perhaps via concepts of mutual information. For example, perhaps we can argue that a smooth label cover is a -wise independent CSP to a certain extent.
This fall I will be teaching CS 121 at Harvard: Introduction to Theoretical Computer Science.
This type of “intro theory” course is taught at many universities, sometimes under the name “introduction to the theory of computation” or “computability and automata”, typically using Sipser’s excellent book.
Sipser’s book is incredibly well-written and liked by students, but there have been many developments in theoretical computer science and CS at large since it was written, both in terms of new results, techniques, and applications, as well as new emphases and points of view on this area.
So what should an “intro TCS” course contain these days?
My sense is that this course is first and foremost about studying computation in its own right, as opposed to being a tool to solve other problems.
This is not just about computability and complexity, but also about looking at different ways to measure the “goodness” of algorithms, especially given the drastic change in the way algorithms interact with the world these days.
So, for example, students in such a course should get at least some awareness of questions such as privacy or incentive-compatibility, even if doing these topics justice requires dedicated courses.
Another point that should be “hammered in” more is of hardness as a resource.
This of course arises in derandomization and cryptography, where recently cryptocurrencies have been quite explicit about equating computational difficulty with “mining” precious metals.
The interaction of computation with the physical world need to also be covered, and these days an “intro TCS” course cannot skip talking about quantum computing, which is our current best approach for modelling physically feasible computation.
But more than anything, considering computation as an object in its own right forces students to change their way of thinking, and truly come to terms with questions such as how do we model computation, how code is also data, and all the self-referential “paradoxes” and complications this entails.
This has always been present in “intro TOC” courses, but students can sometimes lose the big perspective while working on the n-th proof of non-regularity using the pumping lemma, or the m-th NP hardness reduction.
I am of course not alone in thinking of the need for a “more modern” intro TCS course.
See for example, Scott Aaronson’s MIT course. Also CMU’s renowned (or is it infamous? ) Great Ideas in TCS has been following such an approach for quite a while, and the folks at CMU have been very kind with advice and suggestions.
There are also books such as Aaronson’s Quantum Computing Since Democritus and Moore and Mertens’ The nature of computation that take a similar perspective, though neither is quite appropriate as an introductory undergraduate textbook.
Thus, I am working on writing a new set of lecture notes, which are available on http://www.introtcs.org.
These notes are very much a work in progress, and I welcome comments and suggestions.
In particular, the source for these notes is posted on a github repository and people are very much invited to use the issues and pull requests features there to post suggestions, comments, and point out (or even fix!) typos and bugs.
Some of the differences between these notes and the typical treatment include the following:
foo := bar NAND baz
assigns to the variable foo
the result of the NAND of the variables bar
and baz
. There are no loops, if statements, or subroutines, so this corresponds to straightline programs that are of course the same as Boolean circuits. Since students are already familiar with programming, I believe that starting with such an ultra-simple programming language makes things more concrete. Moreover, thanks to the efforts of my wonderful teaching fellow Juan Esteller, there is an implementation of the NAND, NAND++, and NAND<< programming languages that are taught in this course.NAND
programming language to the NAND++
programming language that allows loops by simply stipulating that the program execution goes back to the first line if the loop
variable has the value 1
. We also introduce an index variable i
so that we can use foo_i
to access a potentially unbounded memory. Of course, this is equivalent to Turing machines, and we show this equivalence (as well as equivalence to the lambda calculus and other models) . We also give a further enhancement of the NAND++ language that is equivalent to RAM programs, and so allows us to measure running time in the same it is measured in algorithms courses.
It is definitely an ambitious plan that will not make for an easy course to take or to teach.
Some might say that it’s a “recipe for disaster” but the saving grace is that, with the help of Salil Vadhan, we have assembled a wonderful teaching staff that should be a great help for the students in keeping up.
I will report back on how realistic it was at the end of the term.
Even with our fast pace, there are many areas of theory that are not mentioned above, but students should be exposed to. I hope to cover some of these topics in this course, or at least write lecture notes about them so that curious advanced students could read about on their own.
Once again, much of this is not novel, and courses such as the ones mentioned above have taken similar approaches. However, I still hope these notes can serve a useful resource for other people who teach similar courses.
Acknowledgements: These lecture notes are constantly evolving, and I am getting input from several people, for which I am deeply grateful.
Thanks to Jarosław Błasiok, Chi-Ning Chou, Juan Esteller, Alex Lombardi, Thomas Orton, Juan Perdomo, and Ryan Williams for helpful feedback.
Thanks to Anil Ada, Venkat Guruswami, and Ryan O’Donnell for helpful tips from their experience in teaching CMU 15-251.
Thanks to Juan Esteller for his work on implementing the NAND* languages.
Thanks to David Steurer for writing the scripts that we used for notes on sum of squares and that I am now repurposing to produce these notes and for his endless patience with me as I attempt to learn concepts such as git and makefiles. David’s code itself is based on several other packages, including pandoc, LateX, and the Tufte LaTeX and CSS packages.
Having entered into the organization of TheoryFest 2017 with some trepidation, we organizers were very relieved to see feedback such as “Best. STOC. Ever”. This post shares with you the feedback we got from attendees, and our plans for the next couple of years.
Important: Next year’s Theory Fest will be June 25-29, 2018 in downtown LA. See you there!
Location, location, location. Smack in the middle of lovely Montreal, in the midst of an ongoing street festival/party, and within walking distance to the waterfront and good restaurants. I wouldn’t complain if the conference were held here every year!
No catered lunches. This greatly reduced registration fees, and (together with the weak Canadian dollar) allowed us to be more generous with food and drink during breaks (including an open bar on three nights!). It also encouraged attendees to explore Montreal a bit more.
Every STOC paper also presented at poster session: a hit. Billed as the most controversial aspect of the TheoryFest, this ended up (based upon in-person and online feedback) as the most positive change. Respondents said they expected to hate poster sessions and ended up loving them. Having experienced the energy of poster sessions at non-theory conferences, I am not surprised. (We head the complaint that the poster room was a bit noisy, and will address it.)
Energy level. A day filled with different kinds of events seems to help keep up people’s energy levels. Attendance at the talks was high; I saw hardly any people loitering about in the public areas during talks. Possibly, it helped that STOC talks were in three parallel sessions; 50% more choice of topics at any given moment.
Highest attendance among STOC/FOCS held abroad. 377 attendees, which is quite high for the past decade.
Junior-senior lunches. Via a google doc page, 3 junior researchers (grads/postdocs) got to sign up to go to lunch with a senior researcher. These lunches got rave reviews, though not enough people participated due to the short notice. Next time we’ll be better organized.
117 attendees have responded thus far to our online survey. Their average satisfaction with TheoryFest was 4.5/5 .
Large majorities (>75%) found the amount of time devoted to various sub-components “about right.”
In terms of quality, the poster session was the biggest hit, followed by the plenary long talks. STOC paper talks had somewhat weaker ratings. Here’s a more detailed feedback.
Given the general support for the changes introduced this year, next year’s event will see no more drastic changes; merely tweaks to improve the overall experience.
Some feedback we received include:
“Stoc talks need to be longer.” “Short stoc talks worked well; could shorten even more.”
The alternatives we see are:
Both options have significant minority support but appear to lack majority support, which was the reason behind our split-the-baby decision of 17 min talks in 3 parallel sessions. We welcome further discussion. (The PC Chair Valerie King commented to me that she’d expected to dislike 17 min talks but ended up preferring this format.)
“The middle three days were a bit tiring; stretching from 9am to 10:30pm.”
We realize this and don’t see a good alternative. We hew to the philosophy that a big-tent theory conference should offer a large buffet of content, and attendees are free to decide how much to partake. One major problem with the old setup of STOC was an artificial cap on content —and arguably the day was nevertheless long and monotonous. Nobody complained about monotony this time.
“Some short plenary talks weren’t geared to a STOC audience.”
We’ll address this by assigning each outside speaker a “theory envoy” who will give them feedback on their slides and talk plan. But clearly, the problem will never go away 100%.
Overall, we welcome your feedback! If you didn’t fill out your Theory Fest survey thus far, please do so asap before we close it out, and feel free to also post your feedback as comments to this post.