Windows On Theory

By Boaz Barak and Omer Reingold

Yesterday Hillary Clinton became the first woman to be (presumptively) nominated for president by a major party. But in the eyes of many, the Republican Party was first to make history this election season by breaking the “qualifications ceiling” (or perhaps floor) in their own (presumptive) nomination.

Though already predicted in 2000 by the Simpsons , the possibility of a Trump presidency has rattled enough people so that even mostly technical bloggers such as Terry Tao and Scott Aaronson felt compelled to voice their opinion.

We too have been itching for a while to weigh in and share our opinions and to use every tool in our disposal for that, including this blog. We certainly think it’s very appropriate for scientists to be involved citizens and speak up about their views. But though we debated it, we felt that this being a group (technical) blog, it’s best not to wage into politics (as long as it doesn’t directly touch on issues related to computer science such as the Apple vs. FBI case). Hence we will refrain from future postings about the presidential election. For full disclosure, both of us personally support Hillary Clinton and have been donating to her campaign.

The last few weeks have seen amazing results in additive combinatorics, where following a breakthrough by Croot, Lev and Pach, several longstanding open questions have been resolved using short simple proofs. I haven’t been following this progress, but fortunately Bobby Kleinberg gave an excellent talk yesterday in our reading group about some of these works, and their relations to TCS questions such as approaches for fast matrix multiplication. Since the proofs are short and these results also been extensively blogged about, what follows is mainly for my own benefit.

Among other things, Bobby showed the proof of the following result, that demonstrates much of those ideas:

Theorem: (Ellenberg and Gijswijt, building on Croot-Lev-Pach) There exists an absolute constant such that for every , if then contains a 3-term arithmetic progression.

To put this in perspective, up till a few weeks ago, the best bounds were of the form and were shown using fairly complicated proofs, and it was very reasonable to assume that a bound of the form is the best we can do. Indeed, an old construction of Behrend shows that this is the case in other groups such as the integers modulo some large or where is some large value depending on . The proof generalizes to for every constant prime (and for composite order cyclic groups as well).

The proof is extremely simple. It seems to me that it can be summarized to two observations:

• Due to the algebraic structure of the problem, one can “interpolate” in some sense a -a.p. free set with a polynomial of degree that is a half as small than you would expect otherwise.

• Due to concentration of measure phenomena in finite fields, this constant multiplicative factor makes a huge difference. There are values for which polynomials of degree make up all but an exponentially small fraction of the functions from to , while polynomials of degree only constitute an exponentially small fraction of these functions.

Let’s now show the proof. Assume towards a contradiction that satisfies ( can be some sufficiently small constant, will do) but there do not exist three distinct points that form a -a.p. (i.e., such that or, equivalently, ).

Let be the number of -variate monomials over where each variable has individual degree at most (higher degree can be ignored modulo ) and the total degree is at most . Note that there are possible monomials where each degree is at most two, and their degree ranges from to , where by concentration of measure most of them have degree roughly . Indeed, using the Chernoff bound we can see that if is a sufficiently small constant, we can pick some such that if then but (to get optimal results, one sets to be roughly and derives from this value).

Now, if we choose in that manner, then we can find a polynomial of degree at most that vanishes on but is non zero on at least points. Indeed, finding such a polynomial amounts to solving a set of linear equations in variables.¹ Define the matrix such that . Since the assumption that implies that , the theorem follows immediately from the following two claims:

Claim 1: .

Claim 2: .

Claim 1 is fairly immediate. Since is -a.p. free, for every , is not in and hence is zeros on all the off diagonal elements. On the other hand, by the way we chose it, has at least nonzeroes on the diagonal.

For Claim 2, we expand as a polynomial of degree in the two variables and , and write where corresponds to the part of this polynomial where the degree in is at most and corresponds to the part where the degree in is larger and hence the degree in is at most . We claim that both and are at most . Indeed, we can write as for some coefficients and polynomials , where indexes the monomials in of degree at most . But this shows that is a sum of at most rank one matrices and hence . The same reasoning shows that thus completing the proof of Claim 2 and the theorem itself.

Actually, not really…

Northwestern University held a workshop on semidefinite programming hierarchies and sum of squares.  Videos of the talks by Prasad Raghavendra, David Steurer and myself are available from the link above. The content to unicorns ratio in Prasad and David’s talks is much higher ☺

Tomorrow, Wednesday May 25, is the international Towel Day in honor of Douglas Adams, author of the 5-book trilogy “The hitchhiker’s Guide to the Galaxy”. In his book (and his prior 1978 radio series) Adams gave a nice illustration of computational complexity and non uniform computation in his story about the “deep thought” computer who took 7.5 million years to answer the ultimate question of life, the universe, and everything, though today Google’s calculator can do this in 0.66 seconds.

If you want to celebrate towel day in Cambridge, bring your towel to Tselil Schramm’s talk on refuting random constraint satisfaction problems using Sums of Squares in MIT’s algorithms and complexity seminar (4pm, room 32-G575).

LifeHacker is a website dedicated to tricks, especially technology related, to make people more efficient or productive. The signal to noise ratio in that website is not particularly high, but I think it is a good idea to share tricks that can save time and effort. In particular, are there tools or tricks that made a difference in your life as a theorist?

Here are some of the tools I use. This is obviously a very biased list and reflects my limitations as a Windows user that is too stupid to learn to use emacs  and vim. Please share your better tips in the comments:

LaTeX editor / collaboration platform: I recently discovered Overleaf  which can be described as “Google Docs” for LaTeX. It’s a web-based LaTeX editor that supports several people editing the same document simultaneously, so it’s great for multi-author projects, especially for those last days before the deadline where everyone is editing at once. There are few things as satisfying as watching improvements and additions being added to your paper as you’re reading it. One of the features I like most about it is its git integration. This means that you can also work offline on your favorite editor and pull and push changes from/to the overleaf repository. It also means that authors that don’t want to use overleaf (but can use git) can still easily collaborate with those that do. I actually like the editor enough that I’ve even used it for standalone papers.

Version control: I’ve mentioned git above, and this is the version control I currently use. I found source tree to be an easy GUI to work with git, and (before I switched to overleaf) bitbucket to be a good place to host a git repository. Git can sound intimidating but it takes 5 minutes to learn if someone who knows it explains it to you. The most important thing is to realize that commit and push are two separate commands. The former updates the repository that is on your local machine and the latter synchronizes those changes with the remote repository. The typical workflow is that you first pull updates, then make your edits, then commit and push them. As long as you do this frequently enough you should not have serious conflicts. I am also committing the cardinal sin of putting my git repositories inside my dropbox folder. There are a number of reason why it’s a bad idea, but I find it too convenient to stop. To make this not blow up, I never use the same folder from two different computers and hence have subfolders “Laptop” and “Desktop” for the repositories used by these computers respectively. Update 4/29: Clement Canonne mentions in a comment the Gitobox project that synchronizes a dropbox folder and a git repository and so allows easier collaboration with your non-git-literate colleagues.

Markdown: I recently discovered markdown and particularly its pandoc flavored variety as a quick and easy way to write any technical document – lecture notes, homework assignments, blog posts, technical emails – that is not an actual paper. It’s just much more lightweight than LaTeX and so you type things faster, but it can still handle LaTeX math and (using pdflatex) compile to both html and pdf. All the lecture notes for my crypto course were written in markdown and compiled to both html and pdf using pandoc.

Editor: I was a long time user of winedt but partially because of markdown and other formats, I decided to switch to a more general purpose editor that is not as latex centric. I am currently mostly using the Atom editor that I feel is the “editor of the future” in two senses. First, it is open source, backed by a successful company (github) and has a vibrant community working on extending it. Second, it’s the editor of the future in the sense that it doesn’t work so well in the present, and it sometimes hangs or crashes. If you prefer the “editor of the present” then sublime text might be for you. Some Atom packages I use include Build, Emmet, latextools, language-latex, language-pfm, Markdown Preview Plus,  Mathjax-wrapper, pdf-view, preview-inline, sync-settings (I find it also crucial to enable autosave ).

Remote collaboration: I use several tools to collaborate with people remotely. I’ve used slack as a way to maintain a long-running discussion relating to a project. It works much better than endless email threads. In a technical discussion we’ll sometimes open a google hangouts video chat and in addition use  appear to share a screen of a OneNote pad (assuming one or both of the discussants has a pen-enabled computer such as the Microsoft Surface Pro or Surface Book that seem to become the theorists’ new favorite these days). A good quality camera aimed at the whiteboard also works quite well in my experience.

Presentation: As I’ve written before, Powerpoint has awesome support for math in presentations. One thing which I would love to have – a visual basic script that goes over all my slides and changes all the math in them to a certain color. It’s a pain to do this manually.

Bibliography maintenance:  Here is where I could use some advice. It seems that in every paper I end up spending the last few hours before the deadline scouring DBLP and Google Scholar for bib items and copying/pasting/formatting them. I wish there was an automatic script that would scan my tex sources for things like \ref{Goldwasser-Micali-Rackoff??} and return a bibtex file containing all of the best matches from DBLP/Google Scholar. Bonus points if it can recognize both the conference and journal version and format a bibtex which cites the journal version but adds a note “Preliminary version in STOC ’85”. Update 4/29: A commenter mentions the CryptoBib project that maintains a super-high-quality bibtex file of all crypto conferences and some theory conferences such as ICALP, SODA, STOC, FOCS. Would be great if this was expanded to all theory conferences.

Note taking: I am still a fan of a yellow pad with pen, but I find myself using OneNote quite often these days on my surface book (which is also the “computer of the future” in a sense quite similar to Atom..). It’s useful to take notes in talks, and also to write notes for myself before teaching a class so that they would be available for me the next time I teach it.

p.s. Thanks to David Steurer who I learned many of these tools from, though he shares no responsibility for my failure to learn how to use emacs.

Aleksander Madry asks me to announce that registration for the Highlights of Algorithms conference he posted about is open.
The registration link is http://highlightsofalgorithms.org/registration/. (Early registration is due April 30.)

Also, the preliminary program is available at http://highlightsofalgorithms.org/program/.
The program is packed with 28 invited talks and with even a larger number of short contributions.

Those interested in attending the conference are advised to book accommodation as early as possible (given the high hotel prices in Paris).

(See also pdf version , and these lecture notes)

The divide between frequentists and Bayesians in statistics is one of those interesting cases where questions of philosophical outlook have actual practical implications. At the heart of the debate is Bayes’ theorem:

Both sides agree that it is correct, but they disagree on what the symbols mean. For frequentists, probabilities refer to the fraction that an event happens over repeated samples. They think of probability as counting, or an extension of combinatorics. For Bayesians, probabilities refer to degrees of belief, or, if you want, the odds that you would place on a bet. They see probability as an extension of logic.¹

A Bayesian is like Sherlock Holmes, trying to use all available evidence to guess who committed the crime. A frequentist is like the lawmaker who comes up with the rules how to assign guilt or innocence in future cases, fully accepting that in some small fraction of the cases the answer will be wrong (e.g., that a published result will be false despite having statistical significance). The canonical question for a Bayesian is “what can I infer from the data about the given question?” while for a frequentist it is “what experiment can I set up to answer the question?”. Indeed, for a Bayesian probability is about the degrees of belief in various answers, while for a frequentist probability comes from the random choices in the experiment design.

If we think of algorithmic analogies, then given the task of finding a large clique in graphs, a frequentist would want to design a general procedure that has some assurances of performance on all graphs. A Bayesian would be only interested in the particular graph he’s given. Indeed, Bayesian procedures are often exponential in the worst case, since they want to use all available information, which more often than not will turn out to be computationally costly. Frequentists on the other hand, have more “creative freedom” in the choice of which procedure to use, and often would go for simple efficient ones that still have decent guarantees (think of a general procedure that’s meant to adjudicate many cases as opposed to deploying Sherlock Holmes for each one).

Given all that discussion, it seems fair to place theoretical computer scientists squarely in the frequentist camp of statistics. But today (continuing a previous post) I want to discuss what a Bayesian theory of computation could look like. As an example, I will use my recent paper with Hopkins, Kelner, Kothari, Moitra and Potechin, though my co authors are in no way responsible to my ramblings here.

Peering into the minds of algorithms.

What is wrong with our current theory of algorithms? One issue that bothers me as a cryptographer is that we don’t have many ways to give evidence that an average-case problem is hard beyond saying that “we tried to solve it and we couldn’t”. We don’t have a web of reductions from one central assumption to (almost) everything else as we do in worst-case complexity. But this is just a symptom of a broader lack of understanding.

My hope is to obtain general heuristic methods that, like random models for the primes in number theory or the replica method in statistical physics, would allow us to predict the right answer to many questions in complexity, even if we can’t rigorously prove it. To me such a theory would need to not just focus on questions such as “compute from ” but tackle head-on the question of computational knowledge: how can we model the inferences that computationally bounded observers can make about the world, even if their beliefs are incorrect (or at least incomplete). Once you start talking about observers and beliefs, you find yourself deep in Bayesian territory.

What do I mean by “computational knowledge”? Well, while generally if you stop an arbitrary C program before it finishes its execution then you get (to use a technical term) bubkas, there are some algorithms, such as Monte Carlo Markov Chain, belief propagation, gradient descent, cutting plane, as well as linear and semidefinite programming hierarchies, that have a certain “knob” to tune their running time. The more they run, the higher quality their solution is, but one could try to interpret their intermediate state as saying something about the knowledge that they accumulated about the solution up to this point.

Even more ambitiously, one could hope that in some cases one of those algorithms is the best, and hence its intermediate state can be interpreted as saying something about the knowledge of every computationally bounded observer that has access to the same information and roughly similar computational resources.

Modeling our knowledge of an unknown clique

To be more concrete, suppose that we are given a graph on vertices, and are told that it has a unique maximum clique of size . Let denote the characteristic vector of the (unknown to us) clique. We now ask what is the probability that . This question seems to make no sense. After all, I did not specify any distribution on the graphs, and even if I did, once is given, it completely fixes the vector and so either or .

But if you consider probabilities as encoding beliefs, then it’s quite likely that a computationally bounded observer is not certain whether is in the clique or not. After all, finding a maximum clique is a hard computational problem. So if is much smaller than the time it takes to solve the -clique problem (which is as far as we know), then it might make sense for time observers to assign a probability between and to this event. Can we come up with a coherent theory of such probabilities?

Here is one approach. Since we are given no information on other than that it has an -sized clique, it makes sense for us to model our prior knowledge using the maximum entropy prior of the uniform distribution over -sized sets. But of course once we observe the graph we learn something about this. If the degree of is smaller than then clearly with probability one. Even if the degree of is larger than but significantly smaller than the average degree, we might want to adjust our probability that to something smaller than the a priori value of . Of course by looking not just at the degree but the number of edges, triangles, or maybe some more global parameters of the graph such as connected components, we could adjust the probability further. There is an extra complication as well. Suppose we were lied to, and the graph does not really contain a clique. A computationaly bounded observer cannot really tell the difference, and so would need to assign a probability to the event that even though does not exist. (Similarly, when given a SAT formula close to the satisfiability threshold, a computationally bounded observer cannot tell whether it is satisfiable or not and so would assign probability to events such as “the variable gets assigned true in a satisfying assignment” even if the formula is unsatisfiable.) This is analogous to trying to compute the probability that a unicorn has blue eyes, but indeed computationally bounded observers are in the uncomfortable positions of having to talk about their beliefs even in objects that mathematically cannot exist.

Computational Bayesian probabilities

So, what would a consistent theory of “computational Bayesian probabilities” would look like? Let’s try to stick as closely as possible to standard Bayesian inference. We think that there are some (potentially unknown) parameters (in our case consisting of the planted vector ) that yield some observable (in our case consisting of the graph containing the clique encoded by , say in adjacency matrix representation). As in the Bayesian world, we might denote to say that is sampled from some distribution conditioned on , and to denote the conditional distribution on given , though more often than not, the observed data will completely determine the parameters in the information theoretic sense (as in the planted clique case). Our goal is to infer a value for some “simple” function mapping to a real number (in our case is simply ). We denote by the computational estimate for given . As above, we assume that the estimate is based on some prior distribution

A crucial property we require is calibration: if is truly sampled from the prior distribution then it should hold that

Indeed, the most simple minded computationally bounded observer might ignore completely and simply let to be the a priori expected value of . A computationally unbounded observer will use which in particular means that when (as in the clique case) completely determines , it simply holds that .

But of course we want to also talk about the beliefs of observers with intermediate powers. To do that, we want to say that should respect certain computationally efficient rules of inference, which would in particular rule out things like assigning a positive probability for an isolated vertex to be contained in a clique of size larger than one. For example, if we can infer using this system from that must be zero, then we must define . We also want to satisfy various internal consistency conditions such as linearity between our estimates for different functions (i.e., that ).

Finally, we would also want to ensure that the map is “simple” (i.e., computationally efficient) as well.

Different rules of inference or proof systems lead to different ways of assigning these probabilities. The Sum of Squares algorithm / proof system is one choice I find particularly attractive. Its main advantages are:

• It encapsulates many algorithmic techniques and for many problems it captures the best known algorithm. That makes it a better candidate for capturing (for some restricted subset of all computational problems) the beliefs of all computationally bounded observers.
• It corresponds to a very natural proof system that contains in it many of the types of arguments, such as Cauchy Schwarz and its generalizations, that we use in theoretical computer science. For this reason it has been used to find constructive versions of important results such as the invariance principle.
• It is particularly relevant when the functions we want to estimate are low degree polynomials. If we think of the data that we observe as inherently noisy (e.g., the data is a vector of numbers each of which corresponds to some physical measurement that might have some noise in it), then it is natural to restrict ourselves to that case since high degree polynomials are often very sensitive to noise.
• It is a tractable enough model that we can prove lower bounds for it, and in fact have nice interpretations as to what these “estimates” are, in the sense that they correspond to a distribution-like object that “logically approximates” the Bayesian posterior distribution of given .

SoS lower bounds for the planted clique.

Interestingly, a lower bound showing that SoS fails on some instance amounts to talking about “unicorns”. That is, we need to take an instance that did not arise from a model of the form (e.g., in the planted clique case, a graph that is random and contains no planted clique) and still talk about various estimate of this fictional .

We need to come up with reasonable “pseudo Bayesian” estimates for certain quantities even though in reality these estimates are either completely determined (if came from the model) or simply non-sensical (if didn’t). That is, for every “simple” function , we need to come up with an estimate . In the case of SoS, the notion of “simple” consists of functions that are low degree polynomials in . For every low degree polynomial , we need to give an estimate that estimates from the graph . (Of course if we had unbounded time and really was from the planted distribution then we could simply recover the maximum clique completely from .)

For example, if is not connected to in the graph , then our estimate for should be zero. What might be less clear is what should be our estimate for — i.e., what do we think is the conditional probability that is in the clique given our observation of the graph and our limited time. The a priori probability is simply , but if we observe that, for example, the degree of is a bit bigger than expected, say, then how should we update this probability? The idea is to think like a Bayesian. If does not belong to the clique then its degree is roughly distributed like a normal with mean and standard deviation . On the other hand, if it does belong to the clique then its degree is roughly distributed like a normal with mean and the same standard deviation. So, we can see that if the degree of was then we should update our estimate that is in the clique by a factor of roughly . This turns out to be in the case that the degree was .

We can try to use similar ideas to come up with how we should update our estimate for based on the number of triangles that contain it, and generalize this to updates of more complicated events based on more general statistics. But things get very complex very soon, and indeed prior work has only been able to carry this out for estimates of polynomials up to degree four.

In our new paper we take an alternate route. Rather than trying to work out the updates for each such term individual, we simply declare by fiat that our estimates should:

• Be simple functions of the graph itself. That is will be a low degree function of .
• Respect the calibration condition (*) for all functions that can depend on the graph only in a low degree way.

This condition turns out to imply that our estimates automatically respect all the low degree statistics. The “only” work that is then left is to show that they satisfy the constraint that the estimate of a is always non-negative. This turns out to be quite some work, but it can be thought as following from a recursive structure versus randomness partition. This might seem to have nothing to do with other uses of the “structure vs randomness” approach such as in the setting of the regularity lemma or the prime numbers, but at its core, the general structure vs. randomness argument is really about Bayesian estimates. The idea is that given some complicated object , we separate it to the part containing the structure that we can infer in some computationally bounded way, and then the rest of it, since it contains no discernable structure, can be treated as if it is random even if it is fully deterministic since a Bayesian observer will have uncertainty about it and hence assign it probabilities strictly between zero and one. Thus for example, in the regularity lemma, we can think of a bounded observer that cannot store a matrix in full, and so only remembers the average value in each block, and considers the entry as random inside it. Another example is the case of the set of primes, where a bounded observer can infer that all but finitely many members of do not divide up to some not too large number , but beyond that will simply model as a random set of integers of the appropriate density. Similarly, in the case of the regularity lemma, we split a matrix into a low rank component containing structure that we can infer, with the rest of it treated as random.

I think that a fuller theory of computational Bayesian probabilities, which would be the dual to our standard “frequentist” theory of pseudorandomness, is still waiting to be discovered. Such a theory would go far beyond just looking at sums of squares.