ARV on Abelian Cayley Graphs

Continuing from the previous post, we are going to prove the following result: let {G} be a {d}-regular Cayley graph of an Abelian group, {\phi(G)} be the normalized edge expansion of {G}, {ARV(G)} be the value of the ARV semidefinite programming relaxation of sparsest cut on {G} (we will define it below), and {\lambda_2(G)} be the second smallest normalized Laplacian eigenvalue of {G}. Then we have

\displaystyle   \lambda_2 (G) \leq O(d) \cdot (ARV (G))^2 \ \ \ \ \ (1)

which, together with the fact that {ARV(G) \leq 2 \phi(G)} and {\phi(G) \leq \sqrt{2 \lambda_2}}, implies the Buser inequality

\displaystyle   \lambda_2 (G) \leq O(d) \cdot \phi^2 (G) \ \ \ \ \ (2)

and the approximation bound

\displaystyle   \phi(G) \leq O(\sqrt d) \cdot ARV(G) \ \ \ \ \ (3)

The proof of (1), due to Shayan Oveis Gharan and myself, is very similar to the proof by Bauer et al. of (2).

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Hasselmann, Manabe and Parisi win 2021 Physics Nobel Prize

itscomingrome-lorenza-parisi

Today the Italian academic community, along with lots of other people, was delighted to hear that Giorgio Parisi is one of the three recipients of the 2021 Nobel Prize for Physics.

Parisi has been a giant in the area of understanding “complex” and “disordered” systems. Perhaps, his most influential contribution has been his “replica method” for the analysis of the Sherrington-Kirkpatrick model. His ideas have led to several breakthroughs in statistical physics by Parisi and his collaborators, and they have also found applications in computer science: to tight analyses on a number of questions about combinatorial optimization on random graphs, to results on random constraint satisfaction problems (including the famous connection with random k-SAT analyzed by Mezard, Parisi and Zecchina) and random error correcting codes, and to understanding the solution landscape in optimization problems arising from machine learning. Furthermore these ideas have also led to the development and analysis of algorithms.

The news was particularly well received at Bocconi, where most of the faculty of the future CS department has done work that involved the replica method. (Not to be left out, even I have recently used replica methods.)

Mezard and Montanari have written a book-length treatment on the interplay between ideas from statistical physics, algorithms, optimization, information theory and coding theory that arise from this tradition. Readers of in theory looking for a shorter exposition aimed at theoretical computer scientists will enjoy these notes posted by Boaz Barak, or this even shorter post by Boaz.

In this post, I will try to give a sense to the reader of what the replica method for the Sherrington-Kirkpatrick model looks like when applied to the average-case analysis of optimization problems, stripped of all the physics. Of course, without the physics, nothing makes any sense, and the interested reader should look at Boaz’s posts (and to references that he provides) for an introduction to the context. I did not have time to check too carefully what I wrote, so be aware that several details could be wrong.

What is the typical value of the max cut in a {G_{n,\frac 12}} random graph with {n} vertices?

Working out an upper bound using union bounds and Chernoff bound, and a lower bound by thinking about a greedy algorithm, we can quickly convince ourselves that the answer is {\frac {n^2}{4} + \Theta(n^{1.5})}. Great, but what is the constant in front of the {n^{1.5}}? This question is answered by the Parisi formula, though this fact was not rigorously established by Parisi. (Guerra proved that the formula gives an upper bound, Talagrand proved that it gives a tight bound.)

Some manipulations can reduce the question above to the following question: suppose that I pick a random {n\times n} symmetric matrix {M}, say with zero diagonal, and such that (up to the symmetry requirement) the entries are mutually independent and each entry is equally likely to be {+1} or {-1}, or perhaps each entry is distributed according to a standard normal distribution (the two versions can be proved to be equivalent), what is the typical value of

\displaystyle  \max _{x \in \{+1,1\}^n } \ \ \frac 1{n^{1.5}} x^T M x

up to {o_n(1)} additive terms,?

As a first step, we could replace the maximum with a “soft-max,” and note that, for every choice of {\beta>0}, we have

\displaystyle  \max _{x \in \{+1,1\}^n } \ \ x^T M x \leq \frac 1 \beta \log \sum_{x \in \{+1,1\}^n } e^{\beta x^T Mx}

The above upper bound gets tighter and tighter for larger {\beta}, so if we were able to estimate

\displaystyle  \mathop{\mathbb E} \log \sum_{x \in \{+1,1\}^n } e^{\beta x^T Mx}

for every {\beta} (where the expectation is over the randomness of {M}) then we would be in good shape.

We could definitely use convexity and write

\displaystyle  \mathop{\mathbb E} \max _{x \in \{+1,1\}^n } \ \ x^T M x \leq \frac 1 \beta \mathop{\mathbb E} \log \sum_{x \in \{+1,1\}^n } e^{\beta x^T Mx} \leq \frac 1 \beta \log \mathop{\mathbb E} \sum_{x \in \{+1,1\}^n } e^{\beta x^T Mx}

and then use linearity of expectation and independence of the entries of {M} to get to

\displaystyle  \leq \frac 1 \beta \log \sum_{x \in \{+1,1\}^n } \prod_{1\leq i < j\leq n} \mathop{\mathbb E} e^{2\beta M_{i,j} x_i x_j }

Now things simplify quite a bit, because for all {i<j} the expression {M_{i,j} x_i x_j}, in the Rademacher setting, is equally likely to be {+1} or {-1}, so that, for {\beta = o(1)}, we have

\displaystyle  \mathop{\mathbb E} e^{2\beta M_{i,j} x_i x_j } = cosh (2\beta) \leq 1 + O(\beta^2) \leq e^{O(\beta^2)}

and

\displaystyle  \sum_{x \in \{+1,1\}^n } \prod_{1\leq i < j\leq n} \mathop{\mathbb E} e^{2\beta M_{i,j} x_i x_j } \leq 2^n \cdot e^{O(\beta^2 n^2)}

so that

\displaystyle  \frac 1 \beta \log \sum_{x \in \{+1,1\}^n } \prod_{1\leq i < j\leq n} \mathop{\mathbb E} e^{2\beta M_{i,j} x_i x_j } \leq \frac {O(n)}{\beta} + O(\beta n^2)

which, choosing {\beta = 1/\sqrt n}, gives an {O(n^{1.5})} upper bound which is in the right ballpark. Note that this is exactly the same calculations coming out of a Chernoff bound and union bound. If we optimize the choice of {\beta} we unfortunately do not get the right constant in front of {n^{1.5}}.

So, if we call

\displaystyle  F := \sum_{x \in \{+1,1\}^n } e^{\beta x^T Mx}

we see that we lose too much if we do the step

\displaystyle  \mathop{\mathbb E} \log F \leq \log \mathop{\mathbb E} F

But what else can we do to get rid of the logarithm and to reduce to an expression in which we take expectations of products of independent quantities (if we are not able to exploit the assumption that {M} has mutually independent entries, we will not be able to make progress)?

The idea is that if {k>0} is a small enough quantity (something much smaller than {1/\log F}), then {F^k} is close to 1 and we have the approximation

\displaystyle  \log F^k \approx F^k-1

and we obviously have

\displaystyle  \log F^k = k \log F

so we can use the approximation

\displaystyle  \log F \approx \frac 1k (F^k - 1)

and

\displaystyle  \mathop{\mathbb E} \log F \approx \frac 1k (\mathop{\mathbb E} F^k - 1)

Let’s forget for a moment that we want {k} to be a very small parameter. If {k} was an integer, we would have

\displaystyle  \mathop{\mathbb E} F^k = \mathop{\mathbb E} \left( \sum_{x \in \{+1,1\}^n } e^{\beta x^T Mx} \right)^k = \sum_{x^{(1)},\ldots x^{(k)} \in \{+1,-1\}^n} \mathop{\mathbb E} e^{\beta \cdot ( x^{(1) T} M x^{(1)} + \cdots + x^{(k)T} M x^{(k)}) }

\displaystyle  = \sum_{x^{(1)},\ldots x^{(k)} \in \{+1,-1\}^n} \ \ \prod_{i< j} \ \mathop{\mathbb E} e^{2\beta M_{i,j} \cdot ( x^{(1)}_i x^{(1)}_j + \cdots + x^{(k)}_i x^{(k)}_j )}

Note that the above expression involves choices of {k}-tuples of feasible solutions of our maximization problem. These are the “replicas” in “replica method.”

The above expression does not look too bad, and note how we were fully able to use the independence assumption and “simplify” the expression. Unfortunately, it is actually still very bad. In this case it is preferable to assume the {M_{i,j}} to be Gaussian, write the expectation as an integral, do a change of variable and some tricks so that we reduce to computing the maximum of a certain function, let’s call it {G(z)}, where the input {z} is a {k \times k} matrix, and then we have to guess what is an input of maximum value for this function. If we are lucky, the maximum is equivalent by a {z} in which all entries are identical, the replica symmetric solution. In the Sherrington-Kirkpatrick model we don’t have such luck, and the next guess is that the optimal {z} is a block-diagonal matrix, or a replica symmetry-breaking solution. For large {k}, and large number of blocks, we can approximate the choice of such matrices by writing down a system of differential equations, the Parisi equations, and we are going to assume that such equations do indeed describe an optimal {z} and so a solution to the integral, and so they give as a computation of {(\mathop{\mathbb E} F^k - 1)/k}.

After all this, we get an expression for {(\mathop{\mathbb E} F^k - 1)/k} for every sufficiently large integer {k}, as a function of {k} up to lower order errors. What next? Remember how we wanted {k} to be a tiny real number and not a sufficiently large integer? Well, we take the expression, we forget about the error terms, and we set {k=0\ldots}

A Couple of Announcements

In the second week of July, 2022, there will be a summer school on algorithmic fairness at IPAM, on the UCLA campus, with Cynthia Dwork and Guy Rothblum among the lecturers. Applications (see the above link) are due by March 11, 2022.

We will soon put up a call for nominations for the test of time award to be given at FOCS 2021 (which will take place in Boulder, Colorado, in early 2022). There are three award categories, recognizing, respectively, papers from FOCS 2011, FOCS 2001, and FOCS 1991. In each category, it is also possible to nominate older papers, up to four years before the target conference. For example, for the thirty-year category, it is possible to nominate papers from FOCS 1987, FOCS 1988, FOCS 1989, FOCS 1990, in addition to the target conference FOCS 1991.

Nominations should be sent by October 31, 2021 to focs.tot.2021@gmail.com with a subject line of “FOCS Test of Time Award”. Nominations should contain an explanation of the impact of the nominated paper(s), including references to follow-on work. Self-nominations are discouraged.

In the second week of November, 2021, the Simons Institute will host a workshop on using cryptographic assumptions to prove average-case hardness of problems in high-dimensional statistics. This is such a new topic that the goal of the workshop will be more to explore new directions than to review known results, and we (think that we have) already invited all the authors of recent published work of this type. If you have proved results of this type, and you have not been invited (perhaps because your results are still unpublished?) and you would like to participate in the workshop, there is still space in the schedule so feel free to contact me or one of the other organizers. For both speakers and attendees, physical participation is preferred, but remote participation will be possible.

The Third Annual “Why am I in Italy and you are not?” post

I moved back to Italy exactly two years ago. I was looking for some change and for new challenges and, man, talk about being careful what you wish for!

Last year was characterized by a sudden acceleration of Bocconi’s plans to develop a computer science group. From planning for a slow growth of a couple of people a year until, in 5-7 years, we could have the basis to create a new department, it was decided that a new computer science department would start operating next year — perhaps as soon as February 2022, but definitely, or at least to the extent that one can make definite plans in these crazy times, by September 2022.

Consequently, we went on a hiring spree that was surprisingly successful. Five computer scientists and four statistical physicists have accepted our offers and are coming between now and next summer. In computer science, Andrea Celli (who won the NeurIPS best paper award last year) and Marek Elias started today. Andrea, who is coming from Facebook London, works in algorithmic game theory, and Marek, who is coming TU Eindhoven, works in optimization. Within the next couple of weeks, or as soon as his visa issues are sorted out, Alon Rosen will join us from IDC Herzliya as a full professor. Readers of in theory may know Alon from his work on lattice-based cryptography, or his work on zero-knowledge, or perhaps his work on the cryptographic hardness of finding Nash equilibria. Two other computer science tenured faculty members are going to come, respectively, in February and September 2022, but I am not sure if their moves are public yet.

Meanwhile, I have been under-spending my ERC grant, but perhaps this is going to change and some of my readers will help me out.

If you are interested in coming to Milan for a post-doc, do get in touch with me. A call will be out in a month or so.

After twenty years in Northern California, I am still readjusting to seasonal weather. September is among Milan’s best months: the oppressive heat of the summer gives way to comfortable days and cool nights, but the days are still bright and sunny. Currently, there is no quarantine requirement or other travel restrictions for fully vaccinated international travellers. If you want to visit, this might be your best chance until Spring Break (last year we had a semi-lockdown from late October until after New Year, which might very well happen again; January and February are not Milan’s best months; March features spectacular cherry blossoms, and it is again an ok time to visit).

Benny Chor

I just heard that Benny Chor died this morning. Chor did very important work on computational biology and distributed algorithms, but I (and probably many of my readers) know him primarily for his work on cryptography, for his work on randomness extraction and for introducing the notion of private information retrieval.

I only met him once, at the event for Oded Goldreich’s 60th birthday. On the occasion, he gave a talk on the Chor-Goldreich paper, which introduced the problem of randomness extraction from independent sources, and which introduced min-entropy as the right parameter by which to quantify the randomness content of random sources. He did so using the original slides used for the FOCS 1985 talk.

I took a picture during the talk, which I posted online, and later he sent me an email asking for the original. Sadly, this was the totality of our correspondence. I heard that besides being a brilliant and generous researchers, he was a very playful, likeable and nice person. My thoughts are with his family and his friends.

The Simons Institute Reopens

This coming Fall semester the Simons Institute for the Theory of Computing in Berkeley will have in-person activities, including the really interesting program on the complexity of statistical inference, within which I will co-organize a workshop on cryptography, average-case complexity, and the complexity of statistical problems.

As it had been the case before the pandemic, all Simons Institute events will be streamed and available remotely. This includes a new series of Public Lectures called “Breakthroughs” that starts next week with a talk by Virginia Williams on matrix multiplication.

Bocconi Hired Poorly Qualified Computer Scientist

Today I received an interesting email from our compliance office that is working on the accreditation of our PhD program in Statistics and Computer Science.

One of the requisites for accreditation is to have a certain number of affiliated faculty. To count as an affiliated faculty, however, one must pass certain minimal thresholds of research productivity, the same that are necessary to be promoted to Associate Professor, as quantified according to Italy’s well intentioned but questionably run initiative to conduct research evaluations using quantifiable parameters.

(For context, every Italian professor maintains a list of publications in a site run by the ministry. Although the site is linked to various bibliographic databases, one has to input each publication manually into a local site at one’s own university, then the ministry site fetches the data from the local site. The data in the ministry site is used for these research evaluations. At one point, a secretary and I spent long hours entering my publications from the past ten years, to apply for an Italian grant.)

Be that as it may, the compliance office noted that I did not qualify to be an affiliated faculty (or, for that matter, an Associate Professor) based on my 2016-2020 publication record. That would be seven papers in SoDA and two in FOCS: surely Italian Associate Professors are held to high standards! It turns out, however, that one of the criteria counts only journal publications.

Well, how about the paper in J. ACM and the two papers in SIAM J. on Computing published between 2016 and 2020? That would (barely) be enough, but one SICOMP paper has the same title of a SoDA paper (being, in fact, the same paper) and so the ministry site had rejected it. Luckily, the Bocconi administration was able to remove the SoDA paper from the ministry site, I added again the SICOMP version, and now I finally, if barely, qualify to be an Associate Professor and a PhD program affiliated faculty.

This sounds like the beginning of a long and unproductive relationship between me and the Italian system of research evaluation.

P.S. some colleagues at other Italian universities to whom I told this story argued that the Bocconi administration did not correctly apply the government rules, and that one should count conference proceedings indexed by Scopus; other colleagues said that indeed the government decree n. 589 of August 8, 2018, in article 2, comma 1, part a, only refers to journals. This of course only reinforces my impression that the whole set of evaluation criteria is a dumpster fire that is way too far gone.