Hasselmann, Manabe and Parisi win 2021 Physics Nobel Prize

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 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 ${k=0\ldots}$

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).

What a difference a few months can make

Piazza Duomo, in Milan, on December 26, 2020

Piazza Duomo, in Milan, on July 11, 2021

Finally, a joy

In Rome we have an expression, mai una gioia (literally, “never (a moment of) joy”) that applies well to the present times. Yesterday, there was, finally, something to be joyous about: the announcement that two of my heroes, Laszlo Lovasz and Avi Wigderson, will share the 2021 Abel Prize, one of the highest honors of mathematics.

The reader can find a very good article about them on Quanta Magazine.

Instead of talking about their greatest accomplishment, here I would like to recall two beautiful and somewhat related results, that admit a short treatment.

Silver linings

To put it mildly, 2020 is not shaping up to be a great year, so it is worthwhile to emphasize the good news, wherever we may find them.

Karlin, Klein, and Oveis Gharan have just posted a paper in which, at long last, they improve over the 1.5 approximation ratio for metric TSP which was achieved, in 1974, by Christofides. For a long time, it was suspected that the Held-Karp relaxation of metric TSP had an approximation ratio better than 1.5, but there was no viable approach to prove such a result. In 2011, two different approaches were developed to improve 1.5 in the case of shortest-path metrics on unweighted graphs: one by Oveis Gharan, Saberi and Singh and one by Momke and Svensson. The algorithm of Karlin, Klein and Oveis Gharan (which does not establish that the Held-Karp relaxation has an integrality gap better than 1.5) takes as a starting point ideas from the work of Oveis Gharan, Saberi and Singh.

What’s New

It has been six weeks since I moved to Milan, and I am not yet completely settled in yet.

For example, although, as of yesterday, I finally have working wired internet access in my place, I still do not have a bus card (obtaining the latter has been one of the most stubbornly intractable problems I have encountered) and all the stuff that I did not carry in two bags is still in transit in a container.

Meanwhile, the busyness of handling the move, getting settled, trying to get a bus card, and teaching two courses, has meant that I did not really have time to sit down with my thoughts and process my feelings about such a major life change. If people ask me what I miss about San Francisco I will, truthfully, say something like UberX, or Thai food, or getting a bus card from a vending machine, because I still have not had a chance to miss the bigger stuff. Similarly, this post will be about random small stuff.

加油台灣！

I would like to congratulate my Taiwanese readers for being in the first Asian country to introduce same-sex marriage.

Knuth Prize to Avi Wigderson

Congratulations to the Knuth prize committee chaired by Avrim Blum for the excellent choice of awarding the 2019 Knuth prize to Avi Wigderson.

Avi has worked on all aspects of computational complexity theory, and he has had a transformative influence on the way theoretical computer science relates to pure mathematics. I will not repeat what I wrote about his work on the occasion of his 60th birthday here and here. Long-term readers of in theory will remember that I consider him as one of the saints of computational complexity.

The organizers of the coming FOCS would like me to remind you that the deadline is this Friday, and that someone, for some reason, has set up a fake submission site (on the domain aconf dot org) but the true submission site (that, to be honest, looks less legit than the fake one) is at focs19.cs.utexas.edu.

Also, the deadline to submit nominations for the inaugural FOCS test of time award is in three weeks. There will be three awards, one for papers appeared in FOCS 1985-89, one for FOCS 1995-99 and one for FOCS 2005-09.

On an unrelated note, GMW appeared in FOCS 1986 and the Nisan-Wigderson “Hardness versus randomness” paper appeared in FOCS 1988.

Tested by time

I was delighted (and not at all surprised) to hear that this year’s Turing Award will go to LeCun, Hinton, and Y. Bengio for their work on deep learning.

Like public-key cryptography, deep learning was ahead of its time when first studied, but, thanks to the pioneering efforts of its founders, it was ready to be used when the technology caught up.

Mathematical developments take a long time to mature, so it is essential that applied mathematical research be done ahead of the time of its application, that is, at a time when it is basic research. Maybe quantum computing will be the next example to teach this lesson.

By the way, this summer the Simons Institute will host a program on the foundations of deep learning, co-organized by Samy Bengio, Aleks Madry, Elchanan Mossel and Matus Telgarsky.

Sometimes, it is not just the practical applications of a mathematical advance that take time to develop: the same can be true even for its theoretical applications! Which brings me to the next announcement of this post, namely that the call for nominations for the FOCS test of time award is out. Nominations are due in about four weeks.