FOCS 2024

Junior/Senior lunch in 80°F Chicago

Last summer I attended the Complexity Conference in Ann Arbor for the first time in eight years largely because it was within driving distance. So with FOCS in Chicago this year I didn't have much of an excuse to skip it. I haven't attended a FOCS in person since the 2010 conference in Vegas, though I have been at a few STOCs in that span. 

The 65th IEEE Symposium on Foundations of Computer Science is being held this week at the voco hotel, a beautiful venue in Wolf's Point where the three branches of the Chicago river meet—though you'd never know it from the windowless conference venue on the 14th floor. I'm enjoying reconnecting with colleagues old and new. Sometimes you can go back home again.

Even with triple sessions, we had 12 minute talks for most papers with longer online versions linked from the schedule page. I liked the 12-minute format; with 20 minutes, speakers tend to rush through proofs, while here, they could take their time giving high-level overviews of their papers.

Stats: 274 registrants. 157 students. 16 international. Large compared to recent years but it's not a huge conference. 133 accepted papers chosen among 463 submissions from a program committee of 47 members.

The Knuth Prize winner Rajeev Alur gave his lecture on Specification-guided Reinforcement Learning. 

The Machtey Prize for best student papers goes to Brice Huang for Capacity Threshold for the Ising Perceptron and Meghal Gupta, Mihir Singhal and Hongxun Wu for Optimal Quantile Estimation: Beyond the Comparison Model.

Best paper awards goes to Bernhard Haeupler, Richard Hladík, Václav Rozhoň, Robert Tarjan and Jakub Tětek for Universal Optimality of Dijkstra via Beyond-Worst-Case Heaps (Quanta Article) and Mohsen Ghaffari and Christoph Grunau for Near-Optimal Deterministic Network Decomposition and Ruling Set, and Improved MIS.

Test of time Awards

• FOCS 1994: Given the three papers, Shor's quantum factoring and the Simon paper that inspired him, which I posted about recently, and Expander Codes by Mike Sipser and Dan Spielman. Dan's 1994 talk was a personal favorite, largely because he used pictures without words to effectively convey how to derive error-correcting codes from expander graphs.
• FOCS 2004: Optimal inapproximability results for Max-Cut and other 2-Variable CSPs by Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell (one of my favorite theorems) and Worst-case to average-case reductions based on Gaussian measures by Daniele Micciancio and Oded Regev. 
• FOCS 2014: Popular conjectures imply strong lower bounds for dynamic problems by Amir Abboud and Virginia Vassilevska Williams

In 2025 FOCS goes down under to Sydney, Australia, December 15-17.

Random Thoughts on the Election

 Here are my random thoughts on the election:

1) Here is a list of things I DONT care about

 a) Candidates Gender or Race. The people who say its about time we had a female president might not want to vote for a President Marjorie Taylor Green. (A while back I thought that the first african-american president and/or first female president would be a republican since such a candidates might take some usually-democratic voters into the republican camp. One of many predictions I was wrong about.) I will note here that Kamala has rarely brought up I will be the first female prez.

b) Candidates personal lives. Attempts to tie a candidates affairs to their policies never made sense to me.

c) How well the candidate did in school. I care what they know and don't know now, and also if they know what they don't know. (Who was the last president to not have a college degree? I will tell you at the end of this post.)

d) Their religion. There were people who agreed with JFK on policy issues but did not want to vote for him because he was Catholic. I didn't understand it then nor do I understand it now. Biden is Catholic but this rarely comes up. Romney was Mormon and this only came up in the Republican primary, not in the general.  So I am glad it is no longer an issue. Having said that, we still haven't had a Jewish president, a Muslim President, or an Atheist president. 

e) Do I care if the candidate X will benefit me personally? It is very hard to tell that. Someone like Elon Musk is clearly going to support Kamala since she believes global warming is true and the e-cars will be a part of dealing with it. This is an example of someone supporting someone since it benefits them personally. Oh, bad example.

 Vance thinks people vote this way as he recently said that women past child-bearing age should not care about abortion rights.

f) There are ads that say things like I served in the military defending our country, and now I want to defined your rights to XXX. I don't see how serving in the military makes them a better senator or whatever. 

g) I don't care how they are doing at polls. Some of the articles saying that candidate X is ahead  also tend to say that that shows X  is better. I campaigned for George McGovern in 1972 and he went on to lose by a lot.  Some of my friends told me that I backed a loser and tried to make that an insult (I have better friends now). This puzzled me then since the fact that my candidate lost says NOTHING about how he would do as president.

2) DO I care about if they lie? Depends on how much and about what. That Vance changes his mind about Trump (he was anti-Trump in 2016) or that Kamala changed her mind on fracking are the standard lies that politicians always tell so I don't care about that. This may be a reflection on the low standards we have. More serious are lies that they KNOW are false and might ENDANGER people.

3) DO I care if they changed their mind on an issue. All I care about is how they feel NOW, though if they changed their mind I might not believe them.

4) I  DO care about policy and ability to get things done and to consider all sides of an issue. (I won't say what policies I like since I am trying to keep this post non-partisan).

5) Some of the attacks on Kamala have been a women should not be president. This puzzles me since there will come a time when the Republicans have a female candidate.

6) I am surprised there is anyone who is still undecided. We know both candidates  VERY WELL. What more information do you need?

7) Articles like Five Reasons Kamala Will Win or Five Reasons Trump Will Win are usually crap. For example, one of the reasons Kamala will win is People are tired of  Trump's corruption. But that just means that the author of the article is tired of it, and the author does not speak for the American people. An argument like Kamala is ahead in 5 of the 7 swing states (if it were true) is the kind of argument I want to see. More to the point- an argument that someone will win should not be partisan.

8) Since JD Vance might lose Trump some votes (actually I doubt that) I have heard the old argument that Sarah Palin cost McCain the Presidency, or at least cost him some votes. I began to think do we really know that? so I looked at some articles both new and old about this. I found things like:

The American public did not want Sarah Palin a heartbeat away from the presidency because of her views, or because of her perceived lack of intelligence or blah blah. Hence she cost McCain votes.

NONE of the articles I read pointed to polls or other EVIDENCE for this point of view.

There probably is some evidence on the issue (I do not know which way it would go)  somewhere but the LACK of any INTEREST in it bothers me.

9) I am surprised there are any undecided voters at this point. Both candidates are known to the public, so I don't see what more there is to think about. I accidentally already said this, however (a) I don't want to mess up my numbering and (ii) I do feel strongly about this point, so its worth having twice.

10) Because of Americas only-2-party-system you end up voting for people you agree with on some things but not others. I can imagine Mike Pence saying:

I prefer Trump to Kamala on the abortion issue.

I prefer Kamala to Trump on the  Hang Mike Pence issue.

Gee, who do I vote for?

 Actually he has said he is voting for neither one. 

11) IF Kamala wins then the decision to make Tim Walz her VP will be seen as genius.

      IF Kamala does not win Pennsylvania and loses the election then NOT picking Josh Shapiro (popular governor of PA) will be seen as stupid.

I call bullshit on both of those. There are SO MANY factors  at play here. This reminds me of when a basketball team wins 100-99 and the sportscasters are saying things like

The decision to concentrate on 3-point shots was genius!

12) More generally, after-the-fact pundits all have their own theories about why candidate X won, even those who were confident that candidate Y was going to win, and those theories are either partisan or just uninformed. It is hard to know what works, even after the fact.

13) I had a post, here, about breaking the record for number-of-living-ex-presidents.  I speculated that  if

a) Biden won in 2020 and lost in 2024 (I overlooked the option of his not running.)

b) Carter lives to see the winner of the 2024 election inaugurated

THEN we would have SIX living ex-presidents, which would break the record. They would be 

Carter, Clinton, Bush Jr, Obama, Trump, Biden.

If Trump wins then should Trump count? I think so- he would be a president AND an ex-president. If Kamala wins I won't have that issue. When I wrote that I didn't think Carter would last this long (he is 100), but he may very well. In fact, he has already voted!

14) Allan Lichtman is an American Historian (does that mean he studies American History or that he is American AND a historian?) who has a system to predict who will win the presidential election that has been correct 9 out of the last 10 elections. See here for his system. He predicts Kamala. I personally do not put to much stock in that since its going to be VERY CLOSE (hmmm- that's my prediction, and I could be wrong). While he is a democrat his system is OBJECTIVE-- he would have to be to be right so often. Even so, he is getting hate mail. This makes no sense. Predicting that X will happen does not mean you have an opinion about if X is good or bad. 

15) When people argue passionately about a rules change (e.g., get rid of the electoral college) I do not believe them- they would argue the other way if the current rules favored their candidate. 

16) JD Vance recently came out against the laws that say you can't wear political stuff at the polling place. He said that BOTH Kamala supporters and Trump supporters should be allowed to wear political T-shirts and hats and  what not at the polling place. NO HE DIDN"T. He applauded a Trump Supporter for calling a Poll Worker a XXX and told him to YYY in for asking her to follow the polling place's rules prohibiting political merchandise. See here. (I use XXX and YYY so that this post won't get censored as happened in the past, see here. Oddly enough, this morning the ban was lifted, so for the original post that was banned see here.)  No word yet on if he is going to praise a Texas man for punching a poll worker who told him to remove his Trump hat, see here.

17) Nikki Haley bravel tells Trump that he should not call Kamala the c-word. Actually it was not brave at all- here is here quote:

You've got affiliated PACs that are doing commercials about calling Kamala the c-word. Or you have speakers at Madison Square Garden referreing to her and her pimps. This is not the way to win women.

So being nasty to Kamala is fine in itself, but Trump shouldn't do it since it will lose him votes. 

The last president who did not have a college degree was Harry Truman.

Family Feud vs Pointless

Every now and then I feel like doing a Gasarchian post. This is one of those weeks. I'm going to look at the mathematics behind the American game show Family Feud and the British Pointless. I caught a glimpse of Pointless while I was in Oxford over the summer and then got hooked for a while on old episodes on YouTube. 

In both shows, 100 people are surveyed to give elements in a category, like "Robert Redford Films" and you get points based on how many people match the player's or team's answer. After that the similarity ends. In Family Feud you want to maximize your points, in Pointless you want to get as few as possible.

In Family Feud the categories are vague and not fact checked. If you say "The Hustler" and five other people said "The Hustler" you would get five points, even though Redford did not appear in that film. The only sanity check is that you need at least two matches to get points. The questions are often humorous or risqué like "Things people do right before going to sleep".

In Pointless if you said "The Hustler" you would get 100 points for a wrong answer. To define wrong answers, the category has to have a very formal definition: "Any movie where Robert Redford has a credited role released before 2023. TV movies and documentaries do not count. Voice-over animated films do count."

Pointless often has British-based questions. I can't name any professional darts players but apparently there are several famous ones in the UK. 

Other differences: In Family Feud, each person surveyed gives one answer. In Pointless, each person surveyed has 100 seconds to give as many answers as they can. Family Feud ends with a quick round of five categories where two players need to get 200 points total. Pointless ends where the 2-player team needs to find a pointless answer in a category.

How would AI do in these shows? I asked ChatGPT to come up with an obscure Robert Redford movie and it came up with a good one, Situation Hopeless -- But Not Serious and for a popular one Butch Cassidy and the Sundance Kid. When I asked "give me one thing people do before they go to sleep" it gave me "check their phones". AI wants us to think the last thing we should do at night is talk to AI.

Family Feud has a British version named Family Fortunes. A US version of Pointless never got past the pilot. We're not a country that likes to aim for zero. 

Contrast an Episode of Columbo with the recent Nobel Prizes

 I quote Lance's blog post (here) about Computing and the Nobels

a) On Wednesday October 9th half of the Chemistry Nobel was awarded to computer scientists Demis Hassabis and John Jumper for the protein-folding prediction algorithm AlphaFold, which I (Lance) consider the most impressive and game-changing application of machine learning.

b) The day before John Hopfield and Geoffrey Hinton were awarded the Physics Nobel for the development of neural nets that lead to AlphaFold, Large-Language models, and so much more.

Bottom line: The Nobel Prize in CHEM and PHYSICS went to COMPUTER SCIENTISTS. That's probably not quite right since I am sure that the people involved KNEW lots of Chemistry and Physics. 

In Feb 10, 1974 there was an episode of Columbo called Mind Over Mayhem (see here for a summary) where one of the plot threads was the following: 

a) Carl Finch, a great scientist, dies (of natural causes) and in his files is a ground breaking theory of molecular matter.

b) Neil Cahill who was working with Finch as a computer programmer knows about the file and codes up stuff in it and claims the work as his own.   He is initially not caught and he wins the Scientist of the Year Award. 

c) I won't get into who gets murdered or how Columbo catches them.(Has anyone in the real world been murdered because of an academic dispute?)

When I first saw it I had two thoughts:

1) If Neil had claimed co-authorship that would be more honest and hence would not need to be covered up or lead to murder. AND Neil would STILL get credit

2) More important for now: a computer programmer who coded up stuff was considered NOT part of the research in 1974. And now? Well the description of what the Nobel's did seems far more impressive than what Neil Cahill did, though since Neil Cahill is fictional it's hard to say. 

The question of  how much credit should a programmer on a project get? was unintentionally raised way back in 1974. And now? I have not seen ANY objection to computer scientists winning Nobel Prizes in Chem and Physics so the question seems to not be controversial. I agree with this though I am surprised by the lack of controversy. I also note that I used the term Programmer which is not accurate. They were computer scientists. Having said that, programmers also deserve credit. How much is hard to say. The distinction between computer scientists and programmers is also hard to say. But if programmers were considered part of the research in 1974, a fictional murder could have been prevented. 

(ADDED LATER: Lance saw this post after I posted it and emailed me a link to an article that DID hae some objections to giving a Nobel Prize for AI work. I disagree with the objections, but in the interest of being giving intelligent opposing viewpoints,   the link is here.) 

Computing and the Nobels

Herb Simon

Herbert Simon while a political scientist in the 1940s at my institution, the Illinois Institute of Technology, developed the theory of bounded rationality, realizing that people did not always make the most rational decisions because of lack of information and limited cognitive ability. After Illinois Tech, Herb Simon moved to the Carnegie Institute of Technology and in the 1960s helped found its Computer Science Department, later the Carnegie-Mellon School of Computer Science. With his colleagues at Carnegie he applied his principles to artificial intelligence with Allen Newell and J. C. Shaw leading to an ACM Turing Award in 1975. Bounded rationality would help him win the Nobel Prize in Economics in 1978.

Computing would go on to play an important role in nearly all scientific research. Most notably in 2013, biochemists Martin Karplus, Michael Levitt and Arieh Warshel won the Nobel Prize in Chemistry for their work using computers to model large molecules and simulate chemical reactions. Their entire research was done on computers, not in the lab. But no other computer scientist would win a Nobel Prize for the next 45 years. 

Demis Hassabis

That all changed last week. On Wednesday October 9th half of the Chemistry Nobel was awarded to computer scientists Demis Hassabis and John Jumper for the protein-folding prediction algorithm AlphaFold, which I consider the most impressive and game-changing application of machine learning. 

The day before John Hopfield and Geoffrey Hinton were awarded the Physics Nobel for the development of neural nets that led to AlphaFold, large-language models and so much more. Hinton with his 2018 Turing Award became only the second person, after Herb Simon, to win both prizes.

Is it physics? One podcast tried to answer this question.

Geoffrey Hinton

First of all, [Hopfield's] analogy to spin glasses is a use of a physical model. Certainly, information cannot exist in the absence of matter or energy. So ultimately, information science reduces to matter and energy as much as it reduces to mathematics. And thirdly, Nobel’s will was written in the 1890s. The first prize was awarded in 1901. Things have moved on since then. So the will prescribes physics, chemistry and physiology or medicine as the three science prizes and sometimes various Nobel committees are criticised for being a bit narrow. I think this shows. A certain creativity on the part of the academy to include a very up to date and very important field in physics by a little bit of creative accounting.

As a theorist who deals with information and computation, I disagree that these can only exist with matter and energy. The set of primes, for example, cannot fit into our finite universe.

But the third point is the most important. Nobel's will predates Turing and the development of computation as a new field. The importance of computing and artificial intelligence has take on such an importance that the Nobel Prize committees felt they needed to honor it, even if it means broadening the categories and encompassing computer science as a part of physics.

What does this mean for the Turing Award, now that computer scientists seem more eligible for Nobel prizes? I'll leave that as a open question for now.

A Trip Down Memory Lane: Desc comp, Constant Round Sorting, Division Breakthrough, Derandomization.

I  came across (by accident) the link to all of the BEATCS complexity columns from 1987 to the 2016. See HERE. (If you know a link to a more recent webpage then email me or make a comment. There is a link to all of the issues of BEATCS here; however, I want a page with just the complexity  theory columns.)

This gave me a trip down memory lane and a series of blog posts: Briefly describe an  articles and also get commentary from the original authors on what they think now about the area now.

I did the first in this series, on two articles by Eric Allender, here. Today is the second: articles by Neil Immerman, Gasarch-Golub-Kruskal, Eric Allender (again!), and Valentine Kabanets.

67) Progress in Descriptive Complexity by Neil Immerman,  1999.We usually think of P, NP, and other classes in terms of TIME or SPACE. Descriptive complexity is about defining complexity classes in terms of how they can be described in a logical language. (For reviews of three books on this topic, including Neil Immerman's, see here.) I asked Neil Immerman to comment on the state of Descriptive Complexity now and he said:

Currently the two most promising and exciting areas of progress in Descriptive Complexity are, in my opinion,

• Graph Neural Nets:  see Martin Grohe, "The Descriptive Complexity of Graph Neural Networks'', (see here);    and
  
• Graph Isomorphism: see Martin Grohe,  Descriptive Complexity, Canonisation, and Definable Graph Structure Theory, Cambridge University Press, 2017, and Anuj Dawar, "The Nature and Power of Fixed-Point Logic with Counting", ACM SIGLOG News, Jan. 2015, Vol. 2, No. 1.

 

72) A Survey of Constant Round Sorting by Bill Gasarch,  Evan Golub and Clyde Kruskal in 2000.  The model of computation is that in each round one could do lots of comparisons. Transitive closure was considered free, so the model was not practical and didn't claim to be; however, lower bounds on this model implied lower bounds on more realistic models. I asked Bill Gasarch to  comment on the paper from the perspective of 2023 and he said:

I don't think the paper is out dated in terms of later results, though the model of parallelism used is not studied anymore. A survey on sorting on more realistic models would be a good idea and may actually exist.

74) The Division Breakthrough by Eric Allender, 2001. How complicated are addition, subtraction, multiplication, and division?  The addition and subtraction algorithm that you learned in grade school is a log space algorithm (your teacher should have pointed that out to you). Multiplication is also in log space---that's an easy exercise. But what about division?  When Eric Allender was doing division in grade school he wondered can this be done in log space? This was open for quite some time, but was solved- Division is in Log Space- by Chui, Davida, Litow, in 2001 (actually done by Chui in his MS thesis in 1995 but not published until 2001. Fun Fact: Chui went to law school).  Log space is an upper bound, what about a lower bound? Hesse showed that Division is complete for DLOGTIME-uniform-TC^0.  What is needed is a complete write up of all of this in a uniform notation. Eric Allender has provided that! I asked Eric if there are more open problems in the area of complexity-of-arithmetic. He responded:

What complexity class best captures the complexity of GCD (computing the Greatest Common Divisor)?  GCD is hard for\( TC^0\) (proved in a paper I wrote with Mike Saks and Igor Shparlinski) but we have no upper bound better than P.  Similarly: What is the complexity of modular exponentiation (given x, y, and m, compute x^y mod m)?  Again, we have no upper bound better than P.  Modular exponentiation is an important subroutine in the polynomial-time algorithms for checking primality, and the lack of a better upper bound for this problem is one of the main reasons that we have no better upper bound for primality-checking than P.

...and while we're talking about the primes, let me mention that my SIGACT News article( see here) from early 2023 explains that I'm offering a $1,000 reward to anyone who can provide a modest improvement on the \(TC^0\)-hardness of primality-checking.  We currently only know a non-uniform \(AC^0\) reduction from MAJORITY to primality-checking.  I'm asking for a uniform \(AC^0\) reduction.

76) Derandomization: A Brief Survey by Valentine Kabanets, 2002. A very nice survey. Has there been progress since then? I asked Valentine and he said 

There have been new approaches to the BPP vs P question. See for example a recent survey by Chen and Tell 

New ways of studying the BPP = P conjecture (a survey)

ACM SIGACT News 2023

(available on Tell's webpage https://sites.google.com/site/roeitell/Publications)

 

Fall Jobs Post 2024

In the fall, I write a jobs post predicting the upcoming CS faculty job market and giving suggestions and links. In the spring I used to crowdsource a list of where everyone got jobs but have since outsourced the crowdsource to Grigory Yaroslavtsev. Let's start with a few announcements.

FOCS in Chicago will have a Graduating Bits on Sunday, October 27 from 12-2 PM. If you have job openings for postdocs and researchers the FOCS organizers are collecting them here, The CATCS also maintains a Theoretical Computer Science Jobs posting site. You are also free to add pointers to theory-related job listings as comments to this post. More generally in CS, there is the CRA Database of Job Candidates and the CRA and ACM job postings.

Mohammad Hajiaghayi is organizing a virtual theory jobs panel November 10th with Nicole Immorlica, Samir Khuller and yours truly. 

If you are a bit more senior, the Simons Institute is looking for a new director. 

Last year I suggested AI (which by the way just won two Nobel Prizes) wasn't dramatically affecting the CS faculty job market yet but

Many see programming, rightly or wrongly, as one of the first careers that AI will displace, which may reduce enrollment in the future, as offshoring fears drove CS enrollment down 20 years ago

It didn't take long. In two years we have gone from nearly all of our CS graduates getting jobs in the field to many of them struggling to get internships and jobs in the top companies if at all. If the past is any guide, a weak tech job market leads to fewer majors which leads to fewer slots for CS faculty. We'll start to see these trends this year and they will accelerate quickly if the tech jobs market doesn't recover.

Areas related to data such as Artificial Intelligence, Data Science and Cybersecurity, will draw the most interest. Best if you can tie your research to those areas, or at least that you are open to teaching in them.

Have a well-designed website with all your job materials and links to your papers. Make sure your Google Scholar, LinkedIn and GitHub (if relevant) sites are accurate and up to date. Make a short video describing your research to a general CS crowd. Take advantage of all the links above. Network at FOCS if you can make it. And start early. 

Emil Post Anticipated (more than anticipated) Godel and Turing

 (Thanks to James De Santis for pointing the article that inspired this post on Post. The article is pointed to in this post.)

What is Emil Post known for? I know of him for the following:

a) Post's Problem: Show that there is an r.e. set A that is strictly in between Decidable and Halt using Turing Reductions. He posed the problem in 1944. It was solved in 1956 by the priority method, simultaneously invented by Friedberg and Muchnik . (My students wondered who posted to the web first and if the other one could have seen it there.)

b) He invented Post Tag Systems, a natural problem that is undecidable. Or a model of computation. Depends on how you look at it.

c) The Kleene-Post Theorem which produces a set A that is strictly in between Decidable and Halt. It did not solve Post's Problem since A is not r.e. The proof used forcing and may have been the first or close to the first use of forcing. This was published in 1954.

In summary, I thought of Post as being a recursion theorist. 

 The more I read about Post the more I realize that calling him a recursion theorist is wrong but in an interesting way. Back in the 1950's  I am sure Emil Post was called a logician. The over-specialization that produces Recursion Theorists, Model Theorists, Set Theorists, Proof Theorist was about a decade away. In fact, people who worked in logic often also worked in other parts of math. (Post worked on Laplace Transforms. See The Post Inversion Formula as part of the Wikipedia entry on Laplace Transforms.)

I recently read the article, Emil Post and His Anticipation of Godel and Turing which shows that Post really did have some of the ideas of Godel and Turing at about the same time, and possibly before they did. I will discuss briefly what he did and why it is not better known; however, the article is worth reading for more on this.

What did Post Do? 

Part of Post's Thesis (1920) was showing that the Propositional Logic in Russell-Whitehead was consistent and complete. 

He tried to show that all of RW was consistent and complete. He got RW down to a normal form; however, he realized that rather than reducing the complexity he just localized it. In 1921 he noted the following (I quote the article).

a) Normal Systems can simulate any symbolic logic, indeed any mechanical system for proving theorems.

b) This means, however, that all such systems can be mechanically listed, and the diagonal argument  shows that the general problem of deciding whether a given theorem is produced by a given system is unsolvable.

c) It follows, in turn, that no consistent mechanical system can produce all theorems.

Wow- that sounds like the outline of a proof of Godel's Incompleteness theorem!

Why Didn't Post Publish This? 

 In 1921 Post suffered his first (of many) attacks of manic-depression. He was unable to get an academic job until 1935.  By the time he could have written a paper, his ideas were already known. Note that the items about Post I knew are all post-1940.

The Story is Interesting Because its Boring

There is no interesting conflicts between mathematicians in this story. No Newton vs Leibniz rap battle (see here) no plagiarism. Just bad timing and bad luck. So this story is not even worth  being exaggerated. 

But I find that interesting. Timing and Luck play a big role, perhaps bigger than is commonly thought.

Favorite Theorems: Gradient Descent

September Edition

Who thought the algorithm behind machine learning would have cool complexity implications?

The Complexity of Gradient Descent: CLS = PPAD ∩ PLS

John Fearnley, Paul Goldberg, Alexandros Hollender and Rahul Savani

Let's unpack these classes, subclasses of TFNP, where for every input we know there is an easily verifiable solution and we are looking at the complexity of finding it. PPAD is the class famous for having finding a Nash Equilibrium as a complete set, see this post for details.

PLS is the class of problems where we look for a local minimum. Finding a global minimum is NP-complete--think vertex cover. Finding a local minimum is often easier but still could be hard if you are optimizing over an exponential set of values.

CLS is a bit trickier to define, basically you are finding an approximate local minimum of a function mapping three real variables to one real value.

The authors show that gradient descent is complete for PPAD ∩ PLS even if you only use two input variables. Since gradient descent is in CLS, the equality follows. 

More in my 2021 post. On that post author Paul Goldberg commented

The paper is a fine example of the humorous stereotype of complexity theorists proving a problem "hard" when meanwhile the problem is being routinely solved in practice in the real world.

Nevertheless it's a neat complexity result and now officially one of my favorite theorems.