Archive for April, 2018

How to upper-bound the probability of something bad

Friday, April 13th, 2018

Scott Alexander has a new post decrying how rarely experts encode their knowledge in the form of detailed guidelines with conditional statements and loops—or what one could also call flowcharts or expert systems—rather than just blanket recommendations.  He gives, as an illustration of what he’s looking for, an algorithm that a psychiatrist might use to figure out which antidepressants or other treatments will work for a specific patient—with the huge proviso that you shouldn’t try his algorithm at home, or (most importantly) sue him if it doesn’t work.

Compared to a psychiatrist, I have the huge advantage that if my professional advice fails, normally no one gets hurt or gets sued for malpractice or commits suicide or anything like that.  OK, but what do I actually know that can be encoded in if-thens?

Well, one of the commonest tasks in the day-to-day life of any theoretical computer scientist, or mathematician of the Erdös flavor, is to upper bound the probability that something bad will happen: for example, that your randomized algorithm or protocol will fail, or that your randomly constructed graph or code or whatever it is won’t have the properties needed for your proof.

So without further ado, here are my secrets revealed, my ten-step plan to probability-bounding and computer-science-theorizing success.

Step 1. “1” is definitely an upper bound on the probability of your bad event happening.  Check whether that upper bound is good enough.  (Sometimes, as when this is an inner step in a larger summation over probabilities, the answer will actually be yes.)

Step 2. Try using Markov’s inequality (a nonnegative random variable exceeds its mean by a factor of k at most a 1/k fraction of the time), combined with its close cousin in indispensable obviousness, the union bound (the probability that any of several bad events will happen, is at most the sum of the probabilities of each bad event individually).  About half the time, you can stop right here.

Step 3. See if the bad event you’re worried about involves a sum of independent random variables exceeding some threshold. If it does, hit that sucker with a Chernoff or Hoeffding bound.

Step 4. If your random variables aren’t independent, see if they at least form a martingale: a fancy word for a sum of terms, each of which has a mean of 0 conditioned on all the earlier terms, even though it might depend on the earlier terms in subtler ways.  If so, Azuma your problem into submission.

Step 5. If you don’t have a martingale, but you still feel like your random variables are only weakly correlated, try calculating the variance of whatever combination of variables you care about, and then using Chebyshev’s inequality: the probability that a random variable differs from its mean by at most k times the standard deviation (i.e., the square root of the variance) is at most 1/k2.  If the variance doesn’t work, you can try calculating some higher moments too—just beware that, around the 6th or 8th moment, you and your notebook paper will likely both be exhausted.

Step 6. OK, umm … see if you can upper-bound the variation distance between your probability distribution and a different distribution for which it’s already known (or is easy to see) that it’s unlikely that anything bad happens. A good example of a tool you can use to upper-bound variation distance is Pinsker’s inequality.

Step 7. Now is the time when you start ransacking Google and Wikipedia for things like the Lovász Local Lemma, and concentration bounds for low-degree polynomials, and Hölder’s inequality, and Talagrand’s inequality, and other isoperimetric-type inequalities, and hypercontractive inequalities, and other stuff that you’ve heard your friends rave about, and have even seen successfully used at least twice, but there’s no way you’d remember off the top of your head under what conditions any of this stuff applies, or whether any of it is good enough for your application. (Just between you and me: you may have already visited Wikipedia to refresh your memory about the earlier items in this list, like the Chernoff bound.) “Try a hypercontractive inequality” is surely the analogue of the psychiatrist’s “try electroconvulsive therapy,” for a patient on whom all milder treatments have failed.

Step 8. So, these bad events … how bad are they, anyway? Any chance you can live with them?  (See also: Step 1.)

Step 9. You can’t live with them? Then back up in your proof search tree, and look for a whole different approach or algorithm, which would make the bad events less likely or even kill them off altogether.

Step 10. Consider the possibility that the statement you’re trying to prove is false—or if true, is far beyond any existing tools.  (This might be the analogue of the psychiatrist’s: consider the possibility that evil conspirators really are out to get your patient.)

Amazing progress on longstanding open problems

Wednesday, April 11th, 2018

For those who haven’t seen it:

  1. Aubrey de Grey, better known to the world as a radical life extension researcher, on Sunday posted a preprint on the arXiv claiming to prove that the chromatic number of the plane is at least 5—the first significant progress on the Hadwiger-Nelson problem since 1950.  If you’re tuning in from home, the Hadwiger-Nelson problem asks: what’s the minimum number of colors that you need to color the Euclidean plane, in order to ensure that every two points at distance exactly 1 from each other are colored differently?  It’s not hard to show that at least 4 colors are necessary, or that 7 colors suffice: try convincing yourself by staring at the figure below.  Until a few days ago, nothing better was known.
    This is a problem that’s intrigued me ever since I learned about it at a math camp in 1996, and that I spent at least a day of my teenagerhood trying to solve.
    De Grey constructs an explicit graph with unit distances—originally with 1567 vertices, now with 1585 vertices after after a bug was fixed—and then verifies by computer search (which takes a few hours) that 5 colors are needed for it.  Update: My good friend Marijn Heule, at UT Austin, has now apparently found a smaller such graph, with “only” 874 vertices.  See here.
    So, can we be confident that the proof will stand—i.e., that there are no further bugs?  See the comments of Gil Kalai’s post for discussion.  Briefly, though, it’s now been independently verified, using different SAT-solvers, that the chromatic number of de Grey’s corrected graph is indeed 5.  Paul Phillips emailed to tell me that he’s now independently verified that the graph is unit distance as well.  So I think it’s time to declare the result correct.
    Question for experts: is there a general principle by which we can show that, if the chromatic number of the plane is at least 6, or is 7, then there exists a finite subgraph that witnesses it?  (This is closely related to asking, what’s the logical complexity of the Hadwiger-Nelson problem: is it Π1?)  Update: As de Grey and a commenter pointed out to me, this is the de Bruijn-Erdös Theorem from 1951.  But the proofs inherently require the Axiom of Choice.  Assuming AC, this also gives you that Hadwiger-Nslson is a Π1 statement, since the coordinates of the points in any finite counterexample can be assumed to be algebraic. However, this also raises the strange possibility that the chromatic number of the plane could be smaller assuming AC than not assuming it.
  2. Last week, Urmila Mahadev, a student (as was I, oh so many years ago) of Umesh Vazirani at Berkeley, posted a preprint on the arXiv giving a protocol for a quantum computer to prove the results of any computation it performs to a classical skeptic—assuming a relatively standard cryptographic assumption, namely the quantum hardness of the Learning With Errors (LWE) problem, and requiring only classical communication between the skeptic and the QC.  I don’t know how many readers remember, but way back in 2006, inspired by a $25,000 prize offered by Stephen Wolfram, I decided to offer a $25 prize to anyone who could solve the problem of proving the results of an arbitrary quantum computation to a classical skeptic, or who could give oracle evidence that a solution was impossible.  I had first learned this fundamental problem from Daniel Gottesman.
    Just a year or two later, independent work of Aharonov, Ben-Or, and Eban, and of Broadbent, Fitzsimons, and Kashefi made a major advance on the problem, by giving protocols that were information-theoretically secure.  The downside was that, in contrast to Mahadev’s new protocol, these earlier protocols required the verifier to be a little bit quantum: in particular, to exchange individual unentangled qubits with the QC.  Or, as shown by later work, the verifier could be completely classical, but only if it could send challenges to two or more quantum computers that were entangled but unable to communicate with each other.  In light of these achievements, I decided to award both groups their own checks for half the prize amount ($12.50), to be split among themselves however they chose.
    Neither with Broadbent et al.’s or Aharonov et al.’s earlier work, nor with Mahadev’s new work, is it immediately clear whether the protocols relativize (that is, whether they work relative to an arbitrary oracle), but it’s plausible that they don’t.
    Anyway, assuming that her breakthrough result stands, I look forward to awarding Urmila the full $25 prize when I see her at the Simons Institute in Berkeley this June.

Huge congratulations to Aubrey and Urmila for their achievements!

Update (April 12): My friend Virgi Vassilevska Williams asked me to announce a theoretical computer science women event, which will take during the upcoming STOC in LA.

Another Update: Another friend, Holden Karnofsky of the Open Philanthropy Project, asked me to advertise that OpenPhil is looking to hire a Research Analyst and Senior Research Analyst. See also this Medium piece (“Hiring Analytical Thinkers to Help Give Away Billions”) to learn more about what the job would involve.

Two announcements

Saturday, April 7th, 2018

Before my next main course comes out of the oven, I bring you two palate-cleansing appetizers:

  1. My childhood best friend Alex Halderman, whose heroic exploits helping to secure the world’s voting systems have often been featured on this blog, now has a beautifully produced video for the New York Times, entitled “I Hacked An Election.  So Can The Russians.”  Here Alex lays out the case for an audited paper trail—i.e., for what the world’s cybersecurity experts have been unanimously flailing their arms about for two decades—in terms so simple and vivid that even Congresspeople should be able to understand them.  Please consider sharing the video if you support this important cause.
  2. Jakob Nordstrom asked me to advertise the 5th Swedish Summer School in Computer Science, to be held August 5-11, 2018, in the beautiful Stockholm archipelago at Djuronaset.  This year the focus is on quantum computing, and the lecturers are two of my favorite people in the entire field: Ronald de Wolf (giving a broad intro to QC) and Oded Regev (lecturing on post-quantum cryptography).  The school is mainly for PhD students, but is also open to masters students, postdocs, and faculty.  If you wanted to spend one week getting up to speed on quantum, it’s hard for me to imagine that you’d find any opportunity more excellent.  The application deadline is April 20, so apply now if you’re interested!