SIAM Data Mining 2014: On differential privacy

After my trip to Haverford, I attended the SIAM Data Mining (SDM) conference in Philly. For those who aren't that familiar with the data mining universe, SDM is the SIAM entrant in the data mining conference sweepstakes, along with ACM (KDD) and IEEE (ICDM). SDM is probably also the smallest of the three venues, which makes it comparable in feel to SODA (also because of SIAM organization). The conference attracts the usual data mining suspects, but also more of the applied math folks.

I was the tutorials chair this year, and there were a number of very well-attended tutorials ranging from applications to core mining to theory. In particular, +Moritz Hardt and +Aleksandar Nikolov did a very nice tutorial on differential privacy entitled 'Safer Data Mining'.

SDM is a good venue for theory folks wanting to "test the waters" with data mining: the papers are consistently more mathematically oriented and less "business-heavy", and it's a friendly crowd :).

Shameless plug: I'm the PC co-chair next year along with Jieping Ye and I'd encourage more algorithms folks to submit, and visit Vancouver in April.

In a future post I'll talk more about a panel I also ran at the conference titled 'Ethics in Data Mining'

The Shape of Information

A brief synopsis of my general-audience talk at Haverford College. 

I'm currently visiting Haverford College at the invitation of +Sorelle Friedler as part of Haverford's big data lecture series. Today's talk was a general audience talk about data mining, titled 'The Shape Of Information': (GDrive link)

The Shape Of Information 

What makes data mining so powerful, and so ubiquitous? How can the same set of techniques identify patients at risk for a rare genetic disorder, consumers most likely to like Beyonce's latest album, or even a new star from an sky survey ?  

The answer starts with an idea Descartes had nearly 500 years ago. He suggested expressing geometry in terms of numbers (coordinates). This turned out to be a powerful technique that led (among other things) to the development of the calculus. Data mining returns the favor. It starts with sets of numbers that describe a collection of objects. To find patterns in these objects, we create a geometry in which the numbers are coordinates. And just like that, objects become shapes, and the search for information becomes a quest for common structure in these shapes. 

In this search, we are not limited by the geometry of our world: we can dream up ever more intricate geometries that capture the shape of the information that we seek to find in our data. In this sense, data mining is the best kind of science fiction come to life: we craft a world out of our imagination, and let the laws of this world lead us to fascinating discoveries about the data that inhabits it.

I had a great time visiting with Sorelle's students in their data mining class. Haverford College continues to impress me with the quality of their undergrads (and their faculty !)

danah boyd, Randall Munro, and netizens.

danah boyd, author of 'It's Complicated' just gave a tech talk at Google. Her book has been in the news a lot lately, so I'll skip the details (although Facebook ought to be at least slightly worried).

But what I enjoyed the most about her talk was the feeling that I was listening to a true netizen: someone who lives and breathes on the internet, understands (and has helped build) modern technology extremely well (she is a computer scientist as well as an ethnographer), and is able to deliver a subtle and nuanced perspective on the role and use of technology amidst all the technobabble (I'm looking at you, BIG data) that inundates us.

And she delivers a message that's original and "nontrivial". Both about how teens use and interact with social media, and about how we as a society process technological trends and put them in context of our lives. Her discussion of context collapse was enlightening: apart from explaining why weddings are such fraught experiences (better with alcohol!) it helped me understand incidences of cognitive frisson in my own interactions.

What she shares with Randall Munro in my mind is the ability to speak unselfconsciously and natively in a way that rings true for those of us who inhabit the world of tech, and yet articulate things that we might have felt, but are unable to put into words ourselves. Of course they're wildly different in so many other ways, but in this respect they are like ambassadors of the new world we live in.

STOC 2014 announcement.

Howard Karloff writes in to remind everyone that the STOC 2014 early registration deadline is coming up soon (Apr 30 !). Please make sure to register early and often (ok maybe not the last part). There will be tutorials ! workshops ! posters ! papers ! and an off-off-Broadway production of Let It Go, a tragicomic musical about Dick Lipton's doomed effort to stop working on proving P = NP.

At least a constant fraction of the above statements are true.

And if you are still unconvinced, here's a picture of Columbia University, where the workshops and tutorials will take place:

Directed isoperimetry and Bregman divergences.

A new cell probe lower bound for Bregman near neighbor search via directed hypercontractivity. 

Amirali Abdullah and I have been pondering Bregman divergences for a while now. You can read all about them in my previous post on the topic. While they generalize Euclidean geometry in some nice ways, they are also quite nasty. The most important nastiness that they exhibit is asymmetry:
\[ D_\phi(x, y) \ne D_\phi(y, x)\]
What's worse is that this asymmetry can grow without bound. In particular, we can quantify the degree of asymmetry by the parameter $\mu$:
\[ \mu = \max_{x,y \in \Delta} \frac{D_\phi(x,y)}{D_\phi(y,x)} \]
where $\Delta$ is the domain of interest.

There's been a ton of work on clustering Bregman divergences, and our work on low-dimensional approximate nearest neighbor search. In almost all these results, $\mu$ shows up in the various resources (space and/or time), and it seems hard to get rid of it without making other assumptions.

After our low-D ANN result, we started trying to work on the high-dimensional version, and our thoughts turned to locality-sensitive hashing. After struggling for a while to get something that might be useful, we started pondering lower bounds. In particular, it seemed clear that the worse $\mu$ was, the harder it became to design an algorithm. So could we actually come up with a lower bound that depended on $\mu$ ?

Our first result was via a reduction from the Hamming cube (and $\ell_1$ near neighbors). While these gave us the desired lower bound, it wasn't $\mu$-sensitive. So we went looking for something stronger.

And that's where isoperimetry made an entrance. There's been a long line of results that prove lower bounds for LSH-like data structures for near neighbor search. Roughly speaking, they work in the following manner:

1. Construct a gap distribution: a distribution over inputs and queries such that a query point is very close to its nearest neighbor and very far from any other point in the space. In the $\ell_1$ case, what you want is something like $\epsilon d$ distance to the nearest neighbor, and $\Omega(d)$ distance to the second nearest neighbor. 
2. Imagine your data structure to be a function over the Hamming cube (assigning blocks of points to cells) and look at what happens when you perturb it (formally, by defining the function value to be its average over all nearby points)
3. Use isoperimetry (or hypercontractivity) to argue that the function "expands" a lot. What this implies is that points get scattered everywhere, and so any particular cell you probe doesn't have enough information to determine where the true nearest neighbor actually is. This last step can be proved in different ways, either using Fourier methods, or via the use of Poincaré-type inequalities.

Our initial hope was to use this framework to prove our bound, while incorporating terms relating to the asymmetry of Bregman divergences more directly.

This turned out to be hard. The asymmetry destroys many of the nice algebraic properties associated with the noise operator and related inner products. There's even a deeper problem: if you think of an asymmetric noise operator as pushing things "forward" on a cube with one probability and backwards with another, then it's not hard to imagine a scenario where applying it actually ends up concentrating the mass of the function in a smaller region (which would break hypercontractivity).

We needed two things: a way to get around this not-so-small issue that hypercontractivity isn't true in a directed setting, and a way to analyze the asymmetric noise operator. It turned out that we could circumvent the first problem: we could restrict the class of hash functions we considered in a way that didn't significantly change their entropy (only a constant). Alternatively, we could view hypercontractivity as being true "on average".

The second problem was a bit harder. The solution we came up with was to try and link the directed noise operator to a symmetric operator on a biased space. The intuition here was that the directed operator was creating a nonuniform distribution, so maybe starting off with one might give us what we need.

This actually worked ! There are versions of the Bonami-Beckner inequality in biased spaces that look almost just like their counterparts in uniform space (you merely set the probability of a bit being 1 to $p \ne 1/2$, and the Fourier coefficients are defined in terms of $p$).

There was lots more to address even after the isoperimetry result, but you'll have to read the paper for that :).

I should note that in many ways, this was a "Simons baby": Amir visited during one of the workshops for the program on real analysis, and we had fruitful conversations with Elchanan Mossel and Nathan Keller in early stages of this work.