Communication is now the key to modelling distributed/multicore computations. Jim Demmel has been writing papers and giving talks on this theme for a while now, and as processors get faster, and the cloud becomes a standard computing platform, communication between nodes is turning out to be the major bottleneck.

So suppose you want to learn in this setting ? Suppose you have data sitting on different nodes (you have a data center, or a heterogeneous sensor network, and so on) and you'd like to learn something on the union of the data sets. You can't afford to ship everything to a single server for processing: the data might be too large to store, and the time to ship might be prohibitive.

So can you learn over the (implicit) union of all the data, with as little discussion among nodes as possible ? This was the topic of my Shonan talk, as well as two papers that I've been working on with my student Avishek Saha, in collaboration with Jeff Phillips and Hal Daume. The first one will be presented at AISTATS this week, and the second was just posted to the arxiv.

We started out with the simplest of learning problems: classification. Supppose you have data sitting on two nodes (A and B), and you wish to learn a hypothesis over the union of A and B. What you'd like is a way for the nodes to communicate as little as possible with each other while still generating a hypothesis close to the optimal solution.

It's not hard to see that you could compute an $\epsilon$-sample on A, and ship it over to B. By the usual properties of an $\epsilon$-sample, you guarantee that any classifier on B's data combined with the sample will also classify A correctly to within some $\epsilon$-error. It's also not too hard to show a lower bound that matches this upper bound. The amount of communication is nearly linear in $1/\epsilon$.

But can you do better ? In fact yes, if you let the nodes talk to each other, rather than only allowing one-way communication. One way of gaining intuition for this is that $A$ can generate classifiers, and send them over to $B$, and $B$ can tell $A$ to turn the classifier left or right. Effectively, $B$ acts as an oracle for binary search. The hard part is showing that this is actually a decimation (in that a constant fraction of points are eliminated from consideration as support points in each step), and once we do that, we can show an exponential improvement over one-way communication. There's a trivial way to extend this to more than 2 players, with a $k^2$ blow up in communication for $k$ players.

This binary search intuition only works for points in 2D, because then the search space of classifiers is on the circle, which lends itself naturally to a binary search. In higher dimensions, we have to use what is essentially a generalization of binary search - the multiplicative weight update method. I'll have more to say about this in a later post, but you can think of the MWU as a "confused zombie" binary search, in that you only sort of know "which way to go" when doing the search, and even then points that you dismissed earlier might rise from the dead.

It takes a little more work to bring the overhead for k-players down to a factor k. This comes by selecting one node as a coordinator, and implementing one of the distributed continuous sampling techniques to pass data to the coordinator.

You can read the paper for more details on the method. One thing to note is that the MWU can be "imported" from other methods that use it, which means that we get distributed algorithms for many optimization problems for free. This is great because a number of ML problems essentially reduce to some kind of optimization.

A second design template is multipass streaming: it's fairly easy to see that any multipass sublinear streaming algorithm can be placed in the k-player distributed setting, and so if you want a distributed algorithm, design a multipass streaming algorithm first.

One weakness of our algorithms was that we didn't work in the "agnostic" case, where the optimal solution itself might not be a perfect classifier (or where the data isn't separable, to view it differently). This can be fixed: in an arxiv upload made simultaneously with ours, Blum, Balcan, Fine and Mansour solve this problem very neatly, in addition to proving a number of PAC-learning results in this model.

It's nice to see different groups exploring this view of distributed learning. It shows that the model itself has legs. There are a number of problems that remain to be explored, and I'm hoping we can crack some of them. In all of this, the key is to get from a 'near linear in error' bound to a 'logarithmic in error' bound via replacing sampling by active sampling (or binary search).