## Monday, October 06, 2014

### Algorithms is the new Algorithms...

Hal Daumé wrote a rather provocative post titled ‘Machine Learning is the new algorithms’, and has begged someone to throw his gauntlet back at him. Consider it now thrown !

His thesis is the following quote:
Everything that algorithms was to computer science 15 years ago, machine learning is today
And among the conclusions he draws is the following:
we should yank algorithms out as an UG requirement and replace it with machine learning
Having spent many years having coffees and writing papers with Hal, I know that he truly does understand algorithms and isn’t just trying to be a troll (at least I hope not). So I’m trying to figure out exactly what he’s trying to say. It will help if you read his article first before returning…

First off, I don’t understand the conclusion. Why not (say) replace architecture with ML, or databases with ML. Or why replace anything at all ? the assumption is that ML is a core tool that students will need more than any other topic. Now I have no problem with adding ML to the list of “things a CS person ought to know”, but I don’t see why it’s not important for a CS person to understand how a database works, or how a compiler processes code, or even how we design an algorithm. This fake mutual exclusiveness appears to be without basis.

But maybe he’s saying that algorithms and ML are two flavors of the same object, and hence one can be replaced by the other. If so, what exactly is that object ? In his view, that object is:
given an input, what’s the best way to produce an (optimal) output ?
This seems to be an overly charitable view of ML. In ML, the goal is to, well, learn. Or to be more stodgy about it, ML provides tools for making systematic inferences and predictions from data.

Which suggests that the concerns are fundamentally orthogonal, not in opposition (and Sasho Nikolov makes this point very well in a comment on Hal’s post). As Hal correctly argues, the hard work in ML is front-loaded: modeling, feature extraction, and the like. The algorithm itself is mostly an afterthought.

But what’s ironic is that one of the most important trends in ML of late is the conversion of an ML problem to an optimization problem. The goal is to make good modeling choices that lead to an optimization problem that can be solved well. But wait: what do you need to know how to solve an optimization ? Wait for it…… ALGORITHMS !!

The argument about stability in algorithms is an odd one, especially coming from someone who’s just written a book on ML. Yes, some core algorithms techniques haven’t changed much in the last many years, but did you see that 2014 paper on improvements in recurrence analysis ? Or the new sorting algorithm by Mike Goodrich ? or even the flurry of new results for Hal’s beloved flow problems ?

As for “everything’s in a library”, I’ll take your boost graph library and give you back WEKA, or libSVM, or even scikit-learn. I don’t need to know anything from Hal’s book to do some basic futzing around in ML using libraries: much like I could invoke some standard hashing subroutine without knowing much about universal hash families.

In one sense though, Hal is right: ML is indeed where algorithms was 15 years ago. Because 15 years ago (approximately) is when the streaming revolution started, and with it the new interest in sub linear algorithms, communication complexity, matrix approximations, distributed algorithms with minimal communication, and the whole “theory of big data” effort. And ML is now trying to catch up to all of this, with some help with from algorithms folks :).

What is true is this though: it wouldn’t hurt us to revisit how we construct the core algorithms classes (undergrad and grad). I realize that CLRS is the canon, and it would cause serious heart palpitations to contemplate not stuffing every piece of paper of that book down students’ throats, but maybe we should be adapting algorithms courses to the new and exciting developments in algorithms itself. I bring in heavy doses of approximation and randomization in my grad algorithms class, and before we forked off a whole new class, I used to teach bits of streaming, bloom filters, min-wise hashing and the like as well. My geometry class used to be all about the core concepts from the 3 Marks book, but now I throw in lots of high dimensional geometry, approximate geometry, kernels and the like.

Ultimately, I think a claim of fake mutual exclusivity is lazy, and ignores the true synergy between ML and algorithms. I think algorithms research has a lot to learn from ML about robust solution design and the value of "noise-tolerance", and ML has plenty to learn from algorithms about how to organize problems and solutions, and how deep dives into the structure of a problem can yield insights that are resilient to changes in the problem specification.

## Wednesday, October 01, 2014

### On the master theorem vs Akra-Bazzi

Everyone knows the master theorem.

Or at least everyone reading this blog does.

And I'm almost certain that everyone reading this blog has heard of the generalization of the master theorem due to Akra and Bazzi. It's particularly useful when you have recurrences of the form
$$T(n) = \sum_i a_i T(n/b_i) + g(n)$$
because like the master theorem it gives you a quick way to generate the desired answer (or at least a guess that you can plug in to the recurrence to check).

(And yes, I'm aware of the generalization of A/B due to Drmota and Szpankowski)

When I started teaching grad algorithms this fall, I was convinced that I wanted to teach the Akra-Bazzi method instead of the master theorem. But I didn't, and here's why.

Let's write down the standard formulation that the master theorem applies to
$$T(n) = a T(n/b) + f(n)$$
This recurrence represents the "battle" between the two terms involved in a recursive algorithm: the effort involved in dividing (the $a T(n/b)$) and the effort involved in putting things back together (the $f(n)$).

And the solution mirrors this tension: we look at which term is "stronger" and therefore dominates the resulting running time, or what happens when they balance each other out. In fact this is essentially how the proof works as well.

I have found this to be a useful way to make the master theorem "come alive" as it were, and allow students to see what's likely to happen in a recurrence without actually trying to solve it. And this is very valuable, because it reinforces the point I'm constantly harping on: that the study of recurrences is a way to see how to design a recursive algorithm. That decimation as  a strategy can be seen to work just by looking at the recurrence. And so on.

But the Akra-Bazzi method, even though it's tremendously powerful, admits no such easy intuition. The bound comes from solving the equation
$$\sum a_i b_i^p = 1$$ for $p$, and this is a much more cryptic expression to parse. And the proof doesn't help make it any less cryptic.

Which is not to say you can't see how it works with sufficient experience. But that's the point: with sufficient experience. From a purely pedagogical perspective, I'd much rather teach the master theorem so that students get a more intuitive feel for recurrences, and then tell them about A/B for cases (like in median finding) where the master theorem can only provide an intuitive answer and not a rigorous one.