# Vectorization

When working with learning algorithms, having a faster piece of code often means that you'll make progress faster on your project. For example, if your learning algorithm takes 20 minutes to run to completion, that means you can "try" up to 3 new ideas per hour. But if your code takes 20 hours to run, that means you can "try" only one idea a day, since that's how long you have to wait to get feedback from your program. In this latter case, if you can speed up your code so that it takes only 10 hours to run, that can literally double your personal productivity!

Vectorization refers to a powerful way to speed up your algorithms. Numerical computing and parallel computing researchers have put decades of work into making certain numerical operations (such as matrix-matrix multiplication, matrix-matrix addition, matrix-vector multiplication) fast. The idea of vectorization is that we would like to express our learning algorithms in terms of these highly optimized operations.

For example, if $x \in \Re^{n+1}$ and $\textstyle \theta \in \Re^{n+1}$ are vectors and you need to compute $\textstyle z = \theta^Tx$, you can implement (in Matlab):

z = 0;
for i=1:(n+1),
z = z + theta(i) * x(i);
end;

or you can more simply implement

z = theta' * x;

The second piece of code is not only simpler, but it will also run much faster.

More generally, a good rule-of-thumb for coding Matlab/Octave is:

Whenever possible, avoid using explicit for-loops in your code.

In particular, the first code example used an explicit for loop. By implementing the same functionality without the for loop, we sped it up significantly. A large part of vectorizing our Matlab/Octave code will focus on getting rid of for loops, since this lets Matlab/Octave extract more parallelism from your code, while also incurring less computational overhead from the interpreter.

In terms of a strategy for writing your code, initially you may find that vectorized code is harder to write, read, and/or debug, and that there may be a tradeoff in ease of programming/debugging vs. running time. Thus, for your first few programs, you might choose to first implement your algorithm without too many vectorization tricks, and verify that it is working correctly (perhaps by running on a small problem). Then only after it is working, you can vectorize your code one piece at a time, pausing after each piece to verify that your code is still computing the same result as before. At the end, you'll then hopefully have a correct, debugged, and vectorized/efficient piece of code.

After you become familiar with the most common vectorization methods and tricks, you'll find that it usually isn't much effort to vectorize your code. Doing so will make your code run much faster and, in some cases, simplify it too.