L1 regularized logistic regression is one of the widely used applied machine learning algorithms. Recently I attended an interesting talk by Rajat Raina from Facebook, who is working on both ad placement and new friends recommendation. It seems that Facebook is using logistic regression for ranking which ads to display. I also talked to Abhinav Gputa, co-Founder of RocketFuel, a successful Bay area startup that optimizes ads placement. RocketFuel is also using logistic regression for ranking ads. I further hear that Google is using it as well in Gmail ad placement.

A related problem domain to L1 regularization is compressed sensing - a great resource to learn about compressed sensing is found in Igor Caroon's blog. In the ML community, this problem is called also LASSO.

So why is logistic regression widely used? Some of the reasons are that it rather simple to compute, since it represent a convex optimization problem, that has a global unique minimum. However, when the problem size becomes large, it becomes harder and harder to efficiently compute it. Our new paper Parallel Coordinate Descent for L1-Regularized Loss Minimization (to appear on this year ICML ) gives some insight about how to compute large scale L1 regularized loss efficiently on large scale problems.

As an executive summary, I summarize below our experience in computing large scale logistic regression below. For medium size problems (problem that fit in Matlab), Kim and Boyd interior point method outperforms other algorithms, both in accuracy and speed. It is further robust and converges well on many datasets. Unfortunately, interior point methods do not scale well for large data sets, since they involve the computation of the Hessian, which is quadratic in the number of variables.

I believe that there are two possible approaches for tackling the computational problem on large datasets (i.e. problems that do not fit into a memory of one machine). One of them is our new Shotgun algorithm which works by parallelising iterative coordinate descent (the shooting algorithm). Other useful approach is parallel stochastic gradient descent. The drawback is the L1 sparsity is not obtained using the gradient descent and thus small values have to be truncated to zero.

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