Theoretical Machine Learning Seminar

Linear regression in L_p-norm is a canonical optimization problem that arises in several applications, including sparse recovery, semi-supervised learning, and signal processing. Standard linear regression corresponds to p=2, and p=1 or infinity is equivalent to linear programming. Generic convex optimization algorithms for solving p-regression are slow in practice. Iteratively Reweighted Least Squares (IRLS) is an easy to implement family of algorithms for solving these problems that has been studied for over 50 years. However, these algorithms often diverge for p > 3, and since the work of Osborne (1985), it has been an open problem whether there is an IRLS algorithm that is guaranteed to converge rapidly for p > 3. We propose p-IRLS, the first IRLS algorithm that provably converges geometrically for any p in [2, infty). Our algorithm is simple to implement and is guaranteed to find a high accuracy solution in a sub-linear number of iterations (roughly sqrt(dimension)*log 1/error). Our experiments demonstrate that it performs even better than our theoretical bounds, beats the standard Matlab/CVX implementation for solving these problems by 10–50x, and is the fastest among available implementations in the high-accuracy regime.