A simple LAD-LASSO coordinate descent algorithm for interactive browser-based GPU applications

Abstract: Simultaneous variable selection and robust data fitting are important aspects of many mathematical modelling projects and a wide array of optimisation tools and techniques exist to support them. When the intention is to embed this capability in run-time interactive decision support tools running hundreds of such modelling tasks simultaneously on a GPU, the choices of implementation approach are more limited. Recently, simple and fast Coordinate Descent algorithms have been pro- posed which can implement the LASSO approach to variable selection in conjunction with ordinary least squares (OLS) data fitting. However extending this to use the more robust Least Absolute Deviation (LAD) data fitting has been hampered by the multiple axis wise local minima that occur in the objective function for this LAD-LASSO approach. This paper suggests that these multiple axis wise local minima form a locus which is monotonic in all the axes and that this locus has a convex objective function. Hence allowing the locus to be searched using a ternary chop algorithm that uses Coordinate Descent to identify multiple local minima (points on this locus) as required to find the global minimum. The resulting algorithm is very simple making it practical to implement it as a single thread on a GPU. This opens up the possibility of running many hundreds of such threads in parallel using coarse parallelisation [2]. These are early results in a wider project to explore the use of combi- natorial sub sets of data in interactive mathematical modelling support frameworks.