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Dykstra’s Algorithm, ADMM, and Coordinate Descent: Connections, Insights, and Extensions

Presented by: 
Ryan Tibshirani Carnegie Mellon University
Date: 
Tuesday 26th June 2018 - 11:00 to 11:45
Venue: 
INI Seminar Room 1
Abstract: 
We study connections between Dykstra’s algorithm for projecting onto an intersection of convex sets, the augmented Lagrangian method of multipliers or ADMM, and block coordinate descent. We prove that coordinate descent for a regularized regression problem, in which the penalty is a separable sum of support functions, is exactly equivalent to Dykstra’s algorithm applied to the dual problem. ADMM on the dual problem is also seen to be equivalent, in the special case of two sets, with one being a linear subspace. These connections, aside from being interesting in their own right, suggest new ways of analyzing and extending coordinate descent. For example, from existing convergence theory on Dykstra’s algorithm over polyhedra, we discern that coordinate descent for the lasso problem converges at an (asymptotically) linear rate. We also develop two parallel versions of coordinate descent, based on the Dykstra and ADMM connections. Finally, we discuss the implications of this work for backfitting in additive models.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons