Patrick Johnstone, Rutgers University

Abstract

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and potentially distributed and asynchronous manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. Interestingly, this backtracking procedure also leads to a convergent algorithm even when the operator is only uniformly continuous, but not necessarily Lipschitz, provided it has full domain. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. Convergence rates under various assumptions are also discussed along with preliminary experiments of several kinds of splitting algorithms on the lasso problem.

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