7,839 research outputs found
Alternating Randomized Block Coordinate Descent
Block-coordinate descent algorithms and alternating minimization methods are
fundamental optimization algorithms and an important primitive in large-scale
optimization and machine learning. While various block-coordinate-descent-type
methods have been studied extensively, only alternating minimization -- which
applies to the setting of only two blocks -- is known to have convergence time
that scales independently of the least smooth block. A natural question is
then: is the setting of two blocks special?
We show that the answer is "no" as long as the least smooth block can be
optimized exactly -- an assumption that is also needed in the setting of
alternating minimization. We do so by introducing a novel algorithm AR-BCD,
whose convergence time scales independently of the least smooth (possibly
non-smooth) block. The basic algorithm generalizes both alternating
minimization and randomized block coordinate (gradient) descent, and we also
provide its accelerated version -- AAR-BCD. As a special case of AAR-BCD, we
obtain the first nontrivial accelerated alternating minimization algorithm.Comment: Version 1 appeared Proc. ICML'18. v1 -> v2: added remarks about how
accelerated alternating minimization follows directly from the results that
appeared in ICML'18; no new technical results were needed for thi
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the
commonly used iterated rounding and randomized rounding techniques. In
particular, we show that whenever iterated rounding can be applied to a problem
with some slack, there is a randomized procedure that returns an integral
solution that satisfies the guarantees of iterated rounding and also has
concentration properties. We use this to give new results for several classic
problems where iterated rounding has been useful
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