Wednesday, February 02, 2011

POTD: Continuous Local Search

Inspired by Oded Goldreich's "my choices" page, and my constant and futile attempts to read more, I'm going to attempt to write short (SHORT!) posts summarizing papers that I've been reading lately, with some thoughts. Each such post will be tagged 'potd' for 'paper of the day', in the spirit of John Baez which is to say, not a paper each day, but a paper on the day the post is made :). 

Continuous Local Search
C. Daskalakis and C. Papadimitriou
SODA 2011.

We introduce CLS, for continuous local search, a class of polynomial-time checkable total functions that lies at the intersection of PPAD and PLS, and captures a particularly benign kind of local optimization in which the domain is continuous, as opposed to combinatorial, and the functions involved are continuous. We show that this class contains several well known intriguing problems which were heretofore known to lie in the intersection of PLS and PPAD but were otherwise unclassifiable: Finding fixpoints of contraction maps, the linear complementarity problem for P matrices, finding a stationary point of a low-degree polynomial objective, the simple stochastic games of Shapley and Condon, and finding a mixed Nash equilibrium in congestion, implicit congestion, and network coordination games. The last four problems belong to CCLS, for convex CLS, another subclass of PPAD $\cap$ PLS seeking the componentwise local minimum of a componentwise convex function. It is open whether any or all of these problems are complete for the corresponding classes.
There are many iterative schemes that get used in practice for which time to convergence is unknown. The Weiszfeld algorithm for computing 1-medians is one such method. There are also alternating optimization schemes like k-means, EM and ICP that have resisted convergence analysis for a while - although we now have a pretty good understanding of the behavior of k-means. Classes like CLS appear to capture some numerical iterative schemes like gradient descent, and it might be interesting to establish connections (via reductions) between such iterative methods and other problems of a more game-theoretic nature that appear to crop up in this class.

One catch though is that while the reductions are poly time, the only requirement is that a solution to one map back to a soliution for another. So it's not clear that reductions in CLS or convex CLS preserve "time to convergence" - in fact it seems unlikely.

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