1,338 research outputs found
Convex Rank Tests and Semigraphoids
Convex rank tests are partitions of the symmetric group which have desirable
geometric properties. The statistical tests defined by such partitions involve
counting all permutations in the equivalence classes. Each class consists of
the linear extensions of a partially ordered set specified by data. Our methods
refine existing rank tests of non-parametric statistics, such as the sign test
and the runs test, and are useful for exploratory analysis of ordinal data. We
establish a bijection between convex rank tests and probabilistic conditional
independence structures known as semigraphoids. The subclass of submodular rank
tests is derived from faces of the cone of submodular functions, or from
Minkowski summands of the permutohedron. We enumerate all small instances of
such rank tests. Of particular interest are graphical tests, which correspond
to both graphical models and to graph associahedra
Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization
Nonconvex optimization is central in solving many machine learning problems,
in which block-wise structure is commonly encountered. In this work, we propose
cyclic block coordinate methods for nonconvex optimization problems with
non-asymptotic gradient norm guarantees. Our convergence analysis is based on a
gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a
recent progress on cyclic block coordinate methods. In deterministic settings,
our convergence guarantee matches the guarantee of (full-gradient) gradient
descent, but with the gradient Lipschitz constant being defined w.r.t.~the
Mahalanobis norm. In stochastic settings, we use recursive variance reduction
to decrease the per-iteration cost and match the arithmetic operation
complexity of current optimal stochastic full-gradient methods, with a unified
analysis for both finite-sum and infinite-sum cases. We further prove the
faster, linear convergence of our methods when a Polyak-{\L}ojasiewicz (P{\L})
condition holds for the objective function. To the best of our knowledge, our
work is the first to provide variance-reduced convergence guarantees for a
cyclic block coordinate method. Our experimental results demonstrate the
efficacy of the proposed variance-reduced cyclic scheme in training deep neural
nets
On the global convergence of randomized coordinate gradient descent for non-convex optimization
In this work, we analyze the global convergence property of coordinate
gradient descent with random choice of coordinates and stepsizes for non-convex
optimization problems. Under generic assumptions, we prove that the algorithm
iterate will almost surely escape strict saddle points of the objective
function. As a result, the algorithm is guaranteed to converge to local minima
if all saddle points are strict. Our proof is based on viewing coordinate
descent algorithm as a nonlinear random dynamical system and a quantitative
finite block analysis of its linearization around saddle points
Analysis of casino shelf shuffling machines
Many casinos routinely use mechanical card shuffling machines. We were asked
to evaluate a new product, a shelf shuffler. This leads to new probability, new
combinatorics and to some practical advice which was adopted by the
manufacturer. The interplay between theory, computing, and real-world
application is developed.Comment: Published in at http://dx.doi.org/10.1214/12-AAP884 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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