5,363 research outputs found
A General Analysis of the Convergence of ADMM
We provide a new proof of the linear convergence of the alternating direction
method of multipliers (ADMM) when one of the objective terms is strongly
convex. Our proof is based on a framework for analyzing optimization algorithms
introduced in Lessard et al. (2014), reducing algorithm convergence to
verifying the stability of a dynamical system. This approach generalizes a
number of existing results and obviates any assumptions about specific choices
of algorithm parameters. On a numerical example, we demonstrate that minimizing
the derived bound on the convergence rate provides a practical approach to
selecting algorithm parameters for particular ADMM instances. We complement our
upper bound by constructing a nearly-matching lower bound on the worst-case
rate of convergence.Comment: 10 pages, 6 figure
Local-Aggregate Modeling for Big-Data via Distributed Optimization: Applications to Neuroimaging
Technological advances have led to a proliferation of structured big data
that have matrix-valued covariates. We are specifically motivated to build
predictive models for multi-subject neuroimaging data based on each subject's
brain imaging scans. This is an ultra-high-dimensional problem that consists of
a matrix of covariates (brain locations by time points) for each subject; few
methods currently exist to fit supervised models directly to this tensor data.
We propose a novel modeling and algorithmic strategy to apply generalized
linear models (GLMs) to this massive tensor data in which one set of variables
is associated with locations. Our method begins by fitting GLMs to each
location separately, and then builds an ensemble by blending information across
locations through regularization with what we term an aggregating penalty. Our
so called, Local-Aggregate Model, can be fit in a completely distributed manner
over the locations using an Alternating Direction Method of Multipliers (ADMM)
strategy, and thus greatly reduces the computational burden. Furthermore, we
propose to select the appropriate model through a novel sequence of faster
algorithmic solutions that is similar to regularization paths. We will
demonstrate both the computational and predictive modeling advantages of our
methods via simulations and an EEG classification problem.Comment: 41 pages, 5 figures and 3 table
HIPAD - A Hybrid Interior-Point Alternating Direction algorithm for knowledge-based SVM and feature selection
We consider classification tasks in the regime of scarce labeled training
data in high dimensional feature space, where specific expert knowledge is also
available. We propose a new hybrid optimization algorithm that solves the
elastic-net support vector machine (SVM) through an alternating direction
method of multipliers in the first phase, followed by an interior-point method
for the classical SVM in the second phase. Both SVM formulations are adapted to
knowledge incorporation. Our proposed algorithm addresses the challenges of
automatic feature selection, high optimization accuracy, and algorithmic
flexibility for taking advantage of prior knowledge. We demonstrate the
effectiveness and efficiency of our algorithm and compare it with existing
methods on a collection of synthetic and real-world data.Comment: Proceedings of 8th Learning and Intelligent OptimizatioN (LION8)
Conference, 201
D-ADMM: A Communication-Efficient Distributed Algorithm For Separable Optimization
We propose a distributed algorithm, named Distributed Alternating Direction
Method of Multipliers (D-ADMM), for solving separable optimization problems in
networks of interconnected nodes or agents. In a separable optimization problem
there is a private cost function and a private constraint set at each node. The
goal is to minimize the sum of all the cost functions, constraining the
solution to be in the intersection of all the constraint sets. D-ADMM is proven
to converge when the network is bipartite or when all the functions are
strongly convex, although in practice, convergence is observed even when these
conditions are not met. We use D-ADMM to solve the following problems from
signal processing and control: average consensus, compressed sensing, and
support vector machines. Our simulations show that D-ADMM requires less
communications than state-of-the-art algorithms to achieve a given accuracy
level. Algorithms with low communication requirements are important, for
example, in sensor networks, where sensors are typically battery-operated and
communicating is the most energy consuming operation.Comment: To appear in IEEE Transactions on Signal Processin
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