Fast ADMM Algorithm for Distributed Optimization with Adaptive Penalty

We propose new methods to speed up convergence of the Alternating Direction Method of Multipliers (ADMM), a common optimization tool in the context of large scale and distributed learning. The proposed method accelerates the speed of convergence by automatically deciding the constraint penalty needed for parameter consensus in each iteration. In addition, we also propose an extension of the method that adaptively determines the maximum number of iterations to update the penalty. We show that this approach effectively leads to an adaptive, dynamic network topology underlying the distributed optimization. The utility of the new penalty update schemes is demonstrated on both synthetic and real data, including a computer vision application of distributed structure from motion.Comment: 8 pages manuscript, 2 pages appendix, 5 figure

Adaptive Relaxed ADMM: Convergence Theory and Practical Implementation

Many modern computer vision and machine learning applications rely on solving difficult optimization problems that involve non-differentiable objective functions and constraints. The alternating direction method of multipliers (ADMM) is a widely used approach to solve such problems. Relaxed ADMM is a generalization of ADMM that often achieves better performance, but its efficiency depends strongly on algorithm parameters that must be chosen by an expert user. We propose an adaptive method that automatically tunes the key algorithm parameters to achieve optimal performance without user oversight. Inspired by recent work on adaptivity, the proposed adaptive relaxed ADMM (ARADMM) is derived by assuming a Barzilai-Borwein style linear gradient. A detailed convergence analysis of ARADMM is provided, and numerical results on several applications demonstrate fast practical convergence.Comment: CVPR 201

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

FAASTA: A fast solver for total-variation regularization of ill-conditioned problems with application to brain imaging

The total variation (TV) penalty, as many other analysis-sparsity problems, does not lead to separable factors or a proximal operatorwith a closed-form expression, such as soft thresholding for the $\ell\_1$ penalty. As a result, in a variational formulation of an inverse problem or statisticallearning estimation, it leads to challenging non-smooth optimization problemsthat are often solved with elaborate single-step first-order methods. When thedata-fit term arises from empirical measurements, as in brain imaging, it isoften very ill-conditioned and without simple structure. In this situation, in proximal splitting methods, the computation cost of thegradient step can easily dominate each iteration. Thus it is beneficialto minimize the number of gradient steps.We present fAASTA, a variant of FISTA, that relies on an internal solver forthe TV proximal operator, and refines its tolerance to balance computationalcost of the gradient and the proximal steps. We give benchmarks andillustrations on "brain decoding": recovering brain maps from noisymeasurements to predict observed behavior. The algorithm as well as theempirical study of convergence speed are valuable for any non-exact proximaloperator, in particular analysis-sparsity problems