1,192 research outputs found
Activity Identification and Local Linear Convergence of Forward--Backward-type methods
In this paper, we consider a class of Forward--Backward (FB) splitting
methods that includes several variants (e.g. inertial schemes, FISTA) for
minimizing the sum of two proper convex and lower semi-continuous functions,
one of which has a Lipschitz continuous gradient, and the other is partly
smooth relatively to a smooth active manifold . We propose a
unified framework, under which we show that, this class of FB-type algorithms
(i) correctly identifies the active manifolds in a finite number of iterations
(finite activity identification), and (ii) then enters a local linear
convergence regime, which we characterize precisely in terms of the structure
of the underlying active manifolds. For simpler problems involving polyhedral
functions, we show finite termination. We also establish and explain why FISTA
(with convergent sequences) locally oscillates and can be slower than FB. These
results may have numerous applications including in signal/image processing,
sparse recovery and machine learning. Indeed, the obtained results explain the
typical behaviour that has been observed numerically for many problems in these
fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm
regularization to name only a few.Comment: Full length version of the previous short on
Regularized Nonsmooth Newton Algorithms for Best Approximation
We consider the problem of finding the best approximation point from a
polyhedral set, and its applications, in particular to solving large-scale
linear programs. The classical projection problem has many various and many
applications. We study a regularized nonsmooth Newton type solution method
where the Jacobian is singular; and we compare the computational performance to
that of the classical projection method of Halperin-Lions-Wittmann-Bauschke
(HLWB).
We observe empirically that the regularized nonsmooth method significantly
outperforms the HLWB method. However, the HLWB has a convergence guarantee
while the nonsmooth method is not monotonic and does not guarantee convergence
due in part to singularity of the generalized Jacobian.
Our application to solving large-scale linear programs uses a parametrized
projection problem. This leads to a \emph{stepping stone external path
following} algorithm. Other applications are finding triangles from branch and
bound methods, and generalized constrained linear least squares. We include
scaling methods that improve the efficiency and robustness.Comment: 38 pages, 7 tables, 8 figure
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
Towards ultrahigh dimensional feature selection for big data
In this paper, we present a new adaptive feature scaling scheme for ultrahigh-dimensional feature selection on Big Data, and then reformulate it as a convex semi-infinite programming (SIP) problem. To address the SIP, we propose an eficient feature generating paradigm. Different from traditional gradient-based approaches that conduct optimization on all input features, the proposed paradigm iteratively activates a group of features, and solves a sequence of multiple kernel learning (MKL) subproblems. To further speed up the training, we propose to solve the MKL subproblems in their primal forms through a modified accelerated proximal gradient approach. Due to such optimization scheme, some eficient cache techniques are also developed. The feature generating paradigm is guaranteed to converge globally under mild conditions, and can achieve lower feature selection bias. Moreover, the proposed method can tackle two challenging tasks in feature selection: 1) group-based feature selection with complex structures, and 2) nonlinear feature selection with explicit feature mappings. Comprehensive experiments on a wide range of synthetic and real-world data sets of tens of million data points with O(1014) features demonstrate the competitive performance of the proposed method over state-of-the-art feature selection methods in terms of generalization performance and training eficiency. © 2014 Mingkui Tan, Ivor W. Tsang and Li Wang
Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
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