11,892 research outputs found
Variable Metric Random Pursuit
We consider unconstrained randomized optimization of smooth convex objective
functions in the gradient-free setting. We analyze Random Pursuit (RP)
algorithms with fixed (F-RP) and variable metric (V-RP). The algorithms only
use zeroth-order information about the objective function and compute an
approximate solution by repeated optimization over randomly chosen
one-dimensional subspaces. The distribution of search directions is dictated by
the chosen metric.
Variable Metric RP uses novel variants of a randomized zeroth-order Hessian
approximation scheme recently introduced by Leventhal and Lewis (D. Leventhal
and A. S. Lewis., Optimization 60(3), 329--245, 2011). We here present (i) a
refined analysis of the expected single step progress of RP algorithms and
their global convergence on (strictly) convex functions and (ii) novel
convergence bounds for V-RP on strongly convex functions. We also quantify how
well the employed metric needs to match the local geometry of the function in
order for the RP algorithms to converge with the best possible rate.
Our theoretical results are accompanied by numerical experiments, comparing
V-RP with the derivative-free schemes CMA-ES, Implicit Filtering, Nelder-Mead,
NEWUOA, Pattern-Search and Nesterov's gradient-free algorithms.Comment: 42 pages, 6 figures, 15 tables, submitted to journal, Version 3:
majorly revised second part, i.e. Section 5 and Appendi
Nonlinear Basis Pursuit
In compressive sensing, the basis pursuit algorithm aims to find the sparsest
solution to an underdetermined linear equation system. In this paper, we
generalize basis pursuit to finding the sparsest solution to higher order
nonlinear systems of equations, called nonlinear basis pursuit. In contrast to
the existing nonlinear compressive sensing methods, the new algorithm that
solves the nonlinear basis pursuit problem is convex and not greedy. The novel
algorithm enables the compressive sensing approach to be used for a broader
range of applications where there are nonlinear relationships between the
measurements and the unknowns
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
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
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