3,429 research outputs found
Fast projections onto mixed-norm balls with applications
Joint sparsity offers powerful structural cues for feature selection,
especially for variables that are expected to demonstrate a "grouped" behavior.
Such behavior is commonly modeled via group-lasso, multitask lasso, and related
methods where feature selection is effected via mixed-norms. Several mixed-norm
based sparse models have received substantial attention, and for some cases
efficient algorithms are also available. Surprisingly, several constrained
sparse models seem to be lacking scalable algorithms. We address this
deficiency by presenting batch and online (stochastic-gradient) optimization
methods, both of which rely on efficient projections onto mixed-norm balls. We
illustrate our methods by applying them to the multitask lasso. We conclude by
mentioning some open problems.Comment: Preprint of paper under revie
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
Large-scale Multi-label Learning with Missing Labels
The multi-label classification problem has generated significant interest in
recent years. However, existing approaches do not adequately address two key
challenges: (a) the ability to tackle problems with a large number (say
millions) of labels, and (b) the ability to handle data with missing labels. In
this paper, we directly address both these problems by studying the multi-label
problem in a generic empirical risk minimization (ERM) framework. Our
framework, despite being simple, is surprisingly able to encompass several
recent label-compression based methods which can be derived as special cases of
our method. To optimize the ERM problem, we develop techniques that exploit the
structure of specific loss functions - such as the squared loss function - to
offer efficient algorithms. We further show that our learning framework admits
formal excess risk bounds even in the presence of missing labels. Our risk
bounds are tight and demonstrate better generalization performance for low-rank
promoting trace-norm regularization when compared to (rank insensitive)
Frobenius norm regularization. Finally, we present extensive empirical results
on a variety of benchmark datasets and show that our methods perform
significantly better than existing label compression based methods and can
scale up to very large datasets such as the Wikipedia dataset
Efficient First Order Methods for Linear Composite Regularizers
A wide class of regularization problems in machine learning and statistics
employ a regularization term which is obtained by composing a simple convex
function \omega with a linear transformation. This setting includes Group Lasso
methods, the Fused Lasso and other total variation methods, multi-task learning
methods and many more. In this paper, we present a general approach for
computing the proximity operator of this class of regularizers, under the
assumption that the proximity operator of the function \omega is known in
advance. Our approach builds on a recent line of research on optimal first
order optimization methods and uses fixed point iterations for numerically
computing the proximity operator. It is more general than current approaches
and, as we show with numerical simulations, computationally more efficient than
available first order methods which do not achieve the optimal rate. In
particular, our method outperforms state of the art O(1/T) methods for
overlapping Group Lasso and matches optimal O(1/T^2) methods for the Fused
Lasso and tree structured Group Lasso.Comment: 19 pages, 8 figure
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