2,198 research outputs found
Convex Relaxation for Combinatorial Penalties
In this paper, we propose an unifying view of several recently proposed
structured sparsity-inducing norms. We consider the situation of a model
simultaneously (a) penalized by a set- function de ned on the support of the
unknown parameter vector which represents prior knowledge on supports, and (b)
regularized in Lp-norm. We show that the natural combinatorial optimization
problems obtained may be relaxed into convex optimization problems and
introduce a notion, the lower combinatorial envelope of a set-function, that
characterizes the tightness of our relaxations. We moreover establish links
with norms based on latent representations including the latent group Lasso and
block-coding, and with norms obtained from submodular functions.Comment: 35 pag
Combinatorial Penalties: Which structures are preserved by convex relaxations?
We consider the homogeneous and the non-homogeneous convex relaxations for
combinatorial penalty functions defined on support sets. Our study identifies
key differences in the tightness of the resulting relaxations through the
notion of the lower combinatorial envelope of a set-function along with new
necessary conditions for support identification. We then propose a general
adaptive estimator for convex monotone regularizers, and derive new sufficient
conditions for support recovery in the asymptotic setting
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
Differentially Private Empirical Risk Minimization with Sparsity-Inducing Norms
Differential privacy is concerned about the prediction quality while
measuring the privacy impact on individuals whose information is contained in
the data. We consider differentially private risk minimization problems with
regularizers that induce structured sparsity. These regularizers are known to
be convex but they are often non-differentiable. We analyze the standard
differentially private algorithms, such as output perturbation, Frank-Wolfe and
objective perturbation. Output perturbation is a differentially private
algorithm that is known to perform well for minimizing risks that are strongly
convex. Previous works have derived excess risk bounds that are independent of
the dimensionality. In this paper, we assume a particular class of convex but
non-smooth regularizers that induce structured sparsity and loss functions for
generalized linear models. We also consider differentially private Frank-Wolfe
algorithms to optimize the dual of the risk minimization problem. We derive
excess risk bounds for both these algorithms. Both the bounds depend on the
Gaussian width of the unit ball of the dual norm. We also show that objective
perturbation of the risk minimization problems is equivalent to the output
perturbation of a dual optimization problem. This is the first work that
analyzes the dual optimization problems of risk minimization problems in the
context of differential privacy
Structured sparsity-inducing norms through submodular functions
Sparse methods for supervised learning aim at finding good linear predictors
from as few variables as possible, i.e., with small cardinality of their
supports. This combinatorial selection problem is often turned into a convex
optimization problem by replacing the cardinality function by its convex
envelope (tightest convex lower bound), in this case the L1-norm. In this
paper, we investigate more general set-functions than the cardinality, that may
incorporate prior knowledge or structural constraints which are common in many
applications: namely, we show that for nondecreasing submodular set-functions,
the corresponding convex envelope can be obtained from its \lova extension, a
common tool in submodular analysis. This defines a family of polyhedral norms,
for which we provide generic algorithmic tools (subgradients and proximal
operators) and theoretical results (conditions for support recovery or
high-dimensional inference). By selecting specific submodular functions, we can
give a new interpretation to known norms, such as those based on
rank-statistics or grouped norms with potentially overlapping groups; we also
define new norms, in particular ones that can be used as non-factorial priors
for supervised learning
Iterative Log Thresholding
Sparse reconstruction approaches using the re-weighted l1-penalty have been
shown, both empirically and theoretically, to provide a significant improvement
in recovering sparse signals in comparison to the l1-relaxation. However,
numerical optimization of such penalties involves solving problems with
l1-norms in the objective many times. Using the direct link of reweighted
l1-penalties to the concave log-regularizer for sparsity, we derive a simple
prox-like algorithm for the log-regularized formulation. The proximal splitting
step of the algorithm has a closed form solution, and we call the algorithm
'log-thresholding' in analogy to soft thresholding for the l1-penalty.
We establish convergence results, and demonstrate that log-thresholding
provides more accurate sparse reconstructions compared to both soft and hard
thresholding. Furthermore, the approach can be directly extended to
optimization over matrices with penalty for rank (i.e. the nuclear norm penalty
and its re-weigthed version), where we suggest a singular-value
log-thresholding approach.Comment: 5 pages, 4 figure
Exact solutions to a class of stochastic generalized assignment problems
This paper deals with a stochastic Generalized Assignment Problem with recourse. Only a random subset of the given set of jobs will require to be actually processed. An assignment of each job to an agent is decided a priori, and once the demands are known, reassignments can be performed if there are overloaded agents. We construct a convex approximation of the objective function that is sharp at all feasible solutions. We then present three versions of an exact algorithm to solve this problem, based on branch and bound techniques, optimality cuts, and a special purpose lower bound. numerical results are reported.
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