1,073 research outputs found
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
The Lov\'asz Hinge: A Novel Convex Surrogate for Submodular Losses
Learning with non-modular losses is an important problem when sets of
predictions are made simultaneously. The main tools for constructing convex
surrogate loss functions for set prediction are margin rescaling and slack
rescaling. In this work, we show that these strategies lead to tight convex
surrogates iff the underlying loss function is increasing in the number of
incorrect predictions. However, gradient or cutting-plane computation for these
functions is NP-hard for non-supermodular loss functions. We propose instead a
novel surrogate loss function for submodular losses, the Lov\'asz hinge, which
leads to O(p log p) complexity with O(p) oracle accesses to the loss function
to compute a gradient or cutting-plane. We prove that the Lov\'asz hinge is
convex and yields an extension. As a result, we have developed the first
tractable convex surrogates in the literature for submodular losses. We
demonstrate the utility of this novel convex surrogate through several set
prediction tasks, including on the PASCAL VOC and Microsoft COCO datasets
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
Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization
Creating impact in real-world settings requires artificial intelligence
techniques to span the full pipeline from data, to predictive models, to
decisions. These components are typically approached separately: a machine
learning model is first trained via a measure of predictive accuracy, and then
its predictions are used as input into an optimization algorithm which produces
a decision. However, the loss function used to train the model may easily be
misaligned with the end goal, which is to make the best decisions possible.
Hand-tuning the loss function to align with optimization is a difficult and
error-prone process (which is often skipped entirely).
We focus on combinatorial optimization problems and introduce a general
framework for decision-focused learning, where the machine learning model is
directly trained in conjunction with the optimization algorithm to produce
high-quality decisions. Technically, our contribution is a means of integrating
common classes of discrete optimization problems into deep learning or other
predictive models, which are typically trained via gradient descent. The main
idea is to use a continuous relaxation of the discrete problem to propagate
gradients through the optimization procedure. We instantiate this framework for
two broad classes of combinatorial problems: linear programs and submodular
maximization. Experimental results across a variety of domains show that
decision-focused learning often leads to improved optimization performance
compared to traditional methods. We find that standard measures of accuracy are
not a reliable proxy for a predictive model's utility in optimization, and our
method's ability to specify the true goal as the model's training objective
yields substantial dividends across a range of decision problems.Comment: Full version of paper accepted at AAAI 201
Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions
representable as a difference between submodular functions. Similar to [30],
our new algorithms are guaranteed to monotonically reduce the objective
function at every step. We empirically and theoretically show that the
per-iteration cost of our algorithms is much less than [30], and our algorithms
can be used to efficiently minimize a difference between submodular functions
under various combinatorial constraints, a problem not previously addressed. We
provide computational bounds and a hardness result on the mul- tiplicative
inapproximability of minimizing the difference between submodular functions. We
show, however, that it is possible to give worst-case additive bounds by
providing a polynomial time computable lower-bound on the minima. Finally we
show how a number of machine learning problems can be modeled as minimizing the
difference between submodular functions. We experimentally show the validity of
our algorithms by testing them on the problem of feature selection with
submodular cost features.Comment: 17 pages, 8 figures. A shorter version of this appeared in Proc.
Uncertainty in Artificial Intelligence (UAI), Catalina Islands, 201
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