143 research outputs found
Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration
Several fundamental problems that arise in optimization and computer science
can be cast as follows: Given vectors and a
constraint family , find a set that
maximizes the squared volume of the simplex spanned by the vectors in . A
motivating example is the data-summarization problem in machine learning where
one is given a collection of vectors that represent data such as documents or
images. The volume of a set of vectors is used as a measure of their diversity,
and partition or matroid constraints over are imposed in order to ensure
resource or fairness constraints. Recently, Nikolov and Singh presented a
convex program and showed how it can be used to estimate the value of the most
diverse set when corresponds to a partition matroid. This result was
recently extended to regular matroids in works of Straszak and Vishnoi, and
Anari and Oveis Gharan. The question of whether these estimation algorithms can
be converted into the more useful approximation algorithms -- that also output
a set -- remained open.
The main contribution of this paper is to give the first approximation
algorithms for both partition and regular matroids. We present novel
formulations for the subdeterminant maximization problem for these matroids;
this reduces them to the problem of finding a point that maximizes the absolute
value of a nonconvex function over a Cartesian product of probability
simplices. The technical core of our results is a new anti-concentration
inequality for dependent random variables that allows us to relate the optimal
value of these nonconvex functions to their value at a random point. Unlike
prior work on the constrained subdeterminant maximization problem, our proofs
do not rely on real-stability or convexity and could be of independent interest
both in algorithms and complexity.Comment: in FOCS 201
Constrained Submodular Maximization: Beyond 1/e
In this work, we present a new algorithm for maximizing a non-monotone
submodular function subject to a general constraint. Our algorithm finds an
approximate fractional solution for maximizing the multilinear extension of the
function over a down-closed polytope. The approximation guarantee is 0.372 and
it is the first improvement over the 1/e approximation achieved by the unified
Continuous Greedy algorithm [Feldman et al., FOCS 2011]
A New Framework for Distributed Submodular Maximization
A wide variety of problems in machine learning, including exemplar
clustering, document summarization, and sensor placement, can be cast as
constrained submodular maximization problems. A lot of recent effort has been
devoted to developing distributed algorithms for these problems. However, these
results suffer from high number of rounds, suboptimal approximation ratios, or
both. We develop a framework for bringing existing algorithms in the sequential
setting to the distributed setting, achieving near optimal approximation ratios
for many settings in only a constant number of MapReduce rounds. Our techniques
also give a fast sequential algorithm for non-monotone maximization subject to
a matroid constraint
Algorithms and Hardness for Robust Subspace Recovery
We consider a fundamental problem in unsupervised learning called
\emph{subspace recovery}: given a collection of points in ,
if many but not necessarily all of these points are contained in a
-dimensional subspace can we find it? The points contained in are
called {\em inliers} and the remaining points are {\em outliers}. This problem
has received considerable attention in computer science and in statistics. Yet
efficient algorithms from computer science are not robust to {\em adversarial}
outliers, and the estimators from robust statistics are hard to compute in high
dimensions.
Are there algorithms for subspace recovery that are both robust to outliers
and efficient? We give an algorithm that finds when it contains more than a
fraction of the points. Hence, for say this estimator
is both easy to compute and well-behaved when there are a constant fraction of
outliers. We prove that it is Small Set Expansion hard to find when the
fraction of errors is any larger, thus giving evidence that our estimator is an
{\em optimal} compromise between efficiency and robustness.
As it turns out, this basic problem has a surprising number of connections to
other areas including small set expansion, matroid theory and functional
analysis that we make use of here.Comment: Appeared in Proceedings of COLT 201
Near-Optimal Sensor Scheduling for Batch State Estimation: Complexity, Algorithms, and Limits
In this paper, we focus on batch state estimation for linear systems. This
problem is important in applications such as environmental field estimation,
robotic navigation, and target tracking. Its difficulty lies on that limited
operational resources among the sensors, e.g., shared communication bandwidth
or battery power, constrain the number of sensors that can be active at each
measurement step. As a result, sensor scheduling algorithms must be employed.
Notwithstanding, current sensor scheduling algorithms for batch state
estimation scale poorly with the system size and the time horizon. In addition,
current sensor scheduling algorithms for Kalman filtering, although they scale
better, provide no performance guarantees or approximation bounds for the
minimization of the batch state estimation error. In this paper, one of our
main contributions is to provide an algorithm that enjoys both the estimation
accuracy of the batch state scheduling algorithms and the low time complexity
of the Kalman filtering scheduling algorithms. In particular: 1) our algorithm
is near-optimal: it achieves a solution up to a multiplicative factor 1/2 from
the optimal solution, and this factor is close to the best approximation factor
1/e one can achieve in polynomial time for this problem; 2) our algorithm has
(polynomial) time complexity that is not only lower than that of the current
algorithms for batch state estimation; it is also lower than, or similar to,
that of the current algorithms for Kalman filtering. We achieve these results
by proving two properties for our batch state estimation error metric, which
quantifies the square error of the minimum variance linear estimator of the
batch state vector: a) it is supermodular in the choice of the sensors; b) it
has a sparsity pattern (it involves matrices that are block tri-diagonal) that
facilitates its evaluation at each sensor set.Comment: Correction of typos in proof
Optimal Approximation for Submodular and Supermodular Optimization with Bounded Curvature
We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a monotone increasing submodular function or minimize a monotone decreasing supermodular function with a bounded total curvature c. Intuitively, the parameter c represents how nonlinear a function f is: when c = 0, f is linear, while for c = 1, f may be an arbitrary monotone increasing submodular function. For the case of submodular maximization with total curvature c, we obtain a (1 − c/e)-approximation—the first improvement over the greedy algorithm of of Conforti and Cornuéjols from 1984, which holds for a cardinality constraint, as well as a recent analogous result for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and nonoblivious local search, and allows us to approximately maximize the sum of a nonnegative, monotone increasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show a (1 − c)-approximation for maximization and a 1/(1 − c)-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem
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