2,662 research outputs found
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]
Random Coordinate Descent Methods for Minimizing Decomposable Submodular Functions
Submodular function minimization is a fundamental optimization problem that
arises in several applications in machine learning and computer vision. The
problem is known to be solvable in polynomial time, but general purpose
algorithms have high running times and are unsuitable for large-scale problems.
Recent work have used convex optimization techniques to obtain very practical
algorithms for minimizing functions that are sums of ``simple" functions. In
this paper, we use random coordinate descent methods to obtain algorithms with
faster linear convergence rates and cheaper iteration costs. Compared to
alternating projection methods, our algorithms do not rely on full-dimensional
vector operations and they converge in significantly fewer iterations
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
The Power of Randomization: Distributed Submodular Maximization on Massive Datasets
A wide variety of problems in machine learning, including exemplar
clustering, document summarization, and sensor placement, can be cast as
constrained submodular maximization problems. Unfortunately, the resulting
submodular optimization problems are often too large to be solved on a single
machine. We develop a simple distributed algorithm that is embarrassingly
parallel and it achieves provable, constant factor, worst-case approximation
guarantees. In our experiments, we demonstrate its efficiency in large problems
with different kinds of constraints with objective values always close to what
is achievable in the centralized setting
A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint
We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, 1 - 1/e - epsilon approximation, using (1/epsilon)^{O(1/epsilon^4)} n log^2{n} function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Omega(n^2) running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle
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