2,472 research outputs found
Probably Approximately Correct Greedy Maximization with Efficient Bounds on Information Gain for Sensor Selection
Submodular function maximization finds application in a variety of real-world
decision-making problems. However, most existing methods, based on greedy
maximization, assume it is computationally feasible to evaluate F, the function
being maximized. Unfortunately, in many realistic settings F is too expensive
to evaluate exactly even once. We present probably approximately correct greedy
maximization, which requires access only to cheap anytime confidence bounds on
F and uses them to prune elements. We show that, with high probability, our
method returns an approximately optimal set. We propose novel, cheap confidence
bounds for conditional entropy, which appears in many common choices of F and
for which it is difficult to find unbiased or bounded estimates. Finally,
results on a real-world dataset from a multi-camera tracking system in a
shopping mall demonstrate that our approach performs comparably to existing
methods, but at a fraction of the computational cost
Distributed Submodular Maximization with Limited Information
We consider a class of distributed submodular maximization problems in which
each agent must choose a single strategy from its strategy set. The global
objective is to maximize a submodular function of the strategies chosen by each
agent. When choosing a strategy, each agent has access to only a limited number
of other agents' choices. For each of its strategies, an agent can evaluate its
marginal contribution to the global objective given its information. The main
objective is to investigate how this limitation of information about the
strategies chosen by other agents affects the performance when agents make
choices according to a local greedy algorithm. In particular, we provide lower
bounds on the performance of greedy algorithms for submodular maximization,
which depend on the clique number of a graph that captures the information
structure. We also characterize graph-theoretic upper bounds in terms of the
chromatic number of the graph. Finally, we demonstrate how certain graph
properties limit the performance of the greedy algorithm. Simulations on
several common models for random networks demonstrate our results.Comment: 11 pages, 8 figure
A Memoization Framework for Scaling Submodular Optimization to Large Scale Problems
We are motivated by large scale submodular optimization problems, where
standard algorithms that treat the submodular functions in the \emph{value
oracle model} do not scale. In this paper, we present a model called the
\emph{precomputational complexity model}, along with a unifying memoization
based framework, which looks at the specific form of the given submodular
function. A key ingredient in this framework is the notion of a
\emph{precomputed statistic}, which is maintained in the course of the
algorithms. We show that we can easily integrate this idea into a large class
of submodular optimization problems including constrained and unconstrained
submodular maximization, minimization, difference of submodular optimization,
optimization with submodular constraints and several other related optimization
problems. Moreover, memoization can be integrated in both discrete and
continuous relaxation flavors of algorithms for these problems. We demonstrate
this idea for several commonly occurring submodular functions, and show how the
precomputational model provides significant speedups compared to the value
oracle model. Finally, we empirically demonstrate this for large scale machine
learning problems of data subset selection and summarization.Comment: To Appear in Proc. AISTATS 201
Conditional Gradient Method for Stochastic Submodular Maximization: Closing the Gap
In this paper, we study the problem of \textit{constrained} and
\textit{stochastic} continuous submodular maximization. Even though the
objective function is not concave (nor convex) and is defined in terms of an
expectation, we develop a variant of the conditional gradient method, called
\alg, which achieves a \textit{tight} approximation guarantee. More precisely,
for a monotone and continuous DR-submodular function and subject to a
\textit{general} convex body constraint, we prove that \alg achieves a
[(1-1/e)\text{OPT} -\eps] guarantee (in expectation) with
\mathcal{O}{(1/\eps^3)} stochastic gradient computations. This guarantee
matches the known hardness results and closes the gap between deterministic and
stochastic continuous submodular maximization. By using stochastic continuous
optimization as an interface, we also provide the first tight
approximation guarantee for maximizing a \textit{monotone but stochastic}
submodular \textit{set} function subject to a general matroid constraint
Learning and Optimization with Submodular Functions
In many naturally occurring optimization problems one needs to ensure that
the definition of the optimization problem lends itself to solutions that are
tractable to compute. In cases where exact solutions cannot be computed
tractably, it is beneficial to have strong guarantees on the tractable
approximate solutions. In order operate under these criterion most optimization
problems are cast under the umbrella of convexity or submodularity. In this
report we will study design and optimization over a common class of functions
called submodular functions. Set functions, and specifically submodular set
functions, characterize a wide variety of naturally occurring optimization
problems, and the property of submodularity of set functions has deep
theoretical consequences with wide ranging applications. Informally, the
property of submodularity of set functions concerns the intuitive "principle of
diminishing returns. This property states that adding an element to a smaller
set has more value than adding it to a larger set. Common examples of
submodular monotone functions are entropies, concave functions of cardinality,
and matroid rank functions; non-monotone examples include graph cuts, network
flows, and mutual information.
In this paper we will review the formal definition of submodularity; the
optimization of submodular functions, both maximization and minimization; and
finally discuss some applications in relation to learning and reasoning using
submodular functions.Comment: Tech Report - USC Computer Science CS-599, Convex and Combinatorial
Optimizatio
Differentiable Greedy Submodular Maximization: Guarantees, Gradient Estimators, and Applications
Motivated by, e.g., sensitivity analysis and end-to-end learning, the demand
for differentiable optimization algorithms has been significantly increasing.
In this paper, we establish a theoretically guaranteed versatile framework that
makes the greedy algorithm for monotone submodular function maximization
differentiable. We smooth the greedy algorithm via randomization, and prove
that it almost recovers original approximation guarantees in expectation for
the cases of cardinality and -extensible system constrains. We also
show how to efficiently compute unbiased gradient estimators of any expected
output-dependent quantities. We demonstrate the usefulness of our framework by
instantiating it for various applications
Adaptive Sequence Submodularity
In many machine learning applications, one needs to interactively select a
sequence of items (e.g., recommending movies based on a user's feedback) or
make sequential decisions in a certain order (e.g., guiding an agent through a
series of states). Not only do sequences already pose a dauntingly large search
space, but we must also take into account past observations, as well as the
uncertainty of future outcomes. Without further structure, finding an optimal
sequence is notoriously challenging, if not completely intractable. In this
paper, we view the problem of adaptive and sequential decision making through
the lens of submodularity and propose an adaptive greedy policy with strong
theoretical guarantees. Additionally, to demonstrate the practical utility of
our results, we run experiments on Amazon product recommendation and Wikipedia
link prediction tasks
Maximizing Influence in Social Networks: A Two-Stage Stochastic Programming Approach That Exploits Submodularity
We consider stochastic influence maximization problems arising in social
networks. In contrast to existing studies that involve greedy approximation
algorithms with a 63% performance guarantee, our work focuses on solving the
problem optimally. To this end, we introduce a new class of problems that we
refer to as two-stage stochastic submodular optimization models. We propose a
delayed constraint generation algorithm to find the optimal solution to this
class of problems with a finite number of samples. The influence maximization
problems of interest are special cases of this general problem class. We show
that the submodularity of the influence function can be exploited to develop
strong optimality cuts that are more effective than the standard optimality
cuts available in the literature. Finally, we report our computational
experiments with large-scale real-world datasets for two fundamental influence
maximization problems, independent cascade and linear threshold, and show that
our proposed algorithm outperforms the greedy algorithm
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 nondecreasing
submodular function or minimize a nonincreasing supermodular function in the
setting of bounded total curvature . In the case of submodular maximization
with curvature , we obtain a -approximation --- the first
improvement over the greedy -approximation of Conforti and
Cornuejols from 1984, which holds for a cardinality constraint, as well as
recent approaches that hold for an arbitrary matroid constraint.
Our approach is based on modifications of the continuous greedy algorithm and
non-oblivious local search, and allows us to approximately maximize the sum of
a nonnegative, nondecreasing 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 -approximation for maximization and
-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
Improved Approximation Algorithms for k-Submodular Function Maximization
This paper presents a polynomial-time -approximation algorithm for
maximizing nonnegative -submodular functions. This improves upon the
previous -approximation by Ward and
\v{Z}ivn\'y~(SODA'14), where . We also show that
for monotone -submodular functions there is a polynomial-time
-approximation algorithm while for any a
-approximation algorithm for maximizing monotone
-submodular functions would require exponentially many queries. In
particular, our hardness result implies that our algorithms are asymptotically
tight.
We also extend the approach to provide constant factor approximation
algorithms for maximizing skew-bisubmodular functions, which were recently
introduced as generalizations of bisubmodular functions
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