374 research outputs found
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine
learning: approximating a submodular function everywhere, learning a submodular
function (in a PAC-like setting [53]), and constrained minimization of
submodular functions. We show that the complexity of all three problems depends
on the 'curvature' of the submodular function, and provide lower and upper
bounds that refine and improve previous results [3, 16, 18, 52]. Our proof
techniques are fairly generic. We either use a black-box transformation of the
function (for approximation and learning), or a transformation of algorithms to
use an appropriate surrogate function (for minimization). Curiously, curvature
has been known to influence approximations for submodular maximization [7, 55],
but its effect on minimization, approximation and learning has hitherto been
open. We complete this picture, and also support our theoretical claims by
empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201
Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints
We investigate two new optimization problems -- minimizing a submodular
function subject to a submodular lower bound constraint (submodular cover) and
maximizing a submodular function subject to a submodular upper bound constraint
(submodular knapsack). We are motivated by a number of real-world applications
in machine learning including sensor placement and data subset selection, which
require maximizing a certain submodular function (like coverage or diversity)
while simultaneously minimizing another (like cooperative cost). These problems
are often posed as minimizing the difference between submodular functions [14,
35] which is in the worst case inapproximable. We show, however, that by
phrasing these problems as constrained optimization, which is more natural for
many applications, we achieve a number of bounded approximation guarantees. We
also show that both these problems are closely related and an approximation
algorithm solving one can be used to obtain an approximation guarantee for the
other. We provide hardness results for both problems thus showing that our
approximation factors are tight up to log-factors. Finally, we empirically
demonstrate the performance and good scalability properties of our algorithms.Comment: 23 pages. A short version of this appeared in Advances of NIPS-201
The Maximum Traveling Salesman Problem with Submodular Rewards
In this paper, we look at the problem of finding the tour of maximum reward
on an undirected graph where the reward is a submodular function, that has a
curvature of , of the edges in the tour. This problem is known to be
NP-hard. We analyze two simple algorithms for finding an approximate solution.
Both algorithms require oracle calls to the submodular function. The
approximation factors are shown to be and
, respectively; so the second
method has better bounds for low values of . We also look at how these
algorithms perform for a directed graph and investigate a method to consider
edge costs in addition to rewards. The problem has direct applications in
monitoring an environment using autonomous mobile sensors where the sensing
reward depends on the path taken. We provide simulation results to empirically
evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy
bound with curvature
Resilient Monotone Submodular Function Maximization
In this paper, we focus on applications in machine learning, optimization,
and control that call for the resilient selection of a few elements, e.g.
features, sensors, or leaders, against a number of adversarial
denial-of-service attacks or failures. In general, such resilient optimization
problems are hard, and cannot be solved exactly in polynomial time, even though
they often involve objective functions that are monotone and submodular.
Notwithstanding, in this paper we provide the first scalable,
curvature-dependent algorithm for their approximate solution, that is valid for
any number of attacks or failures, and which, for functions with low curvature,
guarantees superior approximation performance. Notably, the curvature has been
known to tighten approximations for several non-resilient maximization
problems, yet its effect on resilient maximization had hitherto been unknown.
We complement our theoretical analyses with supporting empirical evaluations.Comment: Improved suboptimality guarantees on proposed algorithm and corrected
typo on Algorithm 1's statemen
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