70 research outputs found
Non-Smooth, H\"older-Smooth, and Robust Submodular Maximization
We study the problem of maximizing a continuous DR-submodular function that
is not necessarily smooth. We prove that the continuous greedy algorithm
achieves an [(1-1/e)\OPT-\epsilon] guarantee when the function is monotone
and H\"older-smooth, meaning that it admits a H\"older-continuous gradient. For
functions that are non-differentiable or non-smooth, we propose a variant of
the mirror-prox algorithm that attains an [(1/2)\OPT-\epsilon] guarantee. We
apply our algorithmic frameworks to robust submodular maximization and
distributionally robust submodular maximization under Wasserstein ambiguity. In
particular, the mirror-prox method applies to robust submodular maximization to
obtain a single feasible solution whose value is at least (1/2)\OPT-\epsilon.
For distributionally robust maximization under Wasserstein ambiguity, we deduce
and work over a submodular-convex maximin reformulation whose objective
function is H\"older-smooth, for which we may apply both the continuous greedy
and the mirror-prox algorithms
Adversarially Robust Submodular Maximization under Knapsack Constraints
We propose the first adversarially robust algorithm for monotone submodular
maximization under single and multiple knapsack constraints with scalable
implementations in distributed and streaming settings. For a single knapsack
constraint, our algorithm outputs a robust summary of almost optimal (up to
polylogarithmic factors) size, from which a constant-factor approximation to
the optimal solution can be constructed. For multiple knapsack constraints, our
approximation is within a constant-factor of the best known non-robust
solution.
We evaluate the performance of our algorithms by comparison to natural
robustifications of existing non-robust algorithms under two objectives: 1)
dominating set for large social network graphs from Facebook and Twitter
collected by the Stanford Network Analysis Project (SNAP), 2) movie
recommendations on a dataset from MovieLens. Experimental results show that our
algorithms give the best objective for a majority of the inputs and show strong
performance even compared to offline algorithms that are given the set of
removals in advance.Comment: To appear in KDD 201
Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications
We consider a general conic mixed-binary set where each homogeneous conic
constraint involves an affine function of independent continuous variables and
an epigraph variable associated with a nonnegative function, , of common
binary variables. Sets of this form naturally arise as substructures in a
number of applications including mean-risk optimization, chance-constrained
problems, portfolio optimization, lot-sizing and scheduling, fractional
programming, variants of the best subset selection problem, and
distributionally robust chance-constrained programs. When all of the functions
's are submodular, we give a convex hull description of this set that
relies on characterizing the epigraphs of 's. Our result unifies and
generalizes an existing result in two important directions. First, it considers
\emph{multiple general convex cone} constraints instead of a single
second-order cone type constraint. Second, it takes \emph{arbitrary nonnegative
functions} instead of a specific submodular function obtained from the square
root of an affine function. We close by demonstrating the applicability of our
results in the context of a number of broad problem classes.Comment: 21 page
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