172 research outputs found
Non-monotone Submodular Maximization with Nearly Optimal Adaptivity and Query Complexity
Submodular maximization is a general optimization problem with a wide range
of applications in machine learning (e.g., active learning, clustering, and
feature selection). In large-scale optimization, the parallel running time of
an algorithm is governed by its adaptivity, which measures the number of
sequential rounds needed if the algorithm can execute polynomially-many
independent oracle queries in parallel. While low adaptivity is ideal, it is
not sufficient for an algorithm to be efficient in practice---there are many
applications of distributed submodular optimization where the number of
function evaluations becomes prohibitively expensive. Motivated by these
applications, we study the adaptivity and query complexity of submodular
maximization. In this paper, we give the first constant-factor approximation
algorithm for maximizing a non-monotone submodular function subject to a
cardinality constraint that runs in adaptive rounds and makes
oracle queries in expectation. In our empirical study, we use
three real-world applications to compare our algorithm with several benchmarks
for non-monotone submodular maximization. The results demonstrate that our
algorithm finds competitive solutions using significantly fewer rounds and
queries.Comment: 12 pages, 8 figure
Scalable Influence Maximization for Multiple Products in Continuous-Time Diffusion Networks
A typical viral marketing model identifies influential users in a social network to maximize a single product adoption assuming unlimited user attention, campaign budgets, and time. In reality, multiple products need campaigns, users have limited attention, convincing users incurs costs, and advertisers have limited budgets and expect the adoptions to be maximized soon. Facing these user, monetary, and timing constraints, we formulate the problem as a submodular maximization task in a continuous-time diffusion model under the intersection of a matroid and multiple knapsack constraints. We propose a randomized algorithm estimating the user influence in a network ( nodes, edges) to an accuracy of with randomizations and computations. By exploiting the influence estimation algorithm as a subroutine, we develop an adaptive threshold greedy algorithm achieving an approximation factor of the optimal when out of the knapsack constraints are active. Extensive experiments on networks of millions of nodes demonstrate that the proposed algorithms achieve the state-of-the-art in terms of effectiveness and scalability
Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions
The Adaptive Seeding problem is an algorithmic challenge motivated by
influence maximization in social networks: One seeks to select among certain
accessible nodes in a network, and then select, adaptively, among neighbors of
those nodes as they become accessible in order to maximize a global objective
function. More generally, adaptive seeding is a stochastic optimization
framework where the choices in the first stage affect the realizations in the
second stage, over which we aim to optimize.
Our main result is a -approximation for the adaptive seeding
problem for any monotone submodular function. While adaptive policies are often
approximated via non-adaptive policies, our algorithm is based on a novel
method we call \emph{locally-adaptive} policies. These policies combine a
non-adaptive global structure, with local adaptive optimizations. This method
enables the -approximation for general monotone submodular functions
and circumvents some of the impossibilities associated with non-adaptive
policies.
We also introduce a fundamental problem in submodular optimization that may
be of independent interest: given a ground set of elements where every element
appears with some small probability, find a set of expected size at most
that has the highest expected value over the realization of the elements. We
show a surprising result: there are classes of monotone submodular functions
(including coverage) that can be approximated almost optimally as the
probability vanishes. For general monotone submodular functions we show via a
reduction from \textsc{Planted-Clique} that approximations for this problem are
not likely to be obtainable. This optimization problem is an important tool for
adaptive seeding via non-adaptive policies, and its hardness motivates the
introduction of \emph{locally-adaptive} policies we use in the main result
Submodularity in Action: From Machine Learning to Signal Processing Applications
Submodularity is a discrete domain functional property that can be
interpreted as mimicking the role of the well-known convexity/concavity
properties in the continuous domain. Submodular functions exhibit strong
structure that lead to efficient optimization algorithms with provable
near-optimality guarantees. These characteristics, namely, efficiency and
provable performance bounds, are of particular interest for signal processing
(SP) and machine learning (ML) practitioners as a variety of discrete
optimization problems are encountered in a wide range of applications.
Conventionally, two general approaches exist to solve discrete problems:
relaxation into the continuous domain to obtain an approximate solution, or
development of a tailored algorithm that applies directly in the
discrete domain. In both approaches, worst-case performance guarantees are
often hard to establish. Furthermore, they are often complex, thus not
practical for large-scale problems. In this paper, we show how certain
scenarios lend themselves to exploiting submodularity so as to construct
scalable solutions with provable worst-case performance guarantees. We
introduce a variety of submodular-friendly applications, and elucidate the
relation of submodularity to convexity and concavity which enables efficient
optimization. With a mixture of theory and practice, we present different
flavors of submodularity accompanying illustrative real-world case studies from
modern SP and ML. In all cases, optimization algorithms are presented, along
with hints on how optimality guarantees can be established
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