405 research outputs found
Worst-case Optimal Submodular Extensions for Marginal Estimation
Submodular extensions of an energy function can be used to efficiently
compute approximate marginals via variational inference. The accuracy of the
marginals depends crucially on the quality of the submodular extension. To
identify the best possible extension, we show an equivalence between the
submodular extensions of the energy and the objective functions of linear
programming (LP) relaxations for the corresponding MAP estimation problem. This
allows us to (i) establish the worst-case optimality of the submodular
extension for Potts model used in the literature; (ii) identify the worst-case
optimal submodular extension for the more general class of metric labeling; and
(iii) efficiently compute the marginals for the widely used dense CRF model
with the help of a recently proposed Gaussian filtering method. Using synthetic
and real data, we show that our approach provides comparable upper bounds on
the log-partition function to those obtained using tree-reweighted message
passing (TRW) in cases where the latter is computationally feasible.
Importantly, unlike TRW, our approach provides the first practical algorithm to
compute an upper bound on the dense CRF model.Comment: Accepted to AISTATS 201
Sample Complexity Bounds for Influence Maximization
Influence maximization (IM) is the problem of finding for a given s ? 1 a set S of |S|=s nodes in a network with maximum influence. With stochastic diffusion models, the influence of a set S of seed nodes is defined as the expectation of its reachability over simulations, where each simulation specifies a deterministic reachability function. Two well-studied special cases are the Independent Cascade (IC) and the Linear Threshold (LT) models of Kempe, Kleinberg, and Tardos [Kempe et al., 2003]. The influence function in stochastic diffusion is unbiasedly estimated by averaging reachability values over i.i.d. simulations. We study the IM sample complexity: the number of simulations needed to determine a (1-?)-approximate maximizer with confidence 1-?. Our main result is a surprising upper bound of O(s ? ?^{-2} ln (n/?)) for a broad class of models that includes IC and LT models and their mixtures, where n is the number of nodes and ? is the number of diffusion steps. Generally ? ? n, so this significantly improves over the generic upper bound of O(s n ?^{-2} ln (n/?)). Our sample complexity bounds are derived from novel upper bounds on the variance of the reachability that allow for small relative error for influential sets and additive error when influence is small. Moreover, we provide a data-adaptive method that can detect and utilize fewer simulations on models where it suffices. Finally, we provide an efficient greedy design that computes an (1-1/e-?)-approximate maximizer from simulations and applies to any submodular stochastic diffusion model that satisfies the variance bounds
Near-Optimal Sensor Scheduling for Batch State Estimation: Complexity, Algorithms, and Limits
In this paper, we focus on batch state estimation for linear systems. This
problem is important in applications such as environmental field estimation,
robotic navigation, and target tracking. Its difficulty lies on that limited
operational resources among the sensors, e.g., shared communication bandwidth
or battery power, constrain the number of sensors that can be active at each
measurement step. As a result, sensor scheduling algorithms must be employed.
Notwithstanding, current sensor scheduling algorithms for batch state
estimation scale poorly with the system size and the time horizon. In addition,
current sensor scheduling algorithms for Kalman filtering, although they scale
better, provide no performance guarantees or approximation bounds for the
minimization of the batch state estimation error. In this paper, one of our
main contributions is to provide an algorithm that enjoys both the estimation
accuracy of the batch state scheduling algorithms and the low time complexity
of the Kalman filtering scheduling algorithms. In particular: 1) our algorithm
is near-optimal: it achieves a solution up to a multiplicative factor 1/2 from
the optimal solution, and this factor is close to the best approximation factor
1/e one can achieve in polynomial time for this problem; 2) our algorithm has
(polynomial) time complexity that is not only lower than that of the current
algorithms for batch state estimation; it is also lower than, or similar to,
that of the current algorithms for Kalman filtering. We achieve these results
by proving two properties for our batch state estimation error metric, which
quantifies the square error of the minimum variance linear estimator of the
batch state vector: a) it is supermodular in the choice of the sensors; b) it
has a sparsity pattern (it involves matrices that are block tri-diagonal) that
facilitates its evaluation at each sensor set.Comment: Correction of typos in proof
Greedy Maximization Framework for Graph-based Influence Functions
The study of graph-based submodular maximization problems was initiated in a
seminal work of Kempe, Kleinberg, and Tardos (2003): An {\em influence}
function of subsets of nodes is defined by the graph structure and the aim is
to find subsets of seed nodes with (approximately) optimal tradeoff of size and
influence. Applications include viral marketing, monitoring, and active
learning of node labels. This powerful formulation was studied for
(generalized) {\em coverage} functions, where the influence of a seed set on a
node is the maximum utility of a seed item to the node, and for pairwise {\em
utility} based on reachability, distances, or reverse ranks.
We define a rich class of influence functions which unifies and extends
previous work beyond coverage functions and specific utility functions. We
present a meta-algorithm for approximate greedy maximization with strong
approximation quality guarantees and worst-case near-linear computation for all
functions in our class. Our meta-algorithm generalizes a recent design by Cohen
et al (2014) that was specific for distance-based coverage functions.Comment: 8 pages, 1 figur
Show Me the Money: Dynamic Recommendations for Revenue Maximization
Recommender Systems (RS) play a vital role in applications such as e-commerce
and on-demand content streaming. Research on RS has mainly focused on the
customer perspective, i.e., accurate prediction of user preferences and
maximization of user utilities. As a result, most existing techniques are not
explicitly built for revenue maximization, the primary business goal of
enterprises. In this work, we explore and exploit a novel connection between RS
and the profitability of a business. As recommendations can be seen as an
information channel between a business and its customers, it is interesting and
important to investigate how to make strategic dynamic recommendations leading
to maximum possible revenue. To this end, we propose a novel \model that takes
into account a variety of factors including prices, valuations, saturation
effects, and competition amongst products. Under this model, we study the
problem of finding revenue-maximizing recommendation strategies over a finite
time horizon. We show that this problem is NP-hard, but approximation
guarantees can be obtained for a slightly relaxed version, by establishing an
elegant connection to matroid theory. Given the prohibitively high complexity
of the approximation algorithm, we also design intelligent heuristics for the
original problem. Finally, we conduct extensive experiments on two real and
synthetic datasets and demonstrate the efficiency, scalability, and
effectiveness our algorithms, and that they significantly outperform several
intuitive baselines.Comment: Conference version published in PVLDB 7(14). To be presented in the
VLDB Conference 2015, in Hawaii. This version gives a detailed submodularity
proo
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