307,106 research outputs found
Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods
We investigate online algorithms for maximum (weight) independent set on
graph classes with bounded inductive independence number like, e.g., interval
and disk graphs with applications to, e.g., task scheduling and spectrum
allocation. In the online setting, it is assumed that nodes of an unknown graph
arrive one by one over time. An online algorithm has to decide whether an
arriving node should be included into the independent set. Unfortunately, this
natural and practically relevant online problem cannot be studied in a
meaningful way within a classical competitive analysis as the competitive ratio
on worst-case input sequences is lower bounded by .
As a worst-case analysis is pointless, we study online independent set in a
stochastic analysis. Instead of focussing on a particular stochastic input
model, we present a generic sampling approach that enables us to devise online
algorithms achieving performance guarantees for a variety of input models. In
particular, our analysis covers stochastic input models like the secretary
model, in which an adversarial graph is presented in random order, and the
prophet-inequality model, in which a randomly generated graph is presented in
adversarial order. Our sampling approach bridges thus between stochastic input
models of quite different nature. In addition, we show that our approach can be
applied to a practically motivated admission control setting.
Our sampling approach yields an online algorithm for maximum independent set
with competitive ratio with respect to all of the mentioned
stochastic input models. for graph classes with inductive independence number
. The approach can be extended towards maximum-weight independent set by
losing only a factor of in the competitive ratio with denoting
the (expected) number of nodes
Fast Bounded Online Gradient Descent Algorithms for Scalable Kernel-Based Online Learning
Kernel-based online learning has often shown state-of-the-art performance for
many online learning tasks. It, however, suffers from a major shortcoming, that
is, the unbounded number of support vectors, making it non-scalable and
unsuitable for applications with large-scale datasets. In this work, we study
the problem of bounded kernel-based online learning that aims to constrain the
number of support vectors by a predefined budget. Although several algorithms
have been proposed in literature, they are neither computationally efficient
due to their intensive budget maintenance strategy nor effective due to the use
of simple Perceptron algorithm. To overcome these limitations, we propose a
framework for bounded kernel-based online learning based on an online gradient
descent approach. We propose two efficient algorithms of bounded online
gradient descent (BOGD) for scalable kernel-based online learning: (i) BOGD by
maintaining support vectors using uniform sampling, and (ii) BOGD++ by
maintaining support vectors using non-uniform sampling. We present theoretical
analysis of regret bound for both algorithms, and found promising empirical
performance in terms of both efficacy and efficiency by comparing them to
several well-known algorithms for bounded kernel-based online learning on
large-scale datasets.Comment: ICML201
Network analysis of online bidding activity
With the advent of digital media, people are increasingly resorting to online
channels for commercial transactions. Online auction is a prototypical example.
In such online transactions, the pattern of bidding activity is more complex
than traditional online transactions; this is because the number of bidders
participating in a given transaction is not bounded and the bidders can also
easily respond to the bidding instantaneously. By using the recently developed
network theory, we study the interaction patterns between bidders (items) who
(that) are connected when they bid for the same item (if the item is bid by the
same bidder). The resulting network is analyzed by using the hierarchical
clustering algorithm, which is used for clustering analysis for expression data
from DNA microarrays. A dendrogram is constructed for the item subcategories;
this dendrogram is compared with a traditional classification scheme. The
implication of the difference between the two is discussed.Comment: 8 pages and 11 figure
Diffusion Approximations for Online Principal Component Estimation and Global Convergence
In this paper, we propose to adopt the diffusion approximation tools to study
the dynamics of Oja's iteration which is an online stochastic gradient descent
method for the principal component analysis. Oja's iteration maintains a
running estimate of the true principal component from streaming data and enjoys
less temporal and spatial complexities. We show that the Oja's iteration for
the top eigenvector generates a continuous-state discrete-time Markov chain
over the unit sphere. We characterize the Oja's iteration in three phases using
diffusion approximation and weak convergence tools. Our three-phase analysis
further provides a finite-sample error bound for the running estimate, which
matches the minimax information lower bound for principal component analysis
under the additional assumption of bounded samples.Comment: Appeared in NIPS 201
A Robust AFPTAS for Online Bin Packing with Polynomial Migration
In this paper we develop general LP and ILP techniques to find an approximate
solution with improved objective value close to an existing solution. The task
of improving an approximate solution is closely related to a classical theorem
of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is
often applied in designing robust algorithms for online problems. We apply our
new techniques to the online bin packing problem, where it is allowed to
reassign a certain number of items, measured by the migration factor. The
migration factor is defined by the total size of reassigned items divided by
the size of the arriving item. We obtain a robust asymptotic fully polynomial
time approximation scheme (AFPTAS) for the online bin packing problem with
migration factor bounded by a polynomial in . This answers
an open question stated by Epstein and Levin in the affirmative. As a byproduct
we prove an approximate variant of the sensitivity theorem by Cook at el. for
linear programs
TMsim : an algorithmic tool for the parametric and worst-case simulation of systems with uncertainties
This paper presents a general purpose, algebraic toolânamed TMsimâfor the combined parametric and worst-case analysis of systems with bounded uncertain parameters.The tool is based on the theory of Taylor models and represents uncertain variables on a bounded domain in terms of a Taylor polynomial plus an interval remainder accounting for truncation and round-off errors.This representation is propagated from inputs to outputs by means of a suitable redefinition of the involved calculations, in both scalar and matrix form. The polynomial provides a parametric approximation of the variable, while the remainder gives a conservative bound of the associated error. The combination between the bound of the polynomial and the interval remainder provides an estimation of the overall (worst-case) bound of the variable. After a preliminary theoretical background, the tool (freely available online) is introduced step by step along with the necessary theoretical notions. As a validation, it is applied to illustrative examples as well as to real-life problems of relevance in electrical engineering applications, specifically a quarter-car model and a continuous time linear equalizer
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