23 research outputs found
An Efficient Parallel Algorithm for Spectral Sparsification of Laplacian and SDDM Matrix Polynomials
For "large" class of continuous probability density functions
(p.d.f.), we demonstrate that for every there is mixture of
discrete Binomial distributions (MDBD) with
distinct Binomial distributions that -approximates a
discretized p.d.f. for all , where
. Also, we give two efficient parallel
algorithms to find such MDBD.
Moreover, we propose a sequential algorithm that on input MDBD with
for that induces a discretized p.d.f. ,
that is either Laplacian or SDDM matrix and parameter ,
outputs in time a spectral
sparsifier of a matrix-polynomial, where
notation hides factors.
This improves the Cheng et al.'s [CCLPT15] algorithm whose run time is
.
Furthermore, our algorithm is parallelizable and runs in work
and depth . Our main algorithmic contribution is to
propose the first efficient parallel algorithm that on input continuous p.d.f.
, matrix as above, outputs a spectral sparsifier of
matrix-polynomial whose coefficients approximate component-wise the discretized
p.d.f. .
Our results yield the first efficient and parallel algorithm that runs in
nearly linear work and poly-logarithmic depth and analyzes the long term
behaviour of Markov chains in non-trivial settings. In addition, we strengthen
the Spielman and Peng's [PS14] parallel SDD solver
Two Results on Slime Mold Computations
We present two results on slime mold computations. In wet-lab experiments
(Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum
demonstrated its ability to solve shortest path problems. Biologists proposed a
mathematical model, a system of differential equations, for the slime's
adaption process (J. Theoretical Biology'07). It was shown that the process
convergences to the shortest path (J. Theoretical Biology'12) for all graphs.
We show that the dynamics actually converges for a much wider class of
problems, namely undirected linear programs with a non-negative cost vector.
Combinatorial optimization researchers took the dynamics describing slime
behavior as an inspiration for an optimization method and showed that its
discretization can -approximately solve linear programs with
positive cost vector (ITCS'16). Their analysis requires a feasible starting
point, a step size depending linearly on , and a number of steps
with quartic dependence on , where is
the difference between the smallest cost of a non-optimal basic feasible
solution and the optimal cost ().
We give a refined analysis showing that the dynamics initialized with any
strongly dominating point converges to the set of optimal solutions. Moreover,
we strengthen the convergence rate bounds and prove that the step size is
independent of , and the number of steps depends logarithmically
on and quadratically on
Algorithmic Results for Clustering and Refined Physarum Analysis
In the first part of this thesis, we study the Binary -Rank- problem which given a binary matrix and a positive integer , seeks to find a rank- binary matrix minimizing the number of non-zero entries of . A central open question is whether this problem admits a polynomial time approximation scheme. We give an affirmative answer to this question by designing the first randomized almost-linear time approximation scheme for constant over the reals, , and the Boolean semiring. In addition, we give novel algorithms for important variants of -low rank approximation.
The second part of this dissertation, studies a popular and successful heuristic, known as Approximate Spectral Clustering (ASC), for partitioning the nodes of a graph into clusters with small conductance. We give a comprehensive analysis, showing that ASC runs efficiently and yields a good approximation of an optimal -way node partition of .
In the final part of this thesis, we present two results on slime mold computations: i) the continuous undirected Physarum dynamics converges for undirected linear programs with a non-negative cost vector; and ii) for the discrete directed Physarum dynamics, we give a refined analysis that yields strengthened and close to optimal convergence rate bounds, and shows that the model can be initialized with any strongly dominating point.Im ersten Teil dieser Arbeit untersuchen wir das Binary -Rank- Problem. Hier sind eine bin{\"a}re Matrix und eine positive ganze Zahl gegeben und gesucht wird eine bin{\"a}re Matrix mit Rang , welche die Anzahl von nicht null Eintr{\"a}gen in minimiert. Wir stellen das erste randomisierte, nahezu lineare Aproximationsschema vor konstantes {\"u}ber die reellen Zahlen, und den Booleschen Semiring. Zus{\"a}tzlich erzielen wir neue Algorithmen f{\"u}r wichtige Varianten der -low rank Approximation.
Der zweite Teil dieser Dissertation besch{\"a}ftigt sich mit einer beliebten und erfolgreichen Heuristik, die unter dem Namen Approximate Spectral Cluster (ASC) bekannt ist. ASC partitioniert die Knoten eines gegeben Graphen in Cluster kleiner Conductance. Wir geben eine umfassende Analyse von ASC, die zeigt, dass ASC eine effiziente Laufzeit besitzt und eine gute Approximation einer optimale -Weg-Knoten Partition f{\"u}r berechnet.
Im letzten Teil dieser Dissertation pr{\"a}sentieren wir zwei Ergebnisse {\"u}ber Berechnungen mit Hilfe von Schleimpilzen: i) die kontinuierliche ungerichtete Physarum Dynamik konvergiert f{\"u}r ungerichtete lineare Programme mit einem nicht negativen Kostenvektor; und ii) f{\"u}r die diskrete gerichtete Physikum Dynamik geben wir eine verfeinerte Analyse, die st{\"a}rkere und beinahe optimale Schranken f{\"u}r ihre Konvergenzraten liefert und zeigt, dass das Model mit einem beliebigen stark dominierender Punkt initialisiert werden kann
Online Learning under Adversarial Nonlinear Constraints
In many applications, learning systems are required to process continuous
non-stationary data streams. We study this problem in an online learning
framework and propose an algorithm that can deal with adversarial time-varying
and nonlinear constraints. As we show in our work, the algorithm called
Constraint Violation Velocity Projection (CVV-Pro) achieves regret
and converges to the feasible set at a rate of , despite the fact
that the feasible set is slowly time-varying and a priori unknown to the
learner. CVV-Pro only relies on local sparse linear approximations of the
feasible set and therefore avoids optimizing over the entire set at each
iteration, which is in sharp contrast to projected gradients or Frank-Wolfe
methods. We also empirically evaluate our algorithm on two-player games, where
the players are subjected to a shared constraint
Density Independent Algorithms for Sparsifying k-Step Random Walks
We give faster algorithms for producing sparse approximations of the transition matrices of k-step random walks on undirected and weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with n vertices and m edges, our algorithm produces a graph with about nlog(n) edges that approximates the k-step random walk graph in about m + k^2 nlog^4(n) time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices
Secretary and online matching problems with machine learned advice
The classic analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. In contrast, machine learning approaches shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take these predictions into account. In particular, we study the following online selection problems: (i) the classic secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classic online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases
Secretary and Online Matching Problems with Machine Learned Advice
The classical analysis of online algorithms, due to its worst-case nature,
can be quite pessimistic when the input instance at hand is far from
worst-case. Often this is not an issue with machine learning approaches, which
shine in exploiting patterns in past inputs in order to predict the future.
However, such predictions, although usually accurate, can be arbitrarily poor.
Inspired by a recent line of work, we augment three well-known online settings
with machine learned predictions about the future, and develop algorithms that
take them into account. In particular, we study the following online selection
problems: (i) the classical secretary problem, (ii) online bipartite matching
and (iii) the graphic matroid secretary problem. Our algorithms still come with
a worst-case performance guarantee in the case that predictions are subpar
while obtaining an improved competitive ratio (over the best-known classical
online algorithm for each problem) when the predictions are sufficiently
accurate. For each algorithm, we establish a trade-off between the competitive
ratios obtained in the two respective cases