115,928 research outputs found
Improved Distributed Algorithms for Exact Shortest Paths
Computing shortest paths is one of the central problems in the theory of
distributed computing. For the last few years, substantial progress has been
made on the approximate single source shortest paths problem, culminating in an
algorithm of Becker et al. [DISC'17] which deterministically computes
-approximate shortest paths in time, where
is the hop-diameter of the graph. Up to logarithmic factors, this time
complexity is optimal, matching the lower bound of Elkin [STOC'04].
The question of exact shortest paths however saw no algorithmic progress for
decades, until the recent breakthrough of Elkin [STOC'17], which established a
sublinear-time algorithm for exact single source shortest paths on undirected
graphs. Shortly after, Huang et al. [FOCS'17] provided improved algorithms for
exact all pairs shortest paths problem on directed graphs.
In this paper, we present a new single-source shortest path algorithm with
complexity . For polylogarithmic , this improves
on Elkin's bound and gets closer to the
lower bound of Elkin [STOC'04]. For larger values of
, we present an improved variant of our algorithm which achieves complexity
, and
thus compares favorably with Elkin's bound of in essentially the entire range of parameters. This
algorithm provides also a qualitative improvement, because it works for the
more challenging case of directed graphs (i.e., graphs where the two directions
of an edge can have different weights), constituting the first sublinear-time
algorithm for directed graphs. Our algorithm also extends to the case of exact
-source shortest paths...Comment: 26 page
Distributed algorithms for edge dominating sets
An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimum-size edge dominating sets. The focus is on anonymous port-numbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2, ..., d. The present work shows that in the port-numbering model, edge dominating sets can be approximated as follows: in d-regular graphs, to within 4 − 6/(d + 1) for an odd d and to within 4 − 2/d for an even d; and in graphs with maximum degree Δ, to within 4 − 2/(Δ − 1) for an odd Δ and to within 4 − 2/Δ for an even Δ. These approximation ratios are tight for all values of d and Δ: there are matching lower bounds.Peer reviewe
Nearly optimal robust secret sharing
Abstract: We prove that a known approach to improve Shamir's celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δn, for any constant δ ∈ (0; 1/2). This result holds in the so-called “nonrushing” model in which the n shares are submitted simultaneously for reconstruction. We thus finally obtain a simple, fully explicit, and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k(1+o(1))+O(κ), where k is the secret length and κ is the security parameter. Like Shamir's scheme, in this modified scheme any set of more than δn honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δn(1 + ρ) honest parties, for arbitrarily small ρ > 0, can efficiently reconstruct the secret
FALKON: An Optimal Large Scale Kernel Method
Kernel methods provide a principled way to perform non linear, nonparametric
learning. They rely on solid functional analytic foundations and enjoy optimal
statistical properties. However, at least in their basic form, they have
limited applicability in large scale scenarios because of stringent
computational requirements in terms of time and especially memory. In this
paper, we take a substantial step in scaling up kernel methods, proposing
FALKON, a novel algorithm that allows to efficiently process millions of
points. FALKON is derived combining several algorithmic principles, namely
stochastic subsampling, iterative solvers and preconditioning. Our theoretical
analysis shows that optimal statistical accuracy is achieved requiring
essentially memory and time. An extensive experimental
analysis on large scale datasets shows that, even with a single machine, FALKON
outperforms previous state of the art solutions, which exploit
parallel/distributed architectures.Comment: NIPS 201
Parameterized Distributed Algorithms
In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds.
Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2
Efficient Distributed Online Prediction and Stochastic Optimization with Approximate Distributed Averaging
We study distributed methods for online prediction and stochastic
optimization. Our approach is iterative: in each round nodes first perform
local computations and then communicate in order to aggregate information and
synchronize their decision variables. Synchronization is accomplished through
the use of a distributed averaging protocol. When an exact distributed
averaging protocol is used, it is known that the optimal regret bound of
can be achieved using the distributed mini-batch
algorithm of Dekel et al. (2012), where is the total number of samples
processed across the network. We focus on methods using approximate distributed
averaging protocols and show that the optimal regret bound can also be achieved
in this setting. In particular, we propose a gossip-based optimization method
which achieves the optimal regret bound. The amount of communication required
depends on the network topology through the second largest eigenvalue of the
transition matrix of a random walk on the network. In the setting of stochastic
optimization, the proposed gossip-based approach achieves nearly-linear
scaling: the optimization error is guaranteed to be no more than
after rounds, each of which involves
gossip iterations, when nodes communicate over a
well-connected graph. This scaling law is also observed in numerical
experiments on a cluster.Comment: 30 pages, 2 figure
Minimax PAC bounds on the sample complexity of reinforcement learning with a generative model
International audienceWe consider the problem of learning the optimal action-value function in discounted-reward Markov decision processes (MDPs). We prove new PAC bounds on the sample-complexity of two well-known model-based reinforcement learning (RL) algorithms in the presence of a generative model of the MDP: value iteration and policy iteration. The first result indicates that for an MDP with state-action pairs and the discount factor γin[0, 1) only state-transition samples are required to find an -optimal estimation of the action-value function with the probability (w.p.) 1-δ. Further, we prove that, for small values of , an order of samples is required to find an -optimal policy w.p. 1-δ. We also prove a matching lower bound of on the sample complexity of estimating the optimal action-value function. To the best of our knowledge, this is the first minimax result on the sample complexity of RL: The upper bound matches the lower bound interms of , , δ and 1/(1 -γ) up to a constant factor. Also, both our lower bound and upper bound improve on the state-of-the-art in terms of their dependence on 1/(1-γ)
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