499 research outputs found
Algorithms approaching the threshold for semi-random planted clique
We design new polynomial-time algorithms for recovering planted cliques in
the semi-random graph model introduced by Feige and Kilian~\cite{FK01}. The
previous best algorithms for this model succeed if the planted clique has size
at least in a graph with vertices (Mehta, Mckenzie, Trevisan,
2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for
planted-clique sizes approaching -- the information-theoretic
threshold in the semi-random model~\cite{steinhardt2017does} and a conjectured
computational threshold even in the easier fully-random model. This result
comes close to resolving open questions by Feige and Steinhardt.
Our algorithms are based on higher constant degree sum-of-squares relaxation
and rely on a new conceptual connection that translates certificates of upper
bounds on biclique numbers in \emph{unbalanced} bipartite Erd\H{o}s--R\'enyi
random graphs into algorithms for semi-random planted clique. The use of a
higher-constant degree sum-of-squares is essential in our setting: we prove a
lower bound on the basic SDP for certifying bicliques that shows that the basic
SDP cannot succeed for planted cliques of size . We also provide
some evidence that the information-computation trade-off of our current
algorithms may be inherent by proving an average-case lower bound for
unbalanced bicliques in the low-degree-polynomials model.Comment: 51 pages, the arxiv landing page contains a shortened abstrac
Lagrange Coded Computing: Optimal Design for Resiliency, Security and Privacy
We consider a scenario involving computations over a massive dataset stored
distributedly across multiple workers, which is at the core of distributed
learning algorithms. We propose Lagrange Coded Computing (LCC), a new framework
to simultaneously provide (1) resiliency against stragglers that may prolong
computations; (2) security against Byzantine (or malicious) workers that
deliberately modify the computation for their benefit; and (3)
(information-theoretic) privacy of the dataset amidst possible collusion of
workers. LCC, which leverages the well-known Lagrange polynomial to create
computation redundancy in a novel coded form across workers, can be applied to
any computation scenario in which the function of interest is an arbitrary
multivariate polynomial of the input dataset, hence covering many computations
of interest in machine learning. LCC significantly generalizes prior works to
go beyond linear computations. It also enables secure and private computing in
distributed settings, improving the computation and communication efficiency of
the state-of-the-art. Furthermore, we prove the optimality of LCC by showing
that it achieves the optimal tradeoff between resiliency, security, and
privacy, i.e., in terms of tolerating the maximum number of stragglers and
adversaries, and providing data privacy against the maximum number of colluding
workers. Finally, we show via experiments on Amazon EC2 that LCC speeds up the
conventional uncoded implementation of distributed least-squares linear
regression by up to , and also achieves a
- speedup over the state-of-the-art straggler
mitigation strategies
- …