129 research outputs found

### Simple and Optimal Randomized Fault-Tolerant Rumor Spreading

We revisit the classic problem of spreading a piece of information in a group
of $n$ fully connected processors. By suitably adding a small dose of
randomness to the protocol of Gasienic and Pelc (1996), we derive for the first
time protocols that (i) use a linear number of messages, (ii) are correct even
when an arbitrary number of adversarially chosen processors does not
participate in the process, and (iii) with high probability have the
asymptotically optimal runtime of $O(\log n)$ when at least an arbitrarily
small constant fraction of the processors are working. In addition, our
protocols do not require that the system is synchronized nor that all
processors are simultaneously woken up at time zero, they are fully based on
push-operations, and they do not need an a priori estimate on the number of
failed nodes.
Our protocols thus overcome the typical disadvantages of the two known
approaches, algorithms based on random gossip (typically needing a large number
of messages due to their unorganized nature) and algorithms based on fair
workload splitting (which are either not {time-efficient} or require intricate
preprocessing steps plus synchronization).Comment: This is the author-generated version of a paper which is to appear in
Distributed Computing, Springer, DOI: 10.1007/s00446-014-0238-z It is
available online from
http://link.springer.com/article/10.1007/s00446-014-0238-z This version
contains some new results (Section 6

### Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign
matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For
$d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the $N \times N$ adjacency
matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value
$\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed
version of this matrix is at least $\Delta/\lambda$. We use this connection to
prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC
dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran

### Private Learning Implies Online Learning: An Efficient Reduction

We study the relationship between the notions of differentially private
learning and online learning in games. Several recent works have shown that
differentially private learning implies online learning, but an open problem of
Neel, Roth, and Wu \cite{NeelAaronRoth2018} asks whether this implication is
{\it efficient}. Specifically, does an efficient differentially private learner
imply an efficient online learner? In this paper we resolve this open question
in the context of pure differential privacy. We derive an efficient black-box
reduction from differentially private learning to online learning from expert
advice

### An adaptive nearest neighbor rule for classification

We introduce a variant of the $k$-nearest neighbor classifier in which $k$ is
chosen adaptively for each query, rather than supplied as a parameter. The
choice of $k$ depends on properties of each neighborhood, and therefore may
significantly vary between different points. (For example, the algorithm will
use larger $k$ for predicting the labels of points in noisy regions.)
We provide theory and experiments that demonstrate that the algorithm
performs comparably to, and sometimes better than, $k$-NN with an optimal
choice of $k$. In particular, we derive bounds on the convergence rates of our
classifier that depend on a local quantity we call the `advantage' which is
significantly weaker than the Lipschitz conditions used in previous convergence
rate proofs. These generalization bounds hinge on a variant of the seminal
Uniform Convergence Theorem due to Vapnik and Chervonenkis; this variant
concerns conditional probabilities and may be of independent interest

- …