49 research outputs found
Learning Poisson Binomial Distributions
We consider a basic problem in unsupervised learning: learning an unknown
\emph{Poisson Binomial Distribution}. A Poisson Binomial Distribution (PBD)
over is the distribution of a sum of independent
Bernoulli random variables which may have arbitrary, potentially non-equal,
expectations. These distributions were first studied by S. Poisson in 1837
\cite{Poisson:37} and are a natural -parameter generalization of the
familiar Binomial Distribution. Surprisingly, prior to our work this basic
learning problem was poorly understood, and known results for it were far from
optimal.
We essentially settle the complexity of the learning problem for this basic
class of distributions. As our first main result we give a highly efficient
algorithm which learns to \eps-accuracy (with respect to the total variation
distance) using \tilde{O}(1/\eps^3) samples \emph{independent of }. The
running time of the algorithm is \emph{quasilinear} in the size of its input
data, i.e., \tilde{O}(\log(n)/\eps^3) bit-operations. (Observe that each draw
from the distribution is a -bit string.) Our second main result is a
{\em proper} learning algorithm that learns to \eps-accuracy using
\tilde{O}(1/\eps^2) samples, and runs in time (1/\eps)^{\poly (\log
(1/\eps))} \cdot \log n. This is nearly optimal, since any algorithm {for this
problem} must use \Omega(1/\eps^2) samples. We also give positive and
negative results for some extensions of this learning problem to weighted sums
of independent Bernoulli random variables.Comment: Revised full version. Improved sample complexity bound of O~(1/eps^2
Learning Poisson Binomial Distributions with Differential Privacy
Στη διπλωματική αυτή προσπαθούμε να ενοποιήσουμε δύο ερευνητικά πεδία. Το πρώτο πεδίο αφορά το Distribution Learning ενώ το δεύτερο το Differnetial Privacy. Πιο συγκεκριμένα, δοθέντος ενός learning αλγορίθμου ο οποίος μαθαίνει με ε-accuracy μια Poisson διωνυμική κατανομή προσπαθούμε να βρούμε αν ο αλγόριθμος είναι Differential Private. Δείχνουμε ότι ο αλγόριθμος πετυχαίνει Differential Privacy κάτω από συγκεκριμένες υποθέσεις. Άν η κατανομή είναι κοντά σε μια (n,k) Διωνυμική κατανομή τότε ο αλγόριθμος παραμένει Differential Private. Άν η κατανομή είναι κοντά σε μια κ-Sparse μορφή τότε η ιδιότητα του Differential Privacy εξαρτάται από το πλήθος των στοιχείων του αλγορίθμου.This thesis tries to leverage two major research areas. The first area concerns the Distribution Learning area and the second the Differential Privacy. More specific, given a highly efficient algorithm which learns with ε-accuracy a Poisson Binomial distribution we try to study its Differential Privacy property. We show that if the algorithm is close to a (n,k)-Binomial form the algorithm is differential private. If the PBD is close to a k-Sparse form the algorithm's privacy depends on PBD cardinalit
A Polynomial Time Algorithm for Lossy Population Recovery
We give a polynomial time algorithm for the lossy population recovery
problem. In this problem, the goal is to approximately learn an unknown
distribution on binary strings of length from lossy samples: for some
parameter each coordinate of the sample is preserved with probability
and otherwise is replaced by a `?'. The running time and number of
samples needed for our algorithm is polynomial in and for
each fixed . This improves on algorithm of Wigderson and Yehudayoff that
runs in quasi-polynomial time for any and the polynomial time
algorithm of Dvir et al which was shown to work for by
Batman et al. In fact, our algorithm also works in the more general framework
of Batman et al. in which there is no a priori bound on the size of the support
of the distribution. The algorithm we analyze is implicit in previous work; our
main contribution is to analyze the algorithm by showing (via linear
programming duality and connections to complex analysis) that a certain matrix
associated with the problem has a robust local inverse even though its
condition number is exponentially small. A corollary of our result is the first
polynomial time algorithm for learning DNFs in the restriction access model of
Dvir et al