3 research outputs found
On the Capacity of Private Monomial Computation
In this work, we consider private monomial computation (PMC) for replicated
noncolluding databases. In PMC, a user wishes to privately retrieve an
arbitrary multivariate monomial from a candidate set of monomials in
messages over a finite field , where is a power of a prime
and , replicated over databases. We derive the PMC capacity
under a technical condition on and for asymptotically large . The
condition on is satisfied, e.g., for large enough . Also, we present a
novel PMC scheme for arbitrary that is capacity-achieving in the asymptotic
case above. Moreover, we present formulas for the entropy of a multivariate
monomial and for a set of monomials in uniformly distributed random variables
over a finite field, which are used in the derivation of the capacity
expression.Comment: Accepted for 2020 International Zurich Seminar on Information and
Communicatio
Private Inner Product Retrieval for Distributed Machine Learning
In this paper, we argue that in many basic algorithms for machine learning,
including support vector machine (SVM) for classification, principal component
analysis (PCA) for dimensionality reduction, and regression for dependency
estimation, we need the inner products of the data samples, rather than the
data samples themselves.
Motivated by the above observation, we introduce the problem of private inner
product retrieval for distributed machine learning, where we have a system
including a database of some files, duplicated across some non-colluding
servers. A user intends to retrieve a subset of specific size of the inner
products of the data files with minimum communication load, without revealing
any information about the identity of the requested subset. For achievability,
we use the algorithms for multi-message private information retrieval. For
converse, we establish that as the length of the files becomes large, the set
of all inner products converges to independent random variables with uniform
distribution, and derive the rate of convergence. To prove that, we construct
special dependencies among sequences of the sets of all inner products with
different length, which forms a time-homogeneous irreducible Markov chain,
without affecting the marginal distribution. We show that this Markov chain has
a uniform distribution as its unique stationary distribution, with rate of
convergence dominated by the second largest eigenvalue of the transition
probability matrix. This allows us to develop a converse, which converges to a
tight bound in some cases, as the size of the files becomes large. While this
converse is based on the one in multi-message private information retrieval,
due to the nature of retrieving inner products instead of data itself some
changes are made to reach the desired result
Private Function Computation for Noncolluding Coded Databases
Private computation in a distributed storage system (DSS) is a generalization
of the private information retrieval (PIR) problem. In such setting a user
wishes to compute a function of messages stored in noncolluding coded
databases while revealing no information about the desired function to the
databases. We consider the problem of private polynomial computation (PPC). In
PPC, a user wishes to compute a multivariate polynomial of degree at most
over variables (or messages) stored in multiple databases. First, we
consider the private computation of polynomials of degree , i.e., private
linear computation (PLC) for coded databases. In PLC, a user wishes to compute
a linear combination over the messages while keeping the coefficients of
the desired linear combination hidden from the database. For a linearly encoded
DSS, we present a capacity-achieving PLC scheme and show that the PLC capacity,
which is the ratio of the desired amount of information and the total amount of
downloaded information, matches the maximum distance separable coded capacity
of PIR for a large class of linear storage codes. Then, we consider private
computation of higher degree polynomials, i.e., . For this setup, we
construct two novel PPC schemes. In the first scheme we consider Reed-Solomon
coded databases with Lagrange encoding, which leverages ideas from recently
proposed star-product PIR and Lagrange coded computation. The second scheme
considers the special case of coded databases with systematic Lagrange
encoding. Both schemes yield improved rates compared to the best known schemes
from the literature for a small number of messages, while asymptotically, as
, the systematic scheme gives a significantly better
computation rate compared to all known schemes up to some storage code rate
that depends on the maximum degree of the candidate polynomials.Comment: 34 pages, 4 figures, 7 tables, submitted for publication. Some
overlap with arXiv:1810.04230, arXiv:1901.1028