4 research outputs found
Universal Covertness for Discrete Memoryless Sources
Consider a sequence of length emitted by a Discrete Memoryless
Source (DMS) with unknown distribution . The objective is to construct a
lossless source code that maps to a sequence of length
that is indistinguishable, in terms of Kullback-Leibler divergence, from a
sequence emitted by another DMS with known distribution . The main result
is the existence of a coding scheme that performs this task with an optimal
ratio equal to , the ratio of the Shannon entropies of the two
distributions, as goes to infinity. The coding scheme overcomes the
challenges created by the lack of knowledge about by relying on a
sufficiently fine estimation of , followed by an appropriately designed
type-based source coding that jointly performs source resolvability and
universal lossless source coding. The result recovers and extends previous
results that either assume or uniform, or known. The price
paid for these generalizations is the use of common randomness with vanishing
rate, whose length roughly scales as the square root of . By allowing common
randomness strictly larger than the square root of but still negligible
compared to , a constructive low-complexity encoding and decoding
counterpart to the main result is also provided for binary sources by means of
polar codes.Comment: 36 pages, 2 figure
Estimation of KL Divergence: Optimal Minimax Rate
The problem of estimating the Kullback-Leibler divergence between
two unknown distributions and is studied, under the assumption that the
alphabet size of the distributions can scale to infinity. The estimation is
based on independent samples drawn from and independent samples
drawn from . It is first shown that there does not exist any consistent
estimator that guarantees asymptotically small worst-case quadratic risk over
the set of all pairs of distributions. A restricted set that contains pairs of
distributions, with density ratio bounded by a function is further
considered. {An augmented plug-in estimator is proposed, and its worst-case
quadratic risk is shown to be within a constant factor of
, if
and exceed a constant factor of and , respectively.} Moreover,
the minimax quadratic risk is characterized to be within a constant factor of
, if and exceed a constant factor of
and , respectively. The lower bound on the minimax
quadratic risk is characterized by employing a generalized Le Cam's method. A
minimax optimal estimator is then constructed by employing both the polynomial
approximation and the plug-in approaches.Comment: IEEE Transactions on Information Theor
Instance Based Approximations to Profile Maximum Likelihood
In this paper we provide a new efficient algorithm for approximately
computing the profile maximum likelihood (PML) distribution, a prominent
quantity in symmetric property estimation. We provide an algorithm which
matches the previous best known efficient algorithms for computing approximate
PML distributions and improves when the number of distinct observed frequencies
in the given instance is small. We achieve this result by exploiting new
sparsity structure in approximate PML distributions and providing a new matrix
rounding algorithm, of independent interest. Leveraging this result, we obtain
the first provable computationally efficient implementation of PseudoPML, a
general framework for estimating a broad class of symmetric properties.
Additionally, we obtain efficient PML-based estimators for distributions with
small profile entropy, a natural instance-based complexity measure. Further, we
provide a simpler and more practical PseudoPML implementation that matches the
best-known theoretical guarantees of such an estimator and evaluate this method
empirically.Comment: Accepted at Thirty-fourth Conference on Neural Information Processing
Systems (NeurIPS 2020
Efficient Profile Maximum Likelihood for Universal Symmetric Property Estimation
Estimating symmetric properties of a distribution, e.g. support size,
coverage, entropy, distance to uniformity, are among the most fundamental
problems in algorithmic statistics. While each of these properties have been
studied extensively and separate optimal estimators are known for each, in
striking recent work, Acharya et al. 2016 showed that there is a single
estimator that is competitive for all symmetric properties. This work proved
that computing the distribution that approximately maximizes \emph{profile
likelihood (PML)}, i.e. the probability of observed frequency of frequencies,
and returning the value of the property on this distribution is sample
competitive with respect to a broad class of estimators of symmetric
properties. Further, they showed that even computing an approximation of the
PML suffices to achieve such a universal plug-in estimator. Unfortunately,
prior to this work there was no known polynomial time algorithm to compute an
approximate PML and it was open to obtain a polynomial time universal plug-in
estimator through the use of approximate PML. In this paper we provide a
algorithm (in number of samples) that, given samples from a distribution,
computes an approximate PML distribution up to a multiplicative error of
in time nearly linear in .
Generalizing work of Acharya et al. 2016 on the utility of approximate PML we
show that our algorithm provides a nearly linear time universal plug-in
estimator for all symmetric functions up to accuracy . Further, we show how to extend our work to provide
efficient polynomial-time algorithms for computing a -dimensional
generalization of PML (for constant ) that allows for universal plug-in
estimation of symmetric relationships between distributions.Comment: 68 page