18 research outputs found
Unified lower bounds for interactive high-dimensional estimation under information constraints
We consider the task of distributed parameter estimation using interactive
protocols subject to local information constraints such as bandwidth
limitations, local differential privacy, and restricted measurements. We
provide a unified framework enabling us to derive a variety of (tight) minimax
lower bounds for different parametric families of distributions, both
continuous and discrete, under any loss. Our lower bound framework is
versatile and yields "plug-and-play" bounds that are widely applicable to a
large range of estimation problems. In particular, our approach recovers bounds
obtained using data processing inequalities and Cram\'er--Rao bounds, two other
alternative approaches for proving lower bounds in our setting of interest.
Further, for the families considered, we complement our lower bounds with
matching upper bounds.Comment: Significant improvements: handle sparse parameter estimation,
simplify and generalize argument
Mean Estimation from One-Bit Measurements
We consider the problem of estimating the mean of a symmetric log-concave
distribution under the constraint that only a single bit per sample from this
distribution is available to the estimator. We study the mean squared error as
a function of the sample size (and hence the number of bits). We consider three
settings: first, a centralized setting, where an encoder may release bits
given a sample of size , and for which there is no asymptotic penalty for
quantization; second, an adaptive setting in which each bit is a function of
the current observation and previously recorded bits, where we show that the
optimal relative efficiency compared to the sample mean is precisely the
efficiency of the median; lastly, we show that in a distributed setting where
each bit is only a function of a local sample, no estimator can achieve optimal
efficiency uniformly over the parameter space. We additionally complement our
results in the adaptive setting by showing that \emph{one} round of adaptivity
is sufficient to achieve optimal mean-square error