405 research outputs found
Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler
In 1984, Kurt Mahler posed the following fundamental question: How well can
irrationals in the Cantor set be approximated by rationals in the Cantor set?
Towards development of such a theory, we prove a Dirichlet-type theorem for
this intrinsic diophantine approximation on Cantor-like sets, and discuss
related possible theorems/conjectures. The resulting approximation function is
analogous to that for R^d, but with d being the Hausdorff dimension of the set,
and logarithmic dependence on the denominator instead.Comment: 7 pages, 0 figure
PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference
Generalized linear models (GLMs) -- such as logistic regression, Poisson
regression, and robust regression -- provide interpretable models for diverse
data types. Probabilistic approaches, particularly Bayesian ones, allow
coherent estimates of uncertainty, incorporation of prior information, and
sharing of power across experiments via hierarchical models. In practice,
however, the approximate Bayesian methods necessary for inference have either
failed to scale to large data sets or failed to provide theoretical guarantees
on the quality of inference. We propose a new approach based on constructing
polynomial approximate sufficient statistics for GLMs (PASS-GLM). We
demonstrate that our method admits a simple algorithm as well as trivial
streaming and distributed extensions that do not compound error across
computations. We provide theoretical guarantees on the quality of point (MAP)
estimates, the approximate posterior, and posterior mean and uncertainty
estimates. We validate our approach empirically in the case of logistic
regression using a quadratic approximation and show competitive performance
with stochastic gradient descent, MCMC, and the Laplace approximation in terms
of speed and multiple measures of accuracy -- including on an advertising data
set with 40 million data points and 20,000 covariates.Comment: In Proceedings of the 31st Annual Conference on Neural Information
Processing Systems (NIPS 2017). v3: corrected typos in Appendix
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