  We consider the fundamental learning problem of estimating properties of
distributions over large domains. Using a novel piecewise-polynomial
approximation technique, we derive the first unified methodology for
constructing sample- and time-efficient estimators for all sufficiently smooth,
symmetric and non-symmetric, additive properties. This technique yields
near-linear-time computable estimators whose approximation values are
asymptotically optimal and highly-concentrated, resulting in the first: 1)
estimators achieving the $\mathcal{O}(k/(\varepsilon^2\log k))$ min-max
$\varepsilon$-error sample complexity for all $k$-symbol Lipschitz properties;
2) unified near-optimal differentially private estimators for a variety of
properties; 3) unified estimator achieving optimal bias and near-optimal
variance for five important properties; 4) near-optimal sample-complexity
estimators for several important symmetric properties over both domain sizes
and confidence levels. In addition, we establish a McDiarmid's inequality under
Poisson sampling, which is of independent interest.
