  The study of statistical estimation without distributional assumptions on
data values, but with knowledge of data collection methods was recently
introduced by Chen, Valiant and Valiant (NeurIPS 2020). In this framework, the
goal is to design estimators that minimize the worst-case expected error. Here
the expectation is over a known, randomized data collection process from some
population, and the data values corresponding to each element of the population
are assumed to be worst-case.
  Chen, Valiant and Valiant show that, when data values are
$\ell_{\infty}$-normalized, there is a polynomial time algorithm to compute an
estimator for the mean with worst-case expected error that is within a factor
$\frac{\pi}{2}$ of the optimum within the natural class of semilinear
estimators. However, their algorithm is based on optimizing a somewhat complex
concave objective function over a constrained set of positive semidefinite
matrices, and thus does not come with explicit runtime guarantees beyond being
polynomial time in the input.
  In this paper we design provably efficient algorithms for approximating the
optimal semilinear estimator based on online convex optimization. In the
setting where data values are $\ell_{\infty}$-normalized, our algorithm
achieves a $\frac{\pi}{2}$-approximation by iteratively solving a sequence of
standard SDPs. When data values are $\ell_2$-normalized, our algorithm
iteratively computes the top eigenvector of a sequence of matrices, and does
not lose any multiplicative approximation factor. We complement these positive
results by stating a simple combinatorial condition which, if satisfied by a
data collection process, implies that any (not necessarily semilinear)
estimator for the mean has constant worst-case expected error.
