Online Estimation via Offline Estimation: An Information-Theoretic
  Framework
Dylan J. Foster Yanjun Han Jian Qian Alexander Rakhlin
$ $The classical theory of statistical estimation aims to estimate a
parameter of interest under data generated from a fixed design ("offline
estimation"), while the contemporary theory of online learning provides
algorithms for estimation under adaptively chosen covariates ("online
estimation"). Motivated by connections between estimation and interactive
decision making, we ask: is it possible to convert offline estimation
algorithms into online estimation algorithms in a black-box fashion? We
investigate this question from an information-theoretic perspective by
introducing a new framework, Oracle-Efficient Online Estimation (OEOE), where
the learner can only interact with the data stream indirectly through a
sequence of offline estimators produced by a black-box algorithm operating on
the stream. Our main results settle the statistical and computational
complexity of online estimation in this framework.
  $\bullet$ Statistical complexity. We show that information-theoretically,
there exist algorithms that achieve near-optimal online estimation error via
black-box offline estimation oracles, and give a nearly-tight characterization
for minimax rates in the OEOE framework.
  $\bullet$ Computational complexity. We show that the guarantees above cannot
be achieved in a computationally efficient fashion in general, but give a
refined characterization for the special case of conditional density
estimation: computationally efficient online estimation via black-box offline
estimation is possible whenever it is possible via unrestricted algorithms.
  Finally, we apply our results to give offline oracle-efficient algorithms for
interactive decision making.