Online Estimation via Offline Estimation: An Information-Theoretic Framework
Dylan J. Foster, Yanjun Han, Jian Qian, Alexander Rakhlin
TL;DR
The paper introduces Oracle-Efficient Online Estimation (OEOE), an information-theoretic framework in which online learning is performed through black-box offline estimation oracles. It shows that for finite function classes, one can transform offline estimation into near-optimal online estimation with a minimal statistical cost, but faces computational hardness in general. A key exception appears in conditional density estimation, where online estimation via offline oracles matches the unrestricted computational benchmark through a series of reductions, including delayed online learning. The results extend to interactive decision making (DMSO) and show that offline-to-online conversion is powerful under structural conditions like coverability, enabling oracle-efficient strategies in RL and contextual bandits. Overall, the work delineates when offline information suffices for online guarantees and clarifies the computational limits of such oracle-based approaches, guiding algorithm design in interactive settings.
Abstract
$ $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.