Analyze prefix-averaged progressive mixture estimators

Determine whether the progressive mixture estimator modified by averaging only after a sample-dependent burn-in phase, or analogous sample-dependent-burn-in versions of other online-to-batch estimators, can simultaneously achieve exponential universal rates and minimax optimality.

Background

The paper proves that online-to-batch estimators with a fixed, sample-size-independent burn-in cannot achieve exponential universal rates. It observes that a sample-dependent burn-in, such as beginning averaging halfway through the sample, can restore exponential rates for the progressive mixture estimator, but the resulting estimator's minimax properties are not known.

References

However, to the best of our knowledge, it is currently unknown whether this prefix-averaged progressive mixture estimator retains its minimax optimality in expectation. Resolving whether this estimator, or similarly modified versions of the others, can simultaneously achieve both exponential rates and minimax optimality remains open.

— Reconciling Universal and Uniform Learning with $Q$-Aggregation  (2609.05041 - Høgsgaard et al., 4 Sep 2026) in Section 3, subsection “Online-to-Batch Conversion by Averaging”