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We consider prediction with expert advice when data are generated fromdistributions varying arbitrarily within an unknown constraint set. Thissemi-adversarial setting includes (at the extremes) the classical i.i.d.setting, when the unknown constraint set is restricted to be a singleton, andthe unconstrained adversarial setting, when the constraint set is the set ofall distributions. The Hedge algorithm -- long known to be minimax (rate)optimal in the adversarial regime -- was recently shown to be simultaneouslyminimax optimal for i.i.d. data. In this work, we propose to relax the i.i.d.assumption by seeking adaptivity at all levels of a natural ordering onconstraint sets. We provide matching upper and lower bounds on the minimaxregret at all levels, show that Hedge with deterministic learning rates issuboptimal outside of the extremes, and prove that one can adaptively obtainminimax regret at all levels. We achieve this optimal adaptivity using thefollow-the-regularized-leader (FTRL) framework, with a novel adaptiveregularization scheme that implicitly scales as the square root of the entropyof the current predictive distribution, rather than the entropy of the initialpredictive distribution. Finally, we provide novel technical tools to study thestatistical performance of FTRL along the semi-adversarial spectrum.

Relaxing the I.I.D. Assumption: Adaptively Minimax Optimal Regret via Root-Entropic Regularization