Sequential Probability Assignment against Smoothed Adversaries with Unknown Base Measure

Smoothed online learning has recently been studied as a way to bypass hardness results for the fully adversarial setting, which can be overly pessimistic. In this framework, the adversary is constrained to generate contexts from distributions having a bounded density with respect to some fixed base measure $\mu$. Prior work makes the strong assumption that $\mu$ is known to the learner, with notable exceptions including Block et al. (2024) and Blanchard (2025). In this paper, we study sequential probability assignment (a.k.a., online learning with log loss) with smooth, well-specified data in the more general setting where $\mu$ is \emph{unknown}, without the Lipschitz condition imposed in the work of Block et al. (2024) and Blanchard (2025). We prove regret upper bounds against both adaptive and oblivious smoothed adversaries: the adaptive bound is controlled by empirical $\ell_\infty$ Hellinger entropy, while the oblivious bound improves this dependence to empirical $\ell_2$ Hellinger entropy, the notion that has been shown to characterize the complexity of learning with i.i.d. data (Bilodeau et al., 2023). Both bounds are obtained through algorithms based on truncated empirical Hellinger covers. We complement these results with a general lower bound, showing that our upper bounds are essentially tight for some natural classes, which implies a separation in the difficulty of smoothed online learning between regimes where $\mu$ is known and where it is unknown.