Smoothed Online Classification can be Harder than Batch Classification
This work addresses a theoretical gap in online learning for researchers, revealing limitations in smoothed adversaries for unbounded labels, which is incremental as it builds on prior smoothed learning studies.
The paper tackles the problem of online classification under smoothed adversaries, showing that when the label space is unbounded, smoothed online classification can be harder than i.i.d. batch classification, contrary to previous results for binary cases. It constructs a hypothesis class learnable in batch but not in smoothed online settings and identifies a condition for equivalence.
We study online classification under smoothed adversaries. In this setting, at each time point, the adversary draws an example from a distribution that has a bounded density with respect to a fixed base measure, which is known apriori to the learner. For binary classification and scalar-valued regression, previous works \citep{haghtalab2020smoothed, block2022smoothed} have shown that smoothed online learning is as easy as learning in the iid batch setting under PAC model. However, we show that smoothed online classification can be harder than the iid batch classification when the label space is unbounded. In particular, we construct a hypothesis class that is learnable in the iid batch setting under the PAC model but is not learnable under the smoothed online model. Finally, we identify a condition that ensures that the PAC learnability of a hypothesis class is sufficient for its smoothed online learnability.