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Understanding when learning is possible is a fundamental task in the theory of machine learning. However, many characterizations known from the literature deal with abstract learning as a mathematical object and ignore the crucial question: when can learning be implemented as a computer program? We address this question for universal online learning, a generalist theoretical model of online binary classification, recently characterized by Bousquet et al. (STOC 21). In this model, there is no hypothesis fixed in advance; instead, Adversary--playing the role of Nature--can change their mind as long as local consistency with the given class of hypotheses is maintained. We require Learner to achieve a finite number of mistakes while using a strategy that can be implemented as a computer program. We show that universal online learning does not imply computable universal online learning, even if the class of hypotheses is relatively easy from a computabilitytheoretic perspective. We then study the agnostic variant of computable universal online learning and provide an exact characterization of classes that are learnable in this sense. We also consider a variant of proper universal online learning and show exactly when it is possible. Together, our results give a more realistic perspective on the existing theory of online binary classification and the related problem of inductive inference.

We revisit the problem of private online learning, in which a learner receives a sequence of T data points and has to respond at each time-step a hypothesis. It is required that the entire stream of output hypotheses should satisfy differential privacy. Prior work of Golowich and Livni [2021] established that every concept class H with finite Littlestone dimension d is privately online learnable in the realizable setting. In particular, they proposed an algorithm that achieves an Od(logT) mistake bound against an oblivious adversary. However, their approach yields a suboptimal Od( T) bound against an adaptive adversary. In this work, we present a new algorithm with a mistake bound of Od(logT)against an adaptive adversary, closing this gap. We further investigate the problem in the agnostic setting, which is more general than the realizable setting as it does not impose any assumptions on the data. We give an algorithm that obtains a sublinear regret of Od( T) for generic Littlestone classes, demonstrating that they are also privately online learnable in the agnostic setting.

We resolve a 30-year-old open problem concerning the power of unlabeled data in online learning by tightly quantifying the gap between transductive and standard online learning. In the standard setting, the optimal mistake bound is characterized by the Littlestone dimension dof the concept class H(Littlestone, 1987). We prove that in the transductive setting, the mistake bound is at least Ω d . This constitutes an exponential improvement over previous lower bounds of Ω(loglog(d)), Ω p log(d), and Ω(log(d)), due respectively to Ben-David, Kushilevitz, and Mansour (1995, 1997), and Hanneke, Moran, and Shafer (2023). We also show that this lower bound is tight: for every d, there exists a class of Littlestone dimension d with transductive mistake bound O d . Our upper bound also improves upon the best known upper bound of (2/3) d from Ben-David et al. (1997). These results establish a quadratic gap between transductive and standard online learning, thereby highlighting the benefit of advance access to the unlabeled instance sequence. This contrasts with the PAC setting, where transductive and standard learning exhibit similar sample complexities.