This work explores the connection between differential privacy (DP) and online learning in the context of PAC list learning. In this setting, a $k$-list learner outputs a list of $k$ potential predictions for an instance $x$ and incurs a loss if the true label of $x$ is not included in the list. A basic result in the multiclass PAC framework with a finite number of labels states that private learnability is equivalent to online learnability [Alon, Livni, Malliaris, and Moran (2019); Bun, Livni, and Moran (2020); Jung, Kim, and Tewari (2020)]. Perhaps surprisingly, we show that this equivalence does not hold in the context of list learning. Specifically, we prove that, unlike in the multiclass setting, a finite $k$-Littlestone dimension-a variant of the classical Littlestone dimension that characterizes online $k$-list learnability-is not a sufficient condition for DP $k$-list learnability. However, similar to the multiclass case, we prove that it remains a necessary condition. To demonstrate where the equivalence breaks down, we provide an example showing that the class of monotone functions with $k+1$ labels over $\mathbb{N}$ is online $k$-list learnable, but not DP $k$-list learnable. This leads us to introduce a new combinatorial dimension, the \emph{$k$-monotone dimension}, which serves as a generalization of the threshold dimension. Unlike the multiclass setting, where the Littlestone and threshold dimensions are finite together, for $k>1$, the $k$-Littlestone and $k$-monotone dimensions do not exhibit this relationship. We prove that a finite $k$-monotone dimension is another necessary condition for DP $k$-list learnability, alongside finite $k$-Littlestone dimension. Whether the finiteness of both dimensions implies private $k$-list learnability remains an open question.

We provide a unified framework for characterizing pure and approximate differentially private (DP) learnability. The framework uses the language of graph theory: for a concept class $\mathcal{H}$, we define the contradiction graph $G$ of $\mathcal{H}$. Its vertices are realizable datasets and two datasets $S,S’$ are connected by an edge if they contradict each other (i.e., there is a point $x$ that is labeled differently in $S$ and $S’$). Our main finding is that the combinatorial structure of $G$ is deeply related to learning $\mathcal{H}$ under DP. Learning $\mathcal{H}$ under pure DP is captured by the fractional clique number of $G$. Learning $\mathcal{H}$ under approximate DP is captured by the clique number of $G$. Consequently, we identify graph-theoretic dimensions that characterize DP learnability: the \emph{clique dimension} and \emph{fractional clique dimension}. Along the way, we reveal properties of the contradiction graph which may be of independent interest. We also suggest several open questions and directions for future research.

This work continues to investigate the link between differentially private (DP) and online learning. Alon, Livni, Malliaris, and Moran [4] showed that for binary concept classes, DP learnability of a given class implies that it has a finite Littlestone dimension (equivalently, that it is online learnable). Their proof relies on a model-theoretic result by Hodges [36], which demonstrates that any binary concept class with a large Littlestone dimension contains a large subclass of thresholds. In a follow-up work, Jung, Kim, and Tewari [38] extended this proof to multiclass PAC learning with a bounded number of labels. Unfortunately, Hodges's result does not apply in other natural settings such as multiclass PAC learning with an unbounded label space, and PAC learning of partial concept classes. This naturally raises the question of whether DP learnability continues to imply online learnability in more general scenarios: indeed, Alon, Hanneke, Holzman, and Moran [5] explicitly leave it as an open question in the context of partial concept classes, and the same question is open in the general multiclass setting. In this work, we give a positive answer to these questions showing that for general classification tasks, DP learnability implies online learnability. Our proof reasons directly about Littlestone trees, without relying on thresholds. We achieve this by establishing several Ramsey-type theorems for trees, which might be of independent interest.

We show that many definitions of stability found in the learning theory literature are equivalent to one another. We distinguish between two families of definitions of stability: distribution-dependent and distribution-independent Bayesian stability. Within each family, we establish equivalences between various definitions, encompassing approximate differential privacy, pure differential privacy, replicability, global stability, perfect generalization, TV stability, mutual information stability, KL-divergence stability, and Rényi-divergence stability. Along the way, we prove boosting results that enable the amplification of the stability of a learning rule. This work is a step towards a more systematic taxonomy of stability notions in learning theory, which can promote clarity and an improved understanding of an array of stability concepts that have emerged in recent years.