Ramsey Theorems for Trees and a General 'Private Learning Implies Online Learning' Theorem

This work continues to investigate the link between differentially private (DP) and online learning. Alon, Livni, Malliaris, and Moran (2019) 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 (1997), 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 (2020) 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 (2021) 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.

The Bayesian Stability Zoo

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 R\'enyi-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.

A Unified Characterization of Private Learnability via Graph Theory

We provide a unified framework for characterizing pure and approximate differentially private (DP) learnabiliity. The framework uses the language of graph theory: for a concept class $\mathcal{H}$, we define the contradiction graph $G$ of $\mathcal{H}$. It 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 clique dimension and 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.