The Littlestone dimension of ${\mathcal{H}}$ is $d$. More generally, any class of hypotheses that can be described by polynomial *equalities* of constant degree has finite Littlestone dimension.[^5] This can be generalized even further to classes that are definable in [*stable theories*]{}. This (different, still) notion of stability is deep and well-explored in model theory. We refer the reader to [@Chase19modelmachine], Section 5.1 for more examples of stable theories and the Littlestone classes they correspond to.

#### Organization.

The rest of this manuscript is organized as follows. In Section \[sec:results\] we formally state our main results and discuss some implications. Section \[sec:overview\] overviews some of the main ideas in the proofs. Sections \[sec:preliminaries\] - \[sec:wrapping\] contain complete proofs, and the last section (Section \[sec:conc\]) concludes the paper with some suggestions for future work.

Main Results {#sec:results}
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We next present our main results. We begin with the statements concerning the relationship between online learning and differentially-private learning. In Section \[sec:stability\] we present and discuss the notion of global stability, and finally in Section \[sec:boosting\] we discuss an implication for private boosting. Throughout this section some standard technical terms are used. For definitions of these terms we refer the reader to Section 