On the Equivalence between Online and Private Learnability beyond Binary Classification
arXiv:2006.01980
Abstract
Alon et al. [2019] and Bun et al. [2020] recently showed that online learnability and private PAC learnability are equivalent in binary classification. We investigate whether this equivalence extends to multi-class classification and regression. First, we show that private learnability implies online learnability in both settings. Our extension involves studying a novel variant of the Littlestone dimension that depends on a tolerance parameter and on an appropriate generalization of the concept of threshold functions beyond binary classification. Second, we show that while online learnability continues to imply private learnability in multi-class classification, current proof techniques encounter significant hurdles in the regression setting. While the equivalence for regression remains open, we provide non-trivial sufficient conditions for an online learnable class to also be privately learnable.
An earlier version of this manuscript claimed an upper bound over the sample complexity that is exponential in the Littlestone dimension. The argument contained a technical mistake, and the current version presents a correction that deteriorates the dependence on the Littlestone dimension from exponential to doubly exponential. arXiv admin note: text overlap with arXiv:2003.00563 by other authors