Private Hypothesis Selection
We provide a differentially private algorithm for hypothesis selection. Given samples from an unknown probability distribution P and a set of m probability distributions <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula>, the goal is to output, in a <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>-differentially private manner, a distribution from <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> whose total variation distance to P is comparable to that of the best such distribution (which we denote by <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>). The sample complexity of our basic algorithm is <inline-formula> <tex-math notation="LaTeX">$\text {O}\left ({\frac {\log \text {m}}{\alpha ^{2}} + \frac {\log \text {m}}{\alpha \varepsilon }}\right)$ </tex-math></inline-formula>, representing a minimal cost for privacy when compared to the non-private algorithm. We also can handle infinite hypothesis classes <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> by relaxing to <inline-formula> <tex-math notation="LaTeX">$(\varepsilon,\delta)$ </tex-math></inline-formula>-differential privacy. We apply our hypothesis selection algorithm to give learning algorithms for a number of natural distribution classes, including Gaussians, product distributions, sums of independent random variables, piecewise polynomials, and mixture classes. Our hypothesis selection procedure allows us to generically convert a cover for a class to a learning algorithm, complementing known learning lower bounds which are in terms of the size of the packing number of the class. As the covering and packing numbers are often closely related, for constant <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>, our algorithms achieve the optimal sample complexity for many classes of interest. Finally, we describe an application to private distribution-free PAC learning.