Learning Functional Distributions with Private Labels

We study the problem of learning functional distributions in the presence of noise. A functional is a map from the space of features to distributions over a set of labels, and is often assumed to belong to a known class of hypotheses $\mathcal{F}$. Features are generated by a general random process and labels are sampled independently from feature-dependent distributions. In privacy sensitive applications, labels are passed through a noisy kernel. We consider online learning, where at each time step, a predictor attempts to predict the actual (label) distribution given only the features and noisy labels in prior steps. The performance of the predictor is measured by the expected KL-risk that compares the predicted distributions to the underlying truth. We show that the minimax expected KL-risk is of order $\tilde{\Theta}(\sqrt{T\log|\mathcal{F}|})$ for finite hypothesis class $\mathcal{F}$ and any non-trivial noise level. We then extend this result to general infinite classes via the concept of stochastic sequential covering and provide matching lower and upper bounds for a wide range of natural classes.