Abstract: We provide an end-to-end Renyi DP based-framework for differentially private top-$k$ selection. Unlike previous approaches, which require a data-independent choice on $k$, we propose to privately release a data-dependent choice of $k$ such that the gap between $k$-th and the $(k+1)$st ``quality'' is large. This is achieved by an extension of the Report-Noisy-Max algorithm with a more concentrated Gaussian noise.Not only does this eliminates one hyperparameter, the adaptive choice of $k$ also certifies the stability of the top-$k$ indices in the unordered set so we can release them using a combination of the propose-test-release (PTR) framework and the Distance-to-Stability mechanism. We show that our construction improves the privacy-utility trade-offs compared to the previous top-$k$ selection algorithms theoretically and empirically. Additionally, we apply our algorithm to ``Private Aggregation of Teacher Ensembles (PATE)'' in \emph{multi-label} classification tasks with a large number of labels and show that it leads to significant performance gains.