Abstract: Differential privacy (DP) has emerged as a rigorous notion to quantify data privacy. Subsequently, Renyi differential privacy (RDP) becomes an alternative to the ordinary DP notion in both theoretical and empirical studies, for its convenient compositional rules and flexibility. However, most mechanisms with DP (RDP) guarantees are essentially based on randomizing a fixed, finite-dimensional vector output. In this work, following Hall et al. (2013) we further extend RDP to functional outputs, where the output space can be infinite-dimensional, and develop all necessary tools, *e.g.*, (subsampled) Gaussian mechanism, composition, and post-processing rules, to facilitate its practical adoption. As an illustration, we apply functional RDP (f-RDP) to functions in the reproducing kernel Hilbert space (RKHS) to develop a differentially private generative model (DPGM), where training can be interpreted as iteratively releasing loss functions (in an RKHS) with DP (RDP) guarantees. Empirically, the new training paradigm achieves a significant improvement in privacy-utility trade-off compared to existing alternatives, especially when $\epsilon=0.2$. Our code is available at https://github.com/dihjiang/DP-kernel.