We thank the reviewers for their overall positive and constructive comments. This paper completes the picture in the minimax Hölder setting. Moreover, the Hölder assumption (stated under various names) is standard in this line of work, e.g., in That said, we acknowledge the technicality of the paper. R1: "The horizon T is assumed to be a prior knowledge. This should be stated and commented [...]. Indeed, Subsection 3.3 and Appendix B discuss this and describe how we can get rid this requirement. In the final version we will recall that by "anytime" we mean without the knowledge of T. R2 "Can this algorithmic technique deal with cases in which the function is spatially inhomogenous, This is actually the case in previous papers (Zooming, HOO, Locatelli and Carpentier).