Reviews: Local Minimax Complexity of Stochastic Convex Optimization

While I would certainly agree with the claim that developing local minimax bounds is of interest, I feel that the interest of considering risks for 2-point subproblems is unclear. Note that many (probably, a majority of) known bounds in convex stochastic optimization are obtained on 2-point subproblems (see, e.g. In other words, the minimax risks (up to an absolute factor) for general classes of convex problems are attained already on the hardest 2-point subproblems (1-parametric families), so that, in the paper notation, R_T(\cal F)\sim \sup_{f\in \cal F} R_T(f, \cal F) On the other hand, it is unclear for me if the notion of R_T(f, \cal F) is of interest for its own sake (not as just a technical tool to construct a more general lower bound). For instance, it is not clear why and when the "2-point risk" R_T(f, \cal F) is an adequate measure of local complexity. Note that this is generally not the case in nonparametric estimation (except for few situations such as the case of linear functional estimation in [4-6]).