Optimization of Smooth Functions with Noisy Observations: Local Minimax Rates
We consider the problem of global optimization of an unknown non-convex smooth function with noisy zeroth-order feedback. We propose a local minimax framework to study the fundamental difficulty of optimizing smooth functions with adaptive function evaluations. We show that for functions with fast growth around their global minima, carefully designed optimization algorithms can identify a near global minimizer with many fewer queries than worst-case global minimax theory predicts. For the special case of strongly convex and smooth functions, our implied convergence rates match the ones developed for zeroth-order convex optimization problems. On the other hand, we show that in the worst case no algorithm can converge faster than the minimax rate of estimating an unknown functions in 8 -norm. Finally, we show that non-adaptive algorithms, although optimal in a global minimax sense, do not attain the optimal local minimax rate.
This paper studies local minimax bounds for noisy optimization of smooth functions. In the traditional setting of global optimization, the algorithmic goal is to design an adaptive procedure to find an approximate (in this case, global) optimum of an unknown function f, given access to noisy evaluations  at feasible points. This notion of complexity might be too rough to understand the difficulty of the problem, so it is proposed to study the local minimax complexity, when the algorithm may have additional information on how close is the objective to a fixed function f_0. In practice, the algorithm doesn't need to know f_0, but this parameterization serves as an instance-dependent notion of complexity. This paper additionally considers how the regularity of the function (given by its Holder continuity) improves the complexity, as it is well-known that for Lipschitz functions adaptivity does not help.   The main contributions of the paper can be summarized as follows: 1. An adaptive algorithm whose optimization error decreases at a rate R_n(f_0)=\tildeO(n^{-\alpha/[2\alpha+d-\alpha\beta]}), where \alpha is the Holder exponent and beta is a parameter related to the volume growth of level sets of f_0. This algorithm does not need to know f_0 in advance.  2. A local minimax lower bound matching the previous upper bound up to polylogs, for the case when even f_0 is known in advance. 3. A lower bound of Omega(n^{-\alpha/[2\alpha+d]}) for nonadaptive algorithms, exposing a substantial gap between adaptive and nonadaptive methods. 4. For the case when f_0 is strongly convex they show a local minimax rate of \tilde O(1/\sqrt{n}), and a stricly stronger lower bound when f_0 is merely convex.  The paper is quite interesting, and looks mathematically solid. I didn't have the chance to carefully verify all proofs, but they seem correct.  Detailed comments:  1. The setup of the paper seems comparable to the one of approximately convex optimization (see e.g.  https://papers.nips.cc/paper/5841-information-theoretic-lower-bounds-for-convex-optimization-with-erroneous-oracles or https://papers.nips.cc/paper/6576-algorithms-and-matching-lower-bounds-for-approximately-convex-optimization.pdf). This setting is analog to the one proposed in this paper, but without the extra noise in function evaluations. In particular, they show a phase transition of epsilon_n(f_0) (for some f_0) where no polynomial number of adaptive queries can reach an approximate solution.  Can the results of the paper be translated to this zero noise setting?  -------------- After Rebuttal --------------  I want to thank the authors for clarifying the differences w.r.t. approximate convex optimization. It is very interesting how these subtleties change complexity bounds. As some other reviewer suggested, it is important that the authors also include the dimension dependence in the minimax rates.