This relaxed program is differentiable and can be trained end-to-end, and the resulting training loss is an approximately admissible heuristic that can guide the combinatorial search.

We thank the reviewer for the thorough review. We agree that our discussion of Seung et al. was not However, our contributions go beyond Seung et al.'s work in We kindly ask the reviewer to reconsider the following contributions. Such applications were not available in Seung et al. We indeed applied the results in Seung et al. as a tool to provide necessary conditions of convergence of the dynamics, Reviewer 2: We thank the reviewer for the enthusiastic support! We will provide details in the appendix. Minimax objectives: We thank the author for the inspiring question.

Then, we have that 1. y Fact 2. Let z:= [ x; y ] and z This can be easily proven using the AM-GM inequality. Fact 3. Let z:= [ x; y ] R It is a crucial building block for the algorithms in this work. The following classical theorem holds for AGD. We will start by giving a precise statement of Algorithm 1.Algorithm 1 Alternating Best Response (ABR)Require: g (,), Initial point z The basic idea is the following. The following two lemmas about the inexact APP A algorithm follow from the proof of Theorem 4.1 [ Here we provide their proofs for completeness.

Via reduction, our new bound also implies improved bounds for strongly convex-concave and convex-concave minimax optimization problems.

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A classic problem in machine learning and data analysis is to partition the vertices of a network in such a way that vertices in the same set are densely connected and vertices in different sets are loosely connected. In practice, the most popular approaches rely on local search algorithms; not only for the ease of implementation and the efficiency, but also because of the accuracy of these methods on many real world graphs. For example, the Louvain algorithm - a local search based algorithm - has quickly become the method of choice for clustering in social networks.

We study a secretary problem which captures the task of ranking in online settings.

We thank all reviewers for giving us the insightful comments. Then we collect all positive samples by a Breadth-First Search algorithm. We will add these results to our paper. We also give the qualitative analysis in Figure 2 (b). About the baseline, our baseline is the KNN method, i.e. directly using the nearst We have compared our algorithm with the KNN algorithm in Sec 4.2.

The lower bound in higher dimensions is attained by an orthogonal subspace argument.

Applying BO with GPs on combinatorial spaces is, therefore, not straightforward.