Min-max optimization of an objective function $f: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is an important model for robustness in an adversarial setting, with applications to many areas including optimization, economics, and deep learning. In many of these applications $f$ may be nonconvex-nonconcave, and finding a global min-max point may be computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max optimization model. However, many of the alternative models have solution points which are only guaranteed to exist under strong assumptions on $f$, such as convexity, monotonicity, or special properties of the starting point. In this paper, we propose an optimization model, the $\varepsilon$-greedy adversarial equilibrium, which can serve as a computationally tractable alternative to the min-max optimization model. Roughly, we say a point $(x^\star, y^\star)$ is an $\varepsilon$-greedy adversarial equilibrium if $y^\star$ is an $\varepsilon$-approximate local maximum for $f(x^\star,\cdot)$, and $x^\star$ is an $\varepsilon$-approximate local minimum for a "greedy approximation" to the function $\max_z f(x, z)$ which can be efficiently estimated using second-order optimization algorithms. The existence follows from an algorithm that converges from any starting point to such a point in a number of evaluations to $f$, $\nabla_{y} f(x,y)$, and $\nabla^2_y f(x,y)$, that is polynomial in $1/\varepsilon$, the dimension $d$, and the bounds on $f$ and its Lipschitz constant. In addition to existence, our model retains many desirable properties of the min-max model. For instance, it empowers the min-player to make updates that take into account the max-player's response, and in the case of strong convexity/concavity it corresponds to a global min-max solution with duality gap $O(\epsilon^2)$.

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Greedy Adversarial Equilibrium：非凸-非凹最小-最大优化的有效替代方案

目标函数 $f 的最小-最大优化：\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ 是对抗性环境中稳健性的重要模型，适用于许多领域包括优化、经济学和深度学习。在许多这些应用中，$f$ 可能是非凸非凹的，并且找到全局最小-最大点在计算上可能是棘手的。有很长的工作线寻找计算上易于处理的算法来替代最小-最大优化模型。然而，许多替代模型都有解点，这些解点只能保证在对 $f$ 的强假设下存在，例如凸性、单调性或起点的特殊属性。在本文中，我们提出了一个优化模型，$\varepsilon$-greedy adversarial equilibrium，它可以作为最小-最大优化模型在计算上易于处理的替代方案。粗略地说，如果 $y^\star$ 是 $f( 的近似局部最大值)，我们说点 $(x^\star, y^\star)$ 是 $\varepsilon$-贪婪对抗均衡x^\star,\cdot)$ 和 $x^\star$ 是 $\varepsilon$-近似局部最小值，用于函数 $\max_z f(x, z)$ 的“贪婪逼近”，它可以有效地估计使用二阶优化算法。存在遵循一个算法，该算法在对 $f$、$\nabla_{y} f(x,y)$ 和 $\nabla^2_y f(x, y)$，即 $1/\varepsilon$ 中的多项式，维度 $d$，以及 $f$ 的边界及其 Lipschitz 常数。除了存在之外，我们的模型还保留了 min-max 模型的许多理想属性。