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This study addresses the challenging problem of finding approximate stationary points in nonconvex-nonconcave min-max optimization when the maximization domain is small. The researchers prove that replacing the objective function with its Taylor approximation succeeds when the diameter of the maximization set is sufficiently small relative to the target accuracy, specifically O(ε^(2/(k+1))) for kth order approximations. They demonstrate these diameter bounds are nearly optimal and provide efficient algorithms with convergence guarantees for low-order approximations (k≤2).
Why it matters
Min-max optimization problems arise frequently in machine learning applications including generative adversarial networks and robust optimization. This work provides both theoretical understanding and practical algorithms for solving a broader class of these problems beyond the standard convex-concave case, potentially enabling more robust solutions in adversarial training scenarios.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: We study the problem of finding approximate first-order stationary points in optimization problems of the form $min_{x in X} max_{y in Y} f(x,y)$, where the sets $X,Y$ are convex and $Y$ is compact. The objective function $f$ is smooth, but assumed neither convex in $x$ nor concave in $y$. Our approach relies upon replacing the function $f(x,cdot)$ with its $k$th order Taylor approximation (in $y$) and finding a near-stationary point in the resulting surrogate problem. To guarantee its success, we establish the following result: let the Euclidean diameter of $Y$ be small in terms of the target accuracy $varepsilon$, namely $O(varepsilon^{frac{2}{k+1}})$ for $k in mathbb{N}$ and $O(varepsilon)$ for $k = 0$, with the constant factors controlled by certain regularity parameters of $f$; then any $varepsilon$-stationary point in the surrogate problem remains $O(varepsilon)$-stationary for the initial problem. Moreover, we show that these upper bounds are nearly optimal: the aforementioned reduction provably fails when the diameter of $Y$ is larger. For $0 le k le 2$ the surrogate function can be efficiently maximized in $y$; our general approximation result then leads to efficient algorithms for finding a near-stationary point in nonconvex-nonconcave min-max problems, for which we also provide convergence guarantees.
Source: Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain