We study first-order methods for constrained min-max optimization. Existing methods either require two gradient calls or two projections in each iteration, which may be costly in some applications. In this paper, we first show that a variant of the Optimistic Gradient (OG) method, a single-call single-projection algorithm, has $O(\frac{1}{\sqrt{T}})$ best-iterate convergence rate for inclusion problems with operators that satisfy the weak Minty variation inequality (MVI). Our second result is the first single-call single-projection algorithm -- the Accelerated Reflected Gradient (ARG) method that achieves the optimal $O(\frac{1}{T})$ last-iterate convergence rate for inclusion problems that satisfy negative comonotonicity. Both the weak MVI and negative comonotonicity are well-studied assumptions and capture a rich set of non-convex non-concave min-max optimization problems. Finally, we show that the Reflected Gradient (RG) method, another single-call single-projection algorithm, has $O(\frac{1}{\sqrt{T}})$ last-iterate convergence rate for constrained convex-concave min-max optimization, answering an open problem of [Heish et al, 2019]. Our convergence rates hold for standard measures such as the tangent residual and the natural residual.

翻译：我们研究用于约束极小极大优化的一阶方法。现有方法每次迭代需要两次梯度调用或两次投影，这在某些应用中代价高昂。本文首先证明乐观梯度法（OG）的一个变体——即单次调用单次投影算法——对于满足弱Minty变分不等式（MVI）算子的包含问题具有$O(\frac{1}{\sqrt{T}})$的最优迭代收敛速率。我们的第二个结果是首个单次调用单次投影算法——加速反射梯度法（ARG）——其对于满足负共单调性的包含问题达到了最优的$O(\frac{1}{T})$末次迭代收敛速率。弱MVI与负共单调性均为经过充分研究的假设，能够刻画大量非凸非凹极小极大优化问题。最后，我们证明另一单次调用单次投影算法——反射梯度法（RG）——对于约束凸凹极小极大优化具有$O(\frac{1}{\sqrt{T}})$的末次迭代收敛速率，从而解决了[Heish et al, 2019]中提出的开放问题。我们的收敛速率对诸如切向残差和自然残差等标准度量均成立。