We investigate the convergence of a recently popular class of first-order primal–dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of errors and show that, provided a sufficient decay of these errors, the same convergence rates as for the error-free algorithm can be established. More precisely, we prove the (optimal) \(O\left( 1/N\right)\) convergence to a saddle point in finite dimensions for the class of non-smooth problems considered in this paper, and prove a \(O\left( 1/N^2\right)\) or even linear \(O\left( \theta ^N\right)\) convergence rate if either the primal or dual objective respectively both are strongly convex. Moreover we show that also under a slower decay of errors we can establish rates, however slower and directly depending on the decay of the errors. We demonstrate the performance and practical use of the algorithms on the example of nested algorithms and show how they can be used to split the global objective more efficiently.

中文翻译：

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不精确的一阶原始对偶算法

我们研究了在近端贴图和梯度存在错误的情况下，最近流行的一阶原对偶算法对鞍点问题的收敛性。我们研究了几种类型的错误，并证明，只要充分衰减这些错误，就可以建立与无错误算法相同的收敛速度。更准确地说，对于本文考虑的一类非光滑问题，我们证明了（最优）\（O \ left（1 / N \ right）\）收敛到有限尺寸的鞍点，并证明了\（O \ left（1 / N ^ 2 \ right）\）甚至线性\（O \ left（\ theta ^ N \ right）\）如果原始目标或对偶目标两者都强烈凸，则收敛速度高。此外，我们表明，在误差衰减较慢的情况下，我们也可以建立速率，但速度较慢且直接取决于误差衰减。我们以嵌套算法为例演示算法的性能和实际使用，并展示如何使用它们更有效地拆分全局目标。