The primal-dual hybrid gradient (PDHG) algorithm is a first-order method that splits convex optimization problems with saddle-point structure into smaller subproblems. Those subproblems, unlike those obtained from most other splitting methods, can generally be solved efficiently because they involve simple operations such as matrix-vector multiplications or proximal mappings that are easy to evaluate. In order to work fast, however, the PDHG algorithm requires stepsize parameters fine-tuned for the problem at hand. Unfortunately, the stepsize parameters must often be estimated from quantities that are prohibitively expensive to compute for large-scale optimization problems, such as those in machine learning. In this paper, we introduce accelerated nonlinear variants of the PDHG algorithm that can achieve, for a broad class of optimization problems relevant to machine learning, an optimal rate of convergence with stepsize parameters that are simple to compute. We prove rigorous convergence results, including for problems posed on infinite-dimensional reflexive Banach spaces. We also provide practical implementations of accelerated nonlinear PDHG algorithms for solving several regression tasks in machine learning, including support vector machines without offset, kernel ridge regression, elastic net regularized linear regression, and the least absolute shrinkage selection operator.

翻译：原始- 双混合梯度( PDHG) 算法是一种先行方法, 将螺旋优化问题与马鞍点结构分为小小问题。 这些次问题与大多数其他分解方法不同, 通常可以有效解决, 因为它们涉及简单的操作, 如易于评估的矩阵- 矢量乘法或准度映射。 然而, 为了快速工作, PDHG 算法需要针对手头问题精确调整参数。 不幸的是, 阶梯化参数往往必须从那些对大规模优化问题( 如机器学习中的问题)进行计算过于昂贵的数量来估计。 在本文中, 我们引入了PDHG 算法的加速非线性变方, 这些变方对于与机器学习有关的广泛最优化问题来说, 能够实现与简单测算参数的最佳趋同率。 我们证明了严格的趋同结果, 包括对无限的反射波波段空间上的问题。 我们还提供了快速的非线性 PDHG 算法, 用于解决机器中若干次回归任务( 绝对级级级级), 的递增缩机, 包括不支持 最慢级级级级的 。