We study the block-coordinate forward–backward algorithm in which the blocks are updated in a random and possibly parallel manner, according to arbitrary probabilities. The algorithm allows different stepsizes along the block-coordinates to fully exploit the smoothness properties of the objective function. In the convex case and in an infinite dimensional setting, we establish almost sure weak convergence of the iterates and the asymptotic rate o(1/n) for the mean of the function values. We derive linear rates under strong convexity and error bound conditions. Our analysis is based on an abstract convergence principle for stochastic descent algorithms which allows to extend and simplify existing results.

我们研究了块坐标向前-向后算法，其中，根据任意概率，以随机且可能并行的方式更新块。该算法允许沿着块坐标进行不同的步长调整，以充分利用目标函数的平滑特性。在凸情况下，在无穷维设置中，我们几乎确定了迭代次数的弱收敛性和函数值均值的渐近速率o（1 / n）。我们在强凸性和误差约束条件下得出线性速率。我们的分析基于随机下降算法的抽象收敛原理，该原理可以扩展和简化现有结果。