In this paper, we study the convergence properties of a randomized block-coordinate descent algorithm for the minimization of a composite convex objective function, where the block-coordinates are updated asynchronously and randomly according to an arbitrary probability distribution. We prove that the iterates generated by the algorithm form a stochastic quasi-Fej\'er sequence and thus converge almost surely to a minimizer of the objective function. Moreover, we prove a general sublinear rate of convergence in expectation for the function values and a linear rate of convergence in expectation under an error bound condition of Tseng type.

翻译：本文研究了随机块坐标下降算法在最小化复合凸目标函数时的收敛性质，其中块坐标根据任意概率分布异步随机更新。我们证明了该算法生成的迭代序列构成随机拟Fejér序列，从而几乎必然收敛至目标函数的最小值点。进一步地，我们证明了函数值期望的次线性收敛率，并在Tseng型误差界条件下证明期望线性收敛率。