论文标题
通过自适应采样方法的约束和复合优化
Constrained and Composite Optimization via Adaptive Sampling Methods
论文作者
论文摘要
本文的动机源于开发一种自适应抽样方法来解决目标函数是随机性且约束是确定性的约束优化问题的愿望。本文提出的方法 是一种近端梯度方法，也可以应用于复合优化问题最小f（x） + h（x），其中f是随机的，h是凸（但不一定是可区分的）。自适应抽样方法采用一种机制来逐步提高梯度近似的质量，从而将计算成本保持在最低限度。在受约束或复合优化设置中不再可靠的机制不再可靠，因为它是基于无法正确预测近端梯度步骤质量的重点决策。本文提出的方法衡量了一个完整步骤的结果，以确定梯度近似是否足够准确。否则将生成更准确的梯度，并计算新的步骤。对于强凸和一般凸F，融合结果均被建立。提出了数值实验，以说明该方法的实际行为。
The motivation for this paper stems from the desire to develop an adaptive sampling method for solving constrained optimization problems in which the objective function is stochastic and the constraints are deterministic. The method proposed in this paper is a proximal gradient method that can also be applied to the composite optimization problem min f(x) + h(x), where f is stochastic and h is convex (but not necessarily differentiable). Adaptive sampling methods employ a mechanism for gradually improving the quality of the gradient approximation so as to keep computational cost to a minimum. The mechanism commonly employed in unconstrained optimization is no longer reliable in the constrained or composite optimization settings because it is based on pointwise decisions that cannot correctly predict the quality of the proximal gradient step. The method proposed in this paper measures the result of a complete step to determine if the gradient approximation is accurate enough; otherwise a more accurate gradient is generated and a new step is computed. Convergence results are established both for strongly convex and general convex f. Numerical experiments are presented to illustrate the practical behavior of the method.
