We study a first-order primal-dual subgradient method to optimize risk-constrained risk-penalized optimization problems, where risk is modeled via the popular conditional value at risk (CVaR) measure. The algorithm processes independent and identically distributed samples from the underlying uncertainty in an online fashion and produces an \(\eta /\sqrt{K}\)-approximately feasible and \(\eta /\sqrt{K}\)-approximately optimal point within K iterations with constant step-size, where \(\eta \) increases with tunable risk-parameters of CVaR. We find optimized step sizes using our bounds and precisely characterize the computational cost of risk aversion as revealed by the growth in \(\eta \). Our proposed algorithm makes a simple modification to a typical primal-dual stochastic subgradient algorithm. With this mild change, our analysis surprisingly obviates the need to impose a priori bounds or complex adaptive bounding schemes for dual variables to execute the algorithm as assumed in many prior works. We also draw interesting parallels in sample complexity with that for chance-constrained programs derived in the literature with a very different solution architecture.

我们研究了一阶原始对偶次梯度方法来优化风险约束风险惩罚优化问题，其中风险通过流行的条件风险值 (CVaR) 度量进行建模。该算法以在线方式处理来自潜在不确定性的独立同分布样本，并产生\(\eta /\sqrt{K}\) - 近似可行和\(\eta /\sqrt{K}\) - 近似最优K次迭代中的点，具有恒定的步长，其中\(\eta \)随 CVaR 的可调风险参数而增加。我们使用我们的边界找到优化的步长，并精确地表征风险规避的计算成本，如\(\eta \). 我们提出的算法对典型的原始对偶随机次梯度算法进行了简单的修改。通过这种温和的变化，我们的分析出人意料地避免了对双变量施加先验界限或复杂的自适应界限方案以执行许多先前工作中假设的算法的需要。我们还在样本复杂性与具有非常不同的解决方案架构的文献中得出的机会受限程序的样本复杂性方面进行了有趣的相似。