A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization
A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization
复制标题
DOI：
10.1137/18m1230542
复制
发表时间：
2018-12
期刊：
影响因子：
--
通讯作者：
Saeed Ghadimi;A. Ruszczynski;Mengdi Wang
中科院分区：
文献类型：
--
作者：
Saeed Ghadimi;A. Ruszczynski;Mengdi Wang
研究了约束嵌套随机优化问题，其中目标函数是两个光滑函数的组合，它们的精确值和导数都是不可用的。我们提出了一种单时间尺度随机逼近算法，我们称之为嵌套平均随机逼近（NASA），以找到问题的近似平稳点。该算法有两个辅助平均序列（滤波器）来估计复合目标函数的梯度和内函数的值。通过使用一个特殊的Lyapunov函数，我们证明NASA在寻找$\epsilon$-近似平稳点时实现了${\cal O}(1/\epsilon^{2})$的样本复杂度，从而优于所有现有的嵌套随机近似方法。对于无约束和有约束问题，我们的方法及其分析是相同的，不需要批量样本进行约束非凸随机优化。我们还提出了求解有约束单水平随机优化问题的简化的NASA方法，并证明了无约束和有约束问题具有相同的复杂度结果。
We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single time-scale stochastic approximation algorithm, which we call the Nested Averaged Stochastic Approximation (NASA), to find an approximate stationary point of the problem. The algorithm has two auxiliary averaged sequences (filters) which estimate the gradient of the composite objective function and the inner function value. By using a special Lyapunov function, we show that NASA achieves the sample complexity of ${\cal O}(1/\epsilon^{2})$ for finding an $\epsilon$-approximate stationary point, thus outperforming all extant methods for nested stochastic approximation. Our method and its analysis are the same for both unconstrained and constrained problems, without any need of batch samples for constrained nonconvex stochastic optimization. We also present a simplified variant of the NASA method for solving constrained single level stochastic optimization problems, and we prove the same complexity result for both unconstrained and constrained problems.