Stochastic gradient methods are scalable for solving large-scale optimizationproblems that involve empirical expectations of loss functions. Existingresults mainly apply to optimization problems where the objectives are one- ortwo-level expectations. In this paper, we consider the multi-levelcompositional optimization problem that involves compositions of multi-levelcomponent functions and nested expectations over a random path. It findsapplications in risk-averse optimization and sequential planning. We propose aclass of multi-level stochastic gradient methods that are motivated from themethod of multi-timescale stochastic approximation. First we propose a basic$T$-level stochastic compositional gradient algorithm, establish its almostsure convergence and obtain an $n$-iteration error bound $O (n^{-1/2^T})$. Thenwe develop accelerated multi-level stochastic gradient methods by using anextrapolation-interpolation scheme to take advantage of the smoothness ofindividual component functions. When all component functions are smooth, weshow that the convergence rate improves to $O(n^{-4/(7+T)})$ for generalobjectives and $O (n^{-4/(3+T)})$ for strongly convex objectives. We alsoprovide almost sure convergence and rate of convergence results for nonconvexproblems. The proposed methods and theoretical results are validated usingnumerical experiments.
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机译：随机梯度方法是可扩展的，用于解决涉及损失功能的经验期望的大规模优化问题。现有的结果主要适用于目标的优化问题，其中目标是一级级别的预期。在本文中，我们考虑了多级别致命的优化问题，涉及多级别函数的组成和随机路径上的嵌套期望。它在风险厌恶优化和顺序规划中发现了应用。我们提出了来自多时间尺度随机近似的主题的多级随机梯度方法的aclass。首先，我们提出了一种基本的$ T $ -Level随机成分梯度算法，建立它的差异收敛，获取$ N $〜Mentration错误绑定$ O（n ^ { - 1/2 ^ t}）$。然后，我们通过使用Anextapolation插值方案开发加速的多级随机梯度方法，以利用另外的组件功能的平滑度。当所有组件函数都是平滑的，Weallow offergence率改善了收敛速度（n ^ { - 4 /（7 + t）}）$ for for generalobjectives和$ o（n ^ { - 4 /（3 + t）}） $的强烈凸起目标。我们alsoprovide几乎肯定的融合和融合结果的非谐波问题。验证了拟议的方法和理论结果，使用数值实验进行了验证。
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